Building Lax Integrable Variable-Coefficient Generalizations to Integrable Pdes and Exact Solutions to Nonlinear Pdes

University of Central Florida STARS

Electronic Theses and Dissertations, 2004-2019

2016

Building Lax Integrable Variable-Coefficient Generalizations to Integrable PDEs and Exact Solutions to Nonlinear PDEs

Matthew Russo University of Central Florida

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STARS Citation Russo, Matthew, "Building Lax Integrable Variable-Coefficient Generalizations to Integrable PDEs and Exact Solutions to Nonlinear PDEs" (2016). Electronic Theses and Dissertations, 2004-2019. 4917. https://stars.library.ucf.edu/etd/4917 BUILDING LAX INTEGRABLE VARIABLE-COEFFICIENT GENERALIZATIONS TO INTEGRABLE PDES AND EXACT SOLUTIONS TO NONLINEAR PDES

MATTHEW RUSSO B.S. University of Central Florida, 2012 M.S. University of Central Florida, 2014

A dissertation submitted in partial fulﬁlment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the College of Sciences at the University of Central Florida Orlando, Florida

Spring Term 2016

Major Professor: S. Roy Choudhury c 2016 Matthew Russo

ii ABSTRACT

This dissertation is composed of two parts. In Part I a technique based on extended Lax Pairs is first considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax- or S-integrable nonlinear partial differential equations (nlpdes) with both time- and space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve´ Test, Bell Polynomials, and various similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must ’guess’ a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we present a generalization to the well- known Estabrook-Wahlquist prolongation technique which provides a systematic procedure for the derivation of the Lax representation. In order to obtain a nontrivial Lax representation we must impose differential constraints on the variable coefficients present in the nlpde. The resulting constraints determine a class of equations which represent generalizations to a previously known integrable constant coefficient nlpde. We demonstate the effectiveness of this technique by deriving variable-coefficient generalizations to the nonlinear Schrodinger (NLS) equation, derivative

NLS equation, PT-symmetric NLS, ﬁfth-order KdV, and three equations in the MKdV hierarchy. In Part II of this dissertation, we introduce three types of singular manifold methods which have been successfully used in the literature to derive exact solutions to many nonlinear PDEs extending over a wide range of applications. The singular manifold methods considered are: truncated Painleve´ analysis, Invariant Painleve´ analysis, and a generalized Hirota expansion method. We then consider the KdV and KP-II equations as instructive examples before using each method to derive nontrivial solutions to a microstructure PDE and two generalized Pochhammer-Chree equations.

iii TABLE OF CONTENTS

LIST OF FIGURES vii

PART I: VARIABLE COEFFICIENT LAX-INTEGRABLE SYSTEMS 1

CHAPTER 1: INTRODUCTION 2

CHAPTER 2: KHAWAJA’S LAX PAIR METHOD 7 OutlineofKhawaja’sMethod...... 7

PT-Symmetric and Standard Nonlinear Schrodinger¨ Equations ...... 8 Derivative Nonlinear Schrodinger Equation ...... 15

Deriving a relation between a1 and a2 ...... 17 Fifth-Order Korteweg-de-Vries Equation ...... 20

Deriving a relation between the ai ...... 23 Modiﬁed Korteweg-de-Vries Hierarchy ...... 25 Determining Equations for the First Equation ...... 28

Deriving a relation between the ai ...... 29 Determining Equations for the Second Equation ...... 30

Deriving a relation between the bi ...... 32 Determining Equations for the Third Equation ...... 35

Deriving a relation between the ci ...... 38

iv CHAPTER 3: THE EXTENDED ESTABROOK-WAHLQUIST METHOD 42

Outline of the Extended Estabrook-Wahlquist Method ...... 42 PT-Symmetric and Standard Nonlinear Schrodinger¨ Equation ...... 46

Derivative Nonlinear Schrodinger Equation ...... 59 Fifth-Order Korteweg-de-Vries Equation ...... 67 Modiﬁed Korteweg-de-Vries Equation ...... 88

Cubic-Quintic Nonlinear SchrodingerEquation¨ ...... 95

PART II: SOLUTIONS OF CONSTANT COEFFICIENT INTEGRABLE SYSTEMS 99

CHAPTER 4: EXACT SOLUTIONS OF NONLINEAR PDES 101 Introduction ...... 101

SingularManifoldMethods...... 102 Truncated PainleveAnalysisMethod´ ...... 103

Example:KP-IIEquation...... 104 MicrostructurePDE...... 109

Generalized Pochhammer-Chree Equations ...... 113 Invariant PainleveAnalysisMethod´ ...... 118 Example:KdVEquation ...... 120

MicroStructurePDE ...... 125 Generalized Pochhammer-Chree Equations ...... 130

Generalized Hirota Expansion Method ...... 136 Example:KP-IIEquation...... 137

MicrostructurePDE...... 141 Generalized Pochhammer-Chree Equations ...... 144

CHAPTER 5: CONCLUSION 151

v LIST OF REFERENCES 154

vi LIST OF FIGURES

Figure 4.1: Plots of Real/Imaginary Parts of Solution (4.25) ...... 108 Figure 4.2: Plot of Solution (4.26) ...... 109

Figure 4.3: Plot of Solution (4.37) ...... 113 Figure 4.4: Plot of Solution (4.50) ...... 117

Figure 4.5: Plot of Solution (4.84) ...... 125 Figure 4.6: Plots of Solutions (4.98) and (4.98) ...... 129 Figure 4.7: Plots of Solutions (4.118) and (4.119) ...... 135

Figure 4.8: Plot of Solution (4.120) ...... 136 Figure 4.9: Plots of Real/Imaginary Parts of Solution (4.127) ...... 140

Figure 4.10: Plot of Solution (4.128) ...... 141 Figure 4.11: Plot of Solution (4.135) ...... 144 Figure 4.12: Plot of Solution (4.146) ...... 149

Figure 4.13: Plot of Solution (4.147) ...... 150

vii PART I: VARIABLE COEFFICIENT LAX-INTEGRABLE SYSTEMS

1 CHAPTER 1: INTRODUCTION

For a nonlinear partial differential equation (nlpde) the phase space is inﬁnite dimensional and thus extension of integrability of a Hamiltonian system in the Liouville-Arnold sense becomes

troublesome. As such, a universally accepted deﬁnition of integrability within the context of nlpdes does not exist in the literature [1]- [5]. For the purposes of this dissertation, we will consider a nlpde

to be completely integrable if it can be expressed as the compatibility condition of a nontrivial Lax pair. Here we take a trivial Lax pair to be one for which dependence on the spectral parameter can be removed through a Gauge transformation. Indeed, if a nlpde is shown to possess a Lax pair

then from it one may derive a variety of remarkable properties, including the existence of inﬁnitely

many conserved quantities. More speciﬁcally, consider a nlpde in 1+1

∂m+nu F u, =0 (1.1) ∂xm∂tn where F : A B is a continuous function from a function space A to another function space → n n B. We say that (1.1) possesses a Lax representation if there exist matrices U, V M × (C), ∈ n n where M × (C) is the set of n n matrices with entries in C, such that compatibility of the set × of equations

Φx = UΦ, Φt = VΦ (1.2) is achieved upon satisfaction of (1.1). We say that a Lax pair U, V is nontrivial if it depends { } (nontrivially) on a spectral parameter. The matrices U and V are known as the space and time evolution matrices for the scattering problem (1.2), respectively. Compatability of (1.2) requires

2 the cross-derivative condition Φxt = Φtx be satisﬁed. That is,

0˙ = Φ Φ = U Φ+ UΦ V Φ VΦ xt tx t t − x − x = U Φ+ UVΦ VΦ VUΦ t − − =(U V + UV VU) Φ, (1.3) t − x − where 0˙ is equivalent to the zero matrix upon satisfaction of (1.1). Introducing the commutator operation [A, B]= AB BA the previous equation implies U and V satisfy −

U V + UV VU = 0˙. (1.4) t − x −

This equation is known as the zero-curvature condition and will form the basis of our investigation in Part I.

Variable-coefficient nonlinear partial differential equations have a long history dating from their derivations in a variety of physical contexts [6]- [18]. However, almost all studies, including those which derived exact solutions by a variety of techniques, as well as those which considered integrable sub-cases and various integrability properties by methods such as Painleve´ analysis, Hirota’s method, and Bell Polynomials, treat variable-coefficient nlpdes with coefficients which are functions of the time only. Due to their computational complexity and lack of an efficient method for deriving the conditions for Lax integrability, the question of integrability for equations with time and space dependent coefficients has largely been ignored.

In chapter 2 we introduce a method recently presented in the literature [19, 20] for deriving a Lax

3 pair for space and time dependent coefﬁcient nlpdes. We then illstrate the method by deriving the Lax pair and integrability conditions for the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PT-symmetric NLS, ﬁfth-order KdV, and three equations in the MKdV hierarchy.

This technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must ’guess’ a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. This involves replacing constants in the Lax Pair for the constant coefficient integrable system, including powers of the spectral parameter, by functions. Provided that one has guessed correctly and generalized the constant coefficient system’s Lax Pair sufficiently, and this is of course hard to be sure of ’a priori’, one may then proceed to systematically deduce the Lax Pair for the corresponding variable-coefficient integrable system [21].

Motivated by the somewhat arbitrary nature of the above procedure, we embark in this dissertation on an attempt to systematize the derivation of Lax-integrable sytems with variable coefﬁcients. Of the many techniques which have been employed for constant coefﬁcient integrable systems, the

Estabrook-Wahlquist (EW) prolongation technique [26]- [29] is among the most self-contained. The method directly proceeds to attempt construction of the Lax Pair or linear spectral problem, whose compatibility condition is the integrable system under discussion. While not at all guar- anteed to work, any successful implementation of the technique means that Lax-integrability has already been veriﬁed during the procedure, and in addition the Lax Pair is algorithmically obtained.

We note that failure of the technique does not necessarily imply non-integrability of the equation contained in the compatibility condition of the assumed Lax Pair for other deﬁnitions of integrability. It does however mean that the nlpde is not considered Lax-integrable. Due to the imprecise nature of the deﬁnition of integrability, a nlpde may be considered integrable in one sense but not

4 another. Indeed, consider the viscous Burgers’ equation

ut + uux = νuxx, (1.5) where ν > 0 is the (constant) viscosity. By making use of the Cole-Hopf transformation u(x,t)=

2ν φx Burgers’ equation becomes, after a little manipulation, the heat equation v = νφ . Since − φ t xx any linear partial differential equation is integrable we say that Burgers’ equation is integrable in the sense that it can be linearized. Now consider the (focusing) NLS equation

1 iψ = ψ k ψ 2ψ (1.6) t −2 xx − | |

where k > 0 is a dimensionless constant. There is no transformation which takes this equation to a linear equation, as was the case for Burgers’ equation. However, this equation has been shown to be integrable in many other ways. For example, there exists a nontrivial Lax pair for the NLS and thus it is Lax-integrable. While transformations like the Cole-Hopf transformation are examples of explicit linearizations a Lax pair can be thought of as an implicit linearization of a nlpde.

In applications, the coefficients of a nlpde may include spatial dependence, in addition to the temporal variations that have been extensively considered using a variety of techniques. Both for this reason, as well as for their general mathematical interest, extending integrable hierarchies of nlpdes to include both spatial and temporal dependence of the coefficients is worthwhile. Hence, we attempt to apply the Estabrook-Wahlquist (EW) technique to generate a variety of such integrable systems with such spatiotemporally varying coefficients. However, this immediately requires that the technique be significantly generalized or broadened in several different ways which we outline in chapter 3. We then illustrate the effectiveness of this new and extended method by deriving the

5 Lax pair and integrability condition for the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PT-symmetric NLS, ﬁfth-order KdV, and the ﬁrst equation in the MKdV hierarchy. As an instructive example of when the extended Estabrook-Wahlquist method correctly breaks down we then consider a generalization to the nonintegrable cubic-quintic nonlinear Schrodinger¨ equation.

6 CHAPTER 2: KHAWAJA'S LAX PAIR METHOD

In this chapter we review Khawaja’s Lax pair method for deriving the differential constraints necessary for compatibility of the Lax pair associated with a variable-coefficient nlpde. We then illustrate this method by deriving variable-coefficient generalizations to the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PT-symmetric NLS, fifth-order KdV, and three equations in the MKdV hierarchy.

Outline of Khawaja’s Method

In Khawaja’s Lax pair method [19,20], one seeks to represent a variable-coefficient generalization to a constant-coefficient nonlinear pde as the compatibility condition of a Lax pair. The method is based on the assumption that the Lax pair for the constant-coefficient nlpde is known. A Lax pair for an nlpde can be derived in various ways. Common methods include the Ablowitz-Kaup- Newell-Segur (AKNS) scheme, the Wadati-Konno-Ichikawa (WKI) scheme, and the Estabrook-

Wahlquist method. Once a Lax pair is obtained, it may be decomposed into an expansion about powers of the unknown function and its derivatives where the coefﬁcients are constant coefﬁcient matrices. For example, we may write a Lax pair U, V for the KdV equation { }

ut + uux +6uxxx =0 (2.1)

7 as

ik 1 0 0 U = − + u, (2.2a)    1  0 ik 6 0   −      4ik3 4k2 1 ik 1 0 0 1 0 0 0 3 3 2 6 V = − + − u + u + ux + uxx,  3  2 2 1   1   1 1   1  0 4ik 3 k 3 ik 18 0 3 ik 6 6 0   − −     −              (2.2b)

where here k is the spectral parameter associated with the linear eigenvalue problem (i.e. the Lax pair). In this method one replaces the elements in the constant-coefficient matrices present in the Lax pair expansion with undetermined functions of space and time. Compatibility of this new Lax pair is then enforced, allowing for the determination of the functions in the Lax Pair and derivation of the variable-coefficient constraints. Unfortunately, it is often the case that merely altering the existing form of the constant-coefficient Lax pair is not sufficient. Intuition and a little trial-and- error may be required to determine additional terms which may need to be added to the expansion in order to ensure nontrivial compatibility. In fact, in one of his papers [20] Khawaja alludes to the difficulty in this step of the procedure and further remarks that it took several attempts to find the correct form for the Lax pair he derived.

PT-Symmetric and Standard Nonlinear Schrodinger¨ Equations

We begin with the derivation of the Lax pair and differential constraints for the standard NLS equation. To keep things somewhat general we will consider the system

8 iq (x,t)= f(x,t)q (x,t) g(x,t)q2(x,t)r(x,t) v(x,t)q(x,t) iγ(x,t)q(x,t),(2.3) t − xx − − − ir (x,t)= f(x,t)r (x,t)+ g(x,t)r2(x,t)q(x,t)+ v(x,t)r(x,t) iγ(x,t)r(x,t). (2.4) t xx −

For the choice r(x,t)= q∗(x,t), where ∗ denotes the complex conjugate, this system is equivalent to a variable-coefﬁcient NLS equation. However, with the choices v(x,t) = γ(x,t)=0 and r(x,t)= q∗( x,t) this system is equivalent to the PT-symmetric NLS. Therefore, we may obtain − the results for both the standard cubic NLS and the PT-symmetric NLS simultaneously by studying system (2.3). Following Khawaja’s method we expand the U and V matrices in powers of q, r, and their partial derivatives. We therefore seek a Lax pair of the form

f1 + f2q f3 + f4q U =   (2.5a) f5 + f6r f7 + f8r     g1 + g2q + g3qx + g4qr g5 + g6q + g7qx + g8qr and V =   (2.5b) g9 + g10r + g11rx + g12qr g13 + g14r + g15rx + g16qr     where f1 8 and g1 16 are unknown functions of x and t. Compatibility of U and V requires − −

0 p1(x,t)F1[q,r] U V +[U, V]= 0=˙ (2.6) t − x   p2(x,t)F2[q,r] 0     th where Fi[q,r] represents the i equation in (2.3), and p1,2 are arbitrary real-valued functions. It should be clear that this off-diagonal compatibility condition requires that the coefﬁcients of the q and the r on the off-diagonal of U be zero. Indeed upon plugging U and V into the compatibility

9 condition we immediately ﬁnd that compatibility requires

f = f = f = f = g = g = g = g = g = g = g = g =0,f = ip ,f = ip , 2 3 5 8 2 3 5 8 9 12 14 15 4 1 6 − 2 g = fp ,g = fp ,g = g = ifp p . 7 − 1 11 − 2 4 − 16 − 1 2

The remaining constraints are given by

f g =0 (2.7) 1t − 1x f g =0 (2.8) 7t − 13x

2fp1p2 + g =0 (2.9)

f p fp (f f )+ fp g =0 (2.10) x 1 − 1 1 − 7 1x − 6 f p + fp (f f )+ fp g =0 (2.11) x 2 2 1 − 7 2x − 10 g (f f ) ip (g g iv + γ) g + ip =0 (2.12) 6 1 − 7 − 1 1 − 13 − − 6x 1t g (f f )+ ip (g g iv γ)+ g + ip =0 (2.13) 10 1 − 7 2 1 − 13 − − 10x 2t

(fp1p2)x + g10p1 + g6p2 =0. (2.14)

We will now review the reduction of the system (2.7)-(2.14) to the differential constraints which

may be found in [20]. Solving eq. (2.9) for fp1p2 and substituting the result into (2.14) we obtain

1 g + g p + g p =0. (2.15) −2 x 10 1 6 2

10 Upon multiplying equation (2.10) by p2 and equation (2.11) by p1 and adding, we obtain

(fp p ) + p p f g p g p =0. (2.16) 1 2 x 1 2 x − 6 2 − 10 1

Now utilizing equations (2.9), (2.15), and (2.16) we ﬁnd

f g c(t) x = 2 x f(x,t)= . (2.17) f − g ⇒ g(x,t)2

where c(t) is arbitrary. Now multiplying equation (2.10) by p2 and equation (2.11) by p1 and subtracting, we get

g 1 p fp p f f = 6 + log 1 + 1 2x . (2.18) 1 − 7 −fp 2 g g 1 x

Multiplying equation (2.12) by p2 and equation (2.13) by p1 and adding, we get

(g p + g p )(f f ) 2ip p γ g p + g p + ip p + ip p =0. (2.19) 6 2 10 1 1 − 7 − 1 2 − 6x 2 10x 1 1t 2 2t 1

Substituting for f1,g10,p2, and f using equations (2.18), (2.15), (2.9), and (2.17), respectively, we obtain

g2 gg g2 g g3γ cg˙ 3 3g2g 6 x + xx + i + t =0. (2.20) c p − g 2 c 2c2 − 2c 1 x

11 Solving for g6 we ﬁnd

p cg i (2cγ +c ˙)g g = 1 k + ik x + 3g dx . (2.21) 6 g 1r 1i − 2g2 2 t − c where k1r and k1i are arbitrary real functions of t obtained through integration. On the other hand,

multiplying equation (2.12) by p2 and equation (2.13) by p1 and subtracting, we get

(g p g p )(f f ) 2ip p (g g iv) g p g p + ip p ip p =0. (2.22) 6 2 − 10 1 1 − 7 − 1 2 1 − 13 − − 6x 2 − 10x 1 1t 2 − 2t 1

Once again, substituting for f1,g10,p2, and f using eqs. (2.18), (2.15), (2.9), and (2.17), respectively, we now obtain

cg2 cg g g g2g2 3g p c˙ x xx 6 x 6 t 1t (2.23) i 4 i 3 + i + i 2 (g1 g13 iv) + + =0. g − g gp1 cp1 − − − − 2g p1 2c

Solving for g g we get 1 − 13

g2g2 c˙ 2gp 3p g +2ig g ic(gg 2g2) 6 1t 1 t 6 x xx x (2.24) g1 g13 = iv + i 2 + + − −4 . − cp1 2c 2gp1 − 2g

Now subtracting equation (2.8) from equation (2.7) and substituting for f f and g g using 1 − 7 1 − 13 (2.18) and (2.24), respectively, we obtain

12 2k g g g gγ g v = 1i t (2k γ + k˙ )+ t − 3g (2cγ +c ˙) dx x c − c 1i 1i c t − c g c 1 + c(g (2cγ +c ˙) 3cg )+ g(c(2cγ +¨c) c˙2) dx 2c3 t − tt t − − 2g g x g 2 + i (2k γ + k˙ ) k g . (2.25) c 1r 1r − c 1r t

Since v is assumed to be real the imaginary part of (2.25) must vanish. That is, we require

g(2k γ + k˙ ) 2k g =0 (2.26) 1r 1r − 1r t

from which we obtain

g (x,t) 1 k˙ (t) γ(x,t)= t 1r . (2.27) g(x,t) − 2 k1r(t)

Now substituting c(t) = f(x,t)g(x,t)2 into (2.25), combining integrals, and differentiating with respect to x we obtain the ﬁnal condition

fg3(f (g 2gγ) f g)+ f 2g4 +2f 3g3(gv g v ) 2f 2g4(γ +2γ2) 2f 2g2g2 t t − − tt t xx − x x − t − t + f 2g3(4g γ + g )+ f 4(36g4 48gg2g + 10g2g g + g2(6g2 gg ))=0. (2.28) t tt x − x xx x xxx xx − xxxx

As the PT-symmetric NLS is a special case of the system (2.3) with f = a , g = a , v = γ =0, − 1 − 2 and r(x,t)= q∗( x,t) we can exploit the Lax pair constraints derived above for standard NLS to −

13 obtain the constraints for the PT-symmetric NLS. We therefore ﬁnd, utilizing the same U and U that were given earlier in the section, that compatibility under the requirement that r and q satisfy the PT-symmetric NLS requires

f = f = f = f = g = g = g = g = g = g = g = g =0,f = ip ,f = ip , 2 3 5 8 2 3 5 8 9 12 14 15 4 1 6 − 2 g = a p ,g = a p ,g = g = ia p p . 7 1 1 11 1 2 4 − 16 1 1 2 and the remaining constaints are given by

f g =0, (2.29) 1t − 1x f g =0, (2.30) 7t − 13x

2a1p1p2 + a2 =0, (2.31)

a p + a p (f f ) a p g =0, (2.32) − 1x 1 1 1 1 − 7 − 1 1x − 6 a p a p (f f ) a p g =0, (2.33) − 1x 2 − 1 2 1 − 7 − 1 2x − 10 g (f f ) ip (g g ) g + ip =0, (2.34) 6 1 − 7 − 1 1 − 13 − 6x 1t g (f f )+ ip (g g )+ g + ip =0, (2.35) 10 1 − 7 2 1 − 13 10x 2t (a p p ) + g p + g p =0. (2.36) − 1 1 2 x 10 1 6 2

From equations (2.17) and (2.27) we get

a2(x,t)= f(t)g(x), (2.37)

14 and

c(t) c(t) (2.38) a1(x,t)= 2 = 2 2 . a2(x,t) f(t) g(x)

Plugging these results into (2.28) we obtain

4 4 2 2 2 2 3 c 36(g′) 48g(g′) g′′ + 10g g′g′′′ +6g (g′′) g g′′′′ − − + g8 cf 3c˙f˙ 6c2f 2f˙2 cf 4c¨ +3c2f 3f¨+ f 4c˙2 =0. (2.39) − −

Equation (2.39) may be further reduced to the system

4 2 2 2 2 3 36(g′) 48g(g′) g′′ + 10g g′g′′′ +6g (g′′) g g′′′′ =0, (2.40) − − cf 3c˙f˙ 6c2f 2f˙2 cf 4c¨ +3c2f 3f¨+ f 4c˙2 =0. (2.41) − −

This ﬁnal set of equations represents the conditions on the variable coefﬁcients in (2.3) required for Lax-integrability.

Derivative Nonlinear Schrodinger Equation

In this section we derive the Lax pair and differential constraints for the derivative nonlinear

Schrodinger¨ equation. We consider the equivalent system

15 2 iqt(x,t)+ a1(x,t)qxx(x,t)+ ia2(x,t)(q (x,t)r(x,t))x =0, (2.42)

ir (x,t)+ a (x,t)r (x,t) ia (x,t)(r2(x,t)q(x,t)) =0, (2.43) − t 1 xx − 2 x

where r(x,t) = q∗(x,t) and again ∗ denotes the complex conjugate. The Lax pair U and V are expanded in powers of q and r and their partial derivatives as follows

f1 + f2q f3 + f4q U =   , (2.44) f5 + f6r f7 + f8r     2 g1 + g2q + g3qx + g4qr g5 + g6q + g7qr + g8qx + g9q r V = , (2.45)  2  g10 + g11r + g12qr + g13rx + g14r q g15 + g16r + g17rx + g18qr    

where f1 8 and g1 20 are unknown functions of x and t. Note that the compatibility condition − −

0 p (x,t)F [q,r] ˙ 1 Ut Vx +[U, V]= 0=   (2.46) − 0 0    

we enforce, where F [q,r] represents (2.42) and p1(x,t) is unknown, is chosen out of necessity. Upon considering a more standard compatibility condition as that considered by Khawaja,

0 p1(x,t)F1[q,r] U V +[U, V ]= 0=˙ (2.47) t − x   p2(x,t)F2[q,r] 0    

16 we ﬁnd that the only solution requires p1 or p2 be zero. We chose to let p2 = 0 but it is important to note that the conditions would not change if we had set p1 equal to zero instead. Given this modiﬁcation to the zero-curvature condition compatibility of U and V immediately requires

f2 = f5 = f6 = f8 = g2 = g3 = g4 = g7 = g11 = g12 = g14 = g16 = g17 = g18 =0,

g = p a ,f = ip ,g = ip a . 8 − 1 1 4 1 9 1 2

After substituting these into the compatibility conditions the remaining contraints are then given by

f g =0, (2.48) 1t − 1x f g =0, (2.49) 7t − 15x (p a ) + p a (f f )=0, (2.50) 1 2 x 1 2 7 − 1 (p a ) + p a (f f ) g =0, (2.51) 1 1 x 1 1 7 − 1 − 6 g g (f f )=0, (2.52) − 5x − 5 7 − 1 ip g + ip (g g ) g (f f )=0. (2.53) 1t − 6x 1 15 − 1 − 6 7 − 1

Deriving a relation between a1 and a2

As it turns out, equations (2.48) and (2.50)-(2.53) may be solved exactly for f1,f7,g6,g5 and g15, respectively. Upon solving equations (2.48) and (2.50)-(2.53) for f1,f7,g6,g5 and g15 we obtain

17 f1 = g1xdt + F (x), (2.54) (p1a2)x f7 = + f1, (2.55) − p1a2 R (f1 f7)dx g5 = H(t)e − , (2.56)

(p1a1)xa2 (p1a2)xa1 g6 = − , (2.57) a2 i g15 = (g6x ip1t g6(f1 f7)) + g1. (2.58) −p1 − − −

Plugging these results into the remaining equation, (2.49), we obtain the constraint

2p a2a p 2p2 a2a 2p a a p a + p2a2a +2p a2p a p2a a a 1 2 1x 1xx − 1x 2 1x − 1 2 1x 1x 2x 1 2 1xxx 1x 2 1 1xx − 1 2 1x 2xx +2p2 a a a +2p a a2 p 2p a a p a + p2a a a 2p a a p a 1x 2 1 2x 1 1 2x 1x − 1x 2 1 1 2xx 1 1 2x 2xx − 1 1 2x 1xx 2 p2a a a + ip2(a a a a )=0. (2.59) − 1 2 1 2xxx 1 2 2xt − 2t 2x

In order to have meaningful results we must require that the ai are real-valued functions. Thus we can decouple the last constraint into the following equations

2p a2a p 2p2 a2a 2p a a p a + p2a2a +2p a2p a p2a a a 1 2 1x 1xx − 1x 2 1x − 1 2 1x 1x 2x 1 2 1xxx 1x 2 1 1xx − 1 2 1x 2xx +2p2 a a a +2p a a2 p 2p a a p a + p2a a a 2p a a p a 1x 2 1 2x 1 1 2x 1x − 1x 2 1 1 2xx 1 1 2x 2xx − 1 1 2x 1xx 2 p2a a a =0, (2.60) − 1 2 1 2xxx a a a a =0. (2.61) 2 2xt − 2t 2x

18 Taking p1 = a2 we obtain the ﬁnal constraints

a a a a =0, (2.62) 2t 2x − 2xt 2 a3a 3a2a a 4a3 a +5a a a a +4a2 a a 2 1xxx − 2 2xx 1x − 2x 1 1 2 2x 2xx 2x 2 1x a2a a 2a a2a =0. (2.63) − 2 1 2xxx − 2x 2 1xx

With the aid of MAPLE we ﬁnd that the previous system is exactly solvable for a1 and a2 with solution given by

x dx dx a (x,t) = F (t)F (x)(c + c x) c F (t)F (x) + c xF (t)F (x) (2.64), 1 4 2 1 2 − 1 4 2 F (x) 1 4 2 F (x) 2 2 a2(x,t) = F2(x)F3(t). (2.65)

This final set of expressions represents the forms for the variable coefficients in the variable- coefficient DNLS required for Lax-integrability.

19 Fifth-Order Korteweg-de-Vries Equation

In this section we derive the Lax pair and differential constraints for a generalized variable- coefﬁcient ﬁfth-order KdV equation (vcKdV) given by

2 ut + a1uuxxx + a2uxuxx + a3u ux + a4uux + a5uxxx + a6uxxxxx + a7u + a8ux =0. (2.66)

Following Khawaja’s method the Lax pair for the generalized vcKdV equation is expanded in

powers of u and its derivatives as follows:

f1 + f2u f3 + f4u U =   , (2.67) f5 + f6u f7 + f8u     V1 V2 V =   , (2.68) V3 V4    

2 3 2 where Vi = gk + gk+1u + gk+2u + gk+3u + gk+4ux + gk+5ux + gk+6uxx + gk+7uuxx + gk+8uxxx + gk+9uuxxx + gk+10uxxxx, k = 11(i 1)+1 and f1 8(x,t) and g1 44(x,t) are unknown functions. − − − The compatibility condition

0 p1(x,t)F [u] U V +[U, V]= 0=˙ (2.69) t − x   p2(x,t)F [u] 0    

20 where F [u] represents equation (2.66) and p1 2(x,t) are unknown functions requires immediately − that some of the unknown functions be zero, indicative of a slightly incorrect initial guess. That is, we ﬁnd that g21 = g32 = f2 = g4 = g10 = g11 = f8 = g37 = g43 = g44 = 0. It is instructive to include this incorrect guess rather than remove them beforehand and include only the ﬁnal, correct form. This indeed motivates the need for a new method, which we introduce in the next chapter, which would remove as much human error as possible while still remaining computationally

tractable. We ﬁnd that compatibility under the zero curvature condition requires that the remaining unknown functions satisfy a large coupled system of algebraic and partial differential equations

1 1 f = p ,g = p a ,g = p a ,f = p ,g = p a ,g = p a , 4 1 15 −3 1 3 21 − 1 6 6 2 26 −3 2 6 33 − 2 6

21 2g g =0, g g =0, 41 − 8 39 − 6 p g p g =0, p g p g =0, 2 17 − 1 28 2 19 − 1 30 g +2g = p a , g +2g = p a , 19 17 − 1 2 30 28 − 2 2

g41 +2g39 =0, g8 +2g6 =0,

g + f g f g =0, g + f g f g =0, 39x 3 28 − 5 17 6x 5 17 − 3 28 2g + p (g g )= p a , g + p (g g )= p a , 25 2 38 − 5 − 2 4 19 1 9 − 42 − 1 1 g + p g p g =0, g + p g p g =0, 8 2 20 − 1 31 41 1 31 − 2 20 g + p (g g )= p a , g + a (p f p f )=0, 30 2 42 − 9 − 2 1 9 6 2 3 − 1 5 g + a (p f p f )=0, 2g + p (g g )= p a , 42 6 1 5 − 2 3 14 1 5 − 38 − 1 4 2g + p g p g =0, 2g + p g p g =0, 3 2 16 − 1 27 36 1 27 − 2 16 g + g + f g f g =0, g + g + f g f g =0, 2 5x 5 16 − 3 27 35 38x 3 27 − 5 16 1 1 g + g + f g f g =0, p g + f a p g + f a =0, 7 9x 5 20 − 3 31 1 25 3 5 3 − 2 14 3 3 3 f g + f g f g =0, f g + f g f g =0, 1t − 1x 3 23 − 5 12 7t − 34x 5 12 − 3 23 g + g + f g f g =0, g + g + f g f g =0, 5 7x 5 18 − 3 29 38 40x 3 29 − 5 18 g + g + f g f g =0, (p a ) g + p a (f f )=0, 40 42x 3 31 − 5 20 2 6 x − 31 2 6 1 − 7 g + p g + f g p g f g =0, g + p g + f g p g f g =0, 35x 1 23 3 24 − 2 12 − 5 13 36x 1 24 3 25 − 2 13 − 5 14 g + p g + f g p g f g =0, g + p g + f g p g f g =0, 2x 2 12 5 13 − 1 23 − 3 24 3x 2 13 5 14 − 1 24 − 3 25 g + p g + f g p g f g =0, g + p g + f g p g f g =0, 8x 2 18 5 19 − 1 29 − 3 30 41x 1 29 3 30 − 2 18 − 5 19 g + g (f f )+ f (g g )=0, g + g (f f )+ f (g g )=0, 28x 28 1 − 7 5 39 − 6 17x 17 7 − 1 3 6 − 39

22 1 1 2g (p a ) + p a (f f )=0, (p a ) + p a (f f )+ p (g g )=0, 20 − 1 6 x 1 6 1 − 7 3 2 3 x 3 2 3 1 − 7 2 3 − 36 f g + g (f f )+ f (g g )=0, g + g + g (f f )+ f (g g )= p a , 5t − 23x 23 7 − 1 5 1 − 34 24 27x 27 1 − 7 5 38 − 5 − 2 8 f g + g (f f )+ f (g g )=0, g + g + g (f f )+ f (g g )= p a , 3t − 12x 12 1 − 7 3 34 − 1 29 31x 31 1 − 7 5 42 − 9 − 2 5 1 1 (p a ) + p a (f f )+ p (g g )=0, g + g + g (f f )+ f (g g )= p a , 3 1 3 x 3 1 3 7 − 1 1 36 − 3 18 20x 20 7 − 1 3 9 − 42 − 1 5 g + g + g (f f )+ f (g g )=0, g + g + g (f f )+ f (g g )=0, 27 29x 29 1 − 7 5 40 − 7 16 18x 18 7 − 1 3 7 − 40

g + g + g (f f )+ f (g g )= p a , 13 16x 16 7 − 1 3 5 − 38 − 1 8 g + g (f f )+ p (g g )+ f (g g )=0, 19x 19 7 − 1 1 7 − 40 3 8 − 41 g + g (f f )+ p (g g )+ f (g g )=0, 30x 30 1 − 7 2 40 − 7 5 41 − 8

g + g (f f )+ p (g g )+ f (g g )=0, 14x 14 7 − 1 1 2 − 35 3 3 − 36 g + g (f f )+ p (g g )+ f (g g )=0, 25x 25 1 − 7 2 35 − 2 5 36 − 3 p g + g (f f )+ p (g g )+ f (g g )= p a , 1t − 13x 13 1 − 7 1 34 − 1 3 35 − 2 1 7 p g + g (f f )+ p (g g )+ f (g g )= p a . 2t − 24x 24 7 − 1 2 1 − 34 5 2 − 35 2 7

Deriving a relation between the ai

In this section we reduce the previous system down to equations which depend solely on the ai’s. We ﬁnd that g =0 for k =2, 3, 5 9, 35, 36, 38 42 and k − − f = f ,f = f ,g = g ,g = g ,g = g = p a ,g = g = 1 p a ,g = g = 7 1 5 3 23 12 34 1 30 − 19 1 1 25 14 − 2 1 4 31 − 20 23 (p a ) ,g = g = g ,p = p , − 1 6 x 16 − 27 − 18x 2 1

g = g = p a (p a ) , 18 − 29 − 1 5 − 1 6 xx 1 g = g = (H (t) H (t)), 28 − 17 −2 2 − 1 g = g = (p a ) (p a ) p a , 24 − 13 − 1 5 xx − 1 6 xxxx − 1 8 H1(t) p1 = , a1

a2 4 = H2 4(t)a1, − −

which leads to the PDE

H H a H a H a H a 1 + 1 5 + 1 6 + 1 8 = 1 7 . (2.72) a a a a a 1 t 1 xxx 1 xxxxx 1 x 1

One clear solution to the equation above (for H =0) is 1

a H H a H a H a a = 1 1 + 1 5 + 1 6 + 1 8 , (2.73) 7 H a a a a 1 1 t 1 xxx 1 xxxxx 1 x

where a1,a5,a6,a8 and H1 4 are arbitrary functions in their respective variables. −

24 Modiﬁed Korteweg-de-Vries Hierarchy

In this section we derive the Lax pairs and differential constraints for three equations in a variable- coefﬁcient modiﬁed KdV hierarchy

2 vt + a1vxxx + a2v vx =0, (2.74)

2 3 4 vt + b1vxxxxx + b2v vxxx + b3vvxvxx + b4vx + b5v vx =0, (2.75) and

2 2 vt + c1vxxxxxxx + c2v vxxxxx + c4vvxxvxxx + c5vxvxxx

2 4 3 2 3 6 +c6vxvxx + c7v vxxx + c8v vxvxx + c9v vx + c10v vx =0. (2.76)

Once again following Khawaja’s method the Lax pairs are expanded in powers of u and its derivatives as follows:

f1 + f2v f3 + f4v U =   , (2.77) f5 + f6v f7 + f8v     i i V1 V2 V i = , (i =1, 2, 3) (2.78)  i i V3 V4     25 where

1 2 V1 = g1 + g2v + g3v ,

1 2 3 V2 = g4 + g5v + g6v + g7v + g8vx + g9vxx,

1 2 3 V3 = g10 + g11v + g12v + g13v + g14vx + g15vxx,

1 2 V4 = g16 + g17v + g18v ,

2 2 4 2 2 2 5 V1 = g1vx + g2v + g3vvxx + g4v + g5 + g6vxxxx + g7vvx + g8v vxx + g9v + g10vxxx

2 3 +g11v vx + g12vxx + g13v + g14vx + g15v,

2 2 4 2 2 2 5 V2 = g16vx + g17v + g18vvxx + g19v + g20 + g21vxxxx + g22vvx + g23v vxx + g24v

2 3 +g25vxxx + g26v vx + g27vxx + g28v + g29vx + g30v,

2 2 4 2 2 2 5 V3 = g31vx + g32v + g33vvxx + g34v + g35 + g36vxxxx + g37vvx + g38v vxx + g39v

2 3 +g40vxxx + g41v vx + g42vxx + g43v + g44vx + g45v,

2 2 4 2 2 2 5 V4 = g46vx + g47v + g48vvxx + g49v + g50 + g51vxxxx + g52vvx + g53v vxx + g54v

2 3 +g55vxxx + g56v vx + g57vxx + g58v + g59vx + g60v,

26 3 2 2 2 3 2 2 4 V1 = g1 + g2v + g3vvxx + g4vxvxxx + g5v vx + g6v vxx + g7vxx + g8vvxxxx + g9vx + g10v ,

6 +g11v

3 2 2 2 2 4 2 2 V2 = g12v vxxx + g13vvx + g14v vxx + g15v vx + g16v vx + g17vxvxx + g18vvxx

2 3 2 4 +g19v vxxxx + g20v vx + g21v vxx + g22vvxvxx + g23vvxvxxx + g24v + g25vx + g26vxx

3 3 5 7 +g27vxxx + g28vxxxx + g29vxxxxx + g30vxxxxxx + g31vx + g32v + g33v + g34v + g35,

3 2 2 2 2 4 2 V3 = g36v vxxx + g37vvx + g38v vxx + g39v vx + g40v vx + g41vxvxx

2 2 3 2 4 +g42vvxx + g43v vxxxx + g44v vx + g45v vxx + g46vvxvxx + g47vvxvxxx + g48v + g49vx

3 3 5 7 +g50vxx + g51vxxx + g52vxxxx + g53vxxxxx + g54vxxxxxx + g55vx + g56v + g57v + g58v + g59,

3 2 2 2 3 2 2 4 V4 = g60 + g61v + g62vvxx + g63vxvxxx + g64v vx + g65v vxx + g66vxx + g67vvxxxx + g68vx + g69v

6 +g70v .

It is immediately clear from the latter two V matrices that finding the correct form of the time evolution matrix in the Lax pair via Khawaja’s method can be very difficult. Through various insufficient guesses we arrived at the previous forms, for which we will now give the results. The compatibility condition gives

0 pi(x,t)Fi[v] U V +[U, V ]= 0=˙ (2.79) t − x   qi(x,t)Fi[v] 0     , where F [v] represents the ith equation in the MKDV hierarchy (i =1 3). i −

27 Determining Equations for the First Equation

Requiring the compatibility condition yield a multiple of F1[v] in the off-diagonal as given above yields the following large coupled system of algebraic and partial differential equations f = p ,f = q ,g = 1 p a ,g = p a ,g = 1 q a ,g = q a ,f = f = g = g = 4 1 6 1 7 − 3 1 2 9 − 1 1 13 − 3 1 2 15 − 1 1 2 8 12 6 0,

f q f p =0, g + p g q g =0, 3 1 − 5 1 18x 1 11 − 1 5 g + q g p g =0, g + f g f g =0, 3x 1 5 − 1 11 17 3 14 − 5 8 2g + p g q g =0, g + f g f g =0, 18 1 14 − 1 8 2 5 8 − 3 14 2g + q g p g =0, f (g g )+ q (g g )=0, 3 1 8 − 1 14 5 18 − 3 1 17 − 2 f (g g )+ p (g g )=0, f g + f g f g =0, 3 18 − 3 1 17 − 2 1t − 1x 3 10 − 5 4 f g f g + f g =0, g + g g (f f )=0, 7t − 16x − 3 10 5 4 11 14x − 14 7 − 1 g + g + g (f f )=0, g (p a ) p a (f f )=0, 5 8x 8 7 − 1 8 − 1 1 x − 1 1 7 − 1 g (q a ) + q a (f f )=0, g + f g f g + p g q g =0, 14 − 1 1 x 1 1 7 − 1 17x 3 11 − 5 5 1 10 − 1 4 1 1 g f g + f g p g + q g =0, (p a ) + p a (f f )+ p (g g )=0, 2x − 3 11 5 5 − 1 10 1 4 3 1 2 x 3 1 2 7 − 1 1 18 − 3 1 1 (q a ) q a (f f ) q (g g )=0, f g g (f f ) f (g g )=0, 3 1 2 x − 3 1 2 7 − 1 − 1 18 − 3 3t − 4x − 4 7 − 1 − 3 1 − 16

28 f g + g (f f )+ f (g g )=0, 5t − 10x 10 7 − 1 5 1 − 16 p g g (f f ) p (g g ) f (g g )=0, 1t − 5x − 5 7 − 1 − 1 1 − 16 − 3 2 − 17 p g + g (f f )+ q (g g )+ f (g g )=0. 2t − 11x 11 7 − 1 1 1 − 16 5 2 − 17

Deriving a relation between the ai

In this section we reduce the previous system down to equations which depend solely on the ai’s. In doing so we ﬁnd that

C(t) g16 = g1 = f7 = f1 = g10 = g4 = g17 = g2 = f5 = f3 =0,q1 = p1 = , − − a2

C(t)g8 g18 = g3 = , − − a2 g = g = g , 11 − 5 8x a1 g14 = g8 = C(t)( )x, − − a2 which leads to the condition

K 6a a3 6a a a a + a a2a t a3 + a2a a3a 1 2x − 1 2 2x 2xx 1 2 2xxx − K 2 2 2t − 2 1xxx +3a a2a 6a a a2 +3a a2a =0, (2.81) 1xx 2 2x − 1x 2 2x 1x 2 2xx

where K(t) and C(t) are arbitrary functions of t.

29 Determining Equations for the Second Equation

Requiring the compatability condition yield a multiple of F2[v] in the off-diagonal yields the following large coupled system of algebraic and partial differential equations

q = p ,g = g = g = g = g = g = g = g = 0,g = 2g ,g = 2g ,g = 1 1 51 6 52 7 53 8 54 9 18 − 16 33 − 31 48 2g ,g = g = p b ,g = g = p b ,g = g = p b ,g = g = 1 p b , − 46 36 21 − 1 1 37 22 − 1 4 38 23 − 1 2 39 24 − 5 1 5

b =2b +2b , g + f g f g =0, 3 2 4 1x 3 16 − 2 31 g + p b (f f )=0, g + f g f g =0, 55 1 1 3 − 2 46x 2 31 − 3 16 2g g + g =0, g + p b (f f )=0, 1 − 25 40 10 1 1 2 − 3 g +4g g =0, g 4g g =0, 11 17 − 56 11 − 32 − 56 g 2g g =0, g +2g g =0, 14 − 34 − 59 14 19 − 59 g 4g g =0, g 2g g =0, 26 − 47 − 41 44 − 4 − 29 g 2g g =0, g 4g g =0, 40 − 46 − 25 41 − 2 − 26 g +2g g =0, g 2g g =0, 44 49 − 29 55 − 31 − 10 g +2g g =0, f g + f g f g =0, 55 16 − 10 4t − 50x 3 20 − 2 35 f g + f g f g =0, g + g + f g f g =0, 1t − 5x 2 35 − 3 20 59 57x 2 42 − 3 27 g + g + f g f g =0, g + g + f g f g =0, 60 59x 2 44 − 3 29 14 12x 3 27 − 2 42 g + g + f g f g =0, 3g + g + f g f g =0, 15 14x 3 29 − 2 44 58 56x 2 41 − 3 26 g + g + f g f g =0, 3g + g + f g f g =0, 12 10x 3 25 − 2 40 13 11x 3 26 − 2 41 1 g g + p b (f f )=0, g + g + f g f g =0, 32 − 17 5 1 5 3 − 2 57 55x 2 40 − 3 25

30 g g +2g +2f g 2f g =0, g + g g g (f f ) f (g g )=0, 27 − 42 46x 2 31 − 3 16 50 45x − 5 − 45 4 − 1 − 3 15 − 60 g g g + f g f g =0, g g g g (f f ) f (g g )=0, 19 − 34 − 58x 3 28 − 2 43 58 − 17x − 13 − 17 4 − 1 − 2 2 − 47 g +2g g +2f g 2f g =0, g + g g g (f f ) f (g g )=0, 42 1x − 27 3 16 − 2 31 58 32x − 13 − 32 4 − 1 − 3 47 − 2 g + g g + f g f g =0, g g g + f g f g =0, 45 49x − 30 2 34 − 3 19 34 − 13x − 19 2 43 − 3 28 g + g (f f )+ f (g g )=0, g (p b ) p b (f f )=0, 31x 31 4 − 1 3 46 − 1 25 − 1 1 x − 1 1 4 − 1 g (p b ) + p b (f f )=0, g 2g +2g + p b (f f )=0, 40 − 1 1 x 1 1 4 − 1 11 − 16 31 1 2 2 − 3 2g + g g + p b (f f )=0, g g g + f g f g =0, 11 16 − 31 1 4 2 − 3 20 − 60x − 35 3 30 − 2 45 g g g + f g f g =0, g +2g g p b (f f )=0, 28 − 47x − 43 3 27 − 2 32 31 56 − 16 − 1 4 2 − 3 2g g 2g + p b (f f )=0, g g g + f g f g =0, 31 − 56 − 16 1 2 2 − 3 35 − 15x − 20 2 45 − 3 30 g + g g (f f )+ f (g g )=0, g + g g (f f ) f (g g )=0, 44 42x − 42 4 − 1 3 57 − 12 42 40x − 40 4 − 1 − 3 10 − 55 g + g + g (f f )+ f (g g )=0, g + g + g (f f )+ f (g g )=0, 27 25x 25 4 − 1 2 10 − 55 30 29x 29 4 − 1 2 14 − 59 f g g (f f ) f (g g )=0, f g + g (f f )+ f (g g )=0, 2t − 20x − 20 4 − 1 − 2 5 − 50 3t − 35x 35 4 − 1 3 5 − 50 3g + g g (f f ) f (g g )=0, g + g g (f f ) f (g g )=0, 43 41x − 41 4 − 1 − 3 11 − 56 45 44x − 44 4 − 1 − 3 14 − 59 3g + g + g (f f )+ f (g g )=0, g + g + g (f f )+ f (g g )=0, 28 26x 26 4 − 1 2 11 − 56 29 27x 27 4 − 1 2 12 − 57 1 1 g +2g 2g (p b ) p b (f f )=0, g g + (p b ) + p b (f f )=0, 26 46 − 1 − 1 2 x − 1 2 4 − 1 47 − 2 5 1 5 x 5 1 5 4 − 1 g g g g (f f ) f (g g )=0, g g g + f g f g =0, 49 − 28x − 4 − 28 4 − 1 − 2 13 − 58 43 − 28 − 2x 2 32 − 3 17 g g g g (f f ) f (g g )=0, g + g (f f )+ f (g g )=0, 50 − 30x − 5 − 30 4 − 1 − 2 15 − 60 16x 16 4 − 1 2 1 − 46 g + g g g (f f ) f (g g )=0, g 2g g +(p b ) + p b (f f )=0, 60 34x − 15 − 34 4 − 1 − 3 4 − 49 46 − 26 − 1 1 4 x 1 4 4 − 1 g +2g g (p b ) + p b (f f )=0, 2g g 2g +(p b ) p b (f f )=0, 46 41 − 1 − 1 4 x 1 4 4 − 1 46 − 41 − 1 1 2 x − 1 2 4 − 1 1 1 g g + (p b ) p b (f f )=0, g g g g (f f ) f (g g )=0, 2 − 47 5 1 5 x − 5 1 5 4 − 1 60 − 19x − 15 − 19 4 − 1 − 2 49 − 4 g + g g g (f f ) f (g g )=0,g g g + f g f g =0, 49 43x − 4 − 43 4 − 1 − 3 13 − 58 45 − 4x − 30 2 34 − 3 19

31 g +2g g +2g (f f ) 2f (g g )=0, 57 16x − 12 16 4 − 1 − 2 46 − 1 g +2g g 2g (f f ) 2f (g g )=0. 12 31x − 57 − 31 4 − 1 − 3 1 − 46

Deriving a relation between the bi

In this section we reduce the previous system down to equations which depend solely on the bi’s. In doing so we ﬁnd that g =0 for k =1, 2, 4, 5, 10 17, 29, 31, 32, 44, 46, 47, 49, 50, 55 58, k − − H(t) f4 = f1 = 0,f3 = f2,p2 = ,g34 = g19,g42 = g27,g59 = 2g19,g40 = g25,g41 = g26,g60 = b5 − g19x,

1 g = g +2g f , g = f (g g ), 45 30 19 2 19 2 2 43 − 28 b g = g 2f 2g , g = H(t) 1 , 35 20 − 2 19 25 b 5 x b b g = H(t) 2 , g = H(t) 2 , 26 b 27 − b 5 x 5 xx 1 b b g = H(t) 2 ,g = 1 , 43 −3 b 30 − b 5 xx 5 xxxx and

4 b 1 b g = H(t) 2 H(t) 2 . 28 3 b − 3 b 5 x 5 xx

32 This leads to the following conditions on the bi,

b3 =2b2 +2b4, (2.85)

b = C(t)(2b b ), (2.86) 5 2 − 4

12b b b2 + 12b2b b 3b b b2 +4b2b b 4b b b2 24b b2 b + 24b b b2 2x 2xx 4 2 2xx 4x − 2xx 4x 4 2 2xxx 4 − 2 2xxx 4 − 2 2x 4x 2 2x 4x +12b2b b 12b2b b +4b2b b b b2b 24b2 b b +6b b b2 3b b2b 2 2x 4xx − 2 4x 4xx 2 4 4xxx − 2 4 4xxx − 2x 4 4x 2x 4 4x − 2x 4 4xx +b b3 6b b3 4b3b + 24a3 b 24b b b b +6b b b b =0, (2.87) 2xxx 4 − 2 4x − 2 4xxx 2x 4 − 2 2x 2xx 4 2 4 4x 4xx

and

9600b b4 b + 9600b b3 b2 4800b b2 b3 + 1200b b b4 3840b b b4 240b b b4 + − 1 2x 4x 1 2x 4x − 1 2x 4x 1 2x 4x − 1x 2 2x − 1x 2 4x 1920b b4 b + 120b b b4 + 3840b b5 120b b5 + 7680b b b3 b 5760b b b2 b2 + 1x 2x 4 1x 4 4x 1 2x − 1 4x 1x 2 2x 4x − 1x 2 2x 4x 1920b b b b3 3840b b3 b b + 2880b b2 b b2 960b b b b3 32b b5 + 1x 2 2x 4x − 1x 2x 4 4x 1x 2x 4 4x − 1x 2x 4 4x − 1xxxxx 2 b b5 + 10b b b4 80b b4b 5b b4b + 80b b4b 80b b3b2 + 1xxxxx 4 1xxxx 2x 4 − 1xxxx 2 4x − 1xxxx 4 4x 1xxxxx 2 4 − 1xxxxx 2 4 40b b2b3 10b b b4 + 32b b4b +2b b b4 16b b4b b b4b 1xxxxx 2 4 − 1xxxxx 2 4 1 2 2xxxxx 1 2xxxxx 4 − 1 2 4xxxxx − 1 4 4xxxxx − 960b b3b2 240b b3b2 + 30b b3b2 + 160b b4b 80b b4b + 10b b4b 1x 2 2xx − 1x 2 4xx 1x 4 4xx 1x 2 2xxxx − 1x 2 4xxxx 1x 4 2xxxx − 5b b4b + 1920b b2b3 240b b2b3 + 480b b3 b2 60b b2b3 + 320b b4b 1x 4 4xxxx 1xx 2 2x − 1xx 2 4x 1xx 2x 4 − 1xx 4 4x 1xx 2 2xxx − 160b b4b + 20b b b4 10b b4b 640b b3b2 160b b3b2 + 1xx 2 4xxx 1xx 2xxx 4 − 1xx 4 4xxx − 1xxx 2 2x − 1xxx 2 4x 80b b2 b3 + 20b b3b2 + 320b b4b 160b b4b + 20b b b4 1xxx 2x 4 1xxx 4 4x 1xxx 2 2xx − 1xxx 2 4xx 1xxx 2xx 4 − 10b b4b + 960b b2b b b 480b b b b b2 480b b2b b b + 240b b b b2 + 1xxx 4 4xx 1 2 2xx 2xxx 4 − 1 2 2xx 4xxx 4 − 1 2 4xx 2xxx 4 1 2 2xx 4 33 240b b2b b b 120b b b2b b + 480b b2b b b 240b b b b b2 1 2 4 4xx 4xxx − 1 2 4 4xx 4xxx 1 2 2x 2xxxx 4 − 1 2 2x 2xxxx 4 − 240b b2b b b + 120b b b b2b 240b b2b b b + 120b b b b2b + 1 2 2x 4 4xxxx 1 2 2x 4 4xxxx − 1 2 2xxxx 4 4x 1 2 2xxxx 4 4x 120b b2b b b 60b b b2b b 5760b b2b b b + 2880b b2b b b 1 2 4 4x 4xxxx − 1 2 4 4x 4xxxx − 1x 2 2x 2xx 4x 1x 2 2x 4x 4xx − 5760b b b2 b b + 2880b b b2 b b 1440b b b b b2 + 720b b b b2 b 1x 2 2x 2xx 4 1x 2 2x 4 4xx − 1x 2 2xx 4 4x 1x 2 4 4x 4xx − 1440b b b b2b + 720b b b b2 b 1440b b b b2b + 720b b b2b b + 1x 2x 2xx 4 4x 1x 2 4 4x 4xx − 1x 2x 2xx 4 4x 1x 2x 4 4x 4xx 1920b b2b b b 960b b2b b b 960b b2b b b + 480b b2b b b 1x 2 2x 2xxx 4 − 1x 2 2x 4 4xxx − 1x 2 2xxx 4 4x 1x 2 4 4x 4xxx − 960b b b b b2 + 480b b b b2b + 480b b b b2b 240b b b2b b + 1x 2 2x 2xxx 4 1x 2 2x 4 4xxx 1x 2 2xxx 4 4x − 1x 2 4 4x 4xxx 2880b b2b b b 1440b b2b b b 1440b b2b b b + 720b b2b b b 1xx 2 2x 2xx 4 − 1xx 2 2x 4 4xx − 1xx 2 2xx 4 4x 1xx 2 4 4x 4xx − 1440b b b b b2 + 720b b b b2b + 720b b b b2b 360b b b2b b + 1xx 2 2x 2xx 4 1xx 2 2x 4 4xx 1xx 2 2xx 4 4x − 1xx 2 4 4x 4xx 11520b b b2 b b 5760b b2 b b b 5760b b b2 b b + 2880b b2 b b b 1 2 2x 2xx 4x − 1 2x 2xx 4 4x − 1 2 2x 4x 4xx 1 2x 4 4x 4xx − 5760b b b b b2 + 2880b b b b b2 + 2880b b b b2 b 1440b b b b2 b 1 2 2x 2xx 4x 1 2x 2xx 4 4x 1 2 2x 4x 4xx − 1 2x 4 4x 4xx − 2880b b b b2 b 2880b b2b b b 720b b b b2b 720b b b b b2 + 1 2 2x 2xx 4 − 1 2 2x 2xx 4xx − 1 2x 2xx 4 4xx − 1 2 2x 4 4xx 1440b b b2 b b + 1440b b2b b b + 360b b b2b b + 360b b b b b2 1 2 2xx 4 4x 1 2 2xx 4x 4xx 1 2xx 4 4x 4xx 1 2 4 4x 4xx − 1920b b b3 b b + 960b b b3 b b 1920b b2b b b 480b b b b2b + 1 2 2x 2xxx 4 1 2 2x 4 4xxx − 1 2 2x 2xxx 4x − 1 2x 2xxx 4 4x 960b b2b b b + 240b b b2b b 480b b b b b2 + 240b b b b2 b 1 2 2x 4x 4xxx 1 2x 4 4x 4xxx − 1 2 2xxx 4 4x 1 2 4 4x 4xxx − 1440b b2b b b + 720b b b b2b + 2880b b b2 b b 1440b b b b b2 1x 2 2xx 4 4xx 1x 2 2xx 4 4xx 1xx 2 2x 4 4x − 1xx 2 2x 4 4x − 960b b2b b b + 480b b b b2b + 5760b b b b b b 2880b b b b b b 1xxx 2 2x 4 4x 1xxx 2 2x 4 4x 1x 2 2x 2xx 4 4x − 1x 2 2x 4 4x 4xx − 1440b b b b b b + 1920b b b b b b 960b b b b b b 640b b3b b + 1 2 2xx 4 4x 4xx 1 2 2x 2xxx 4 4x − 1 2 2x 4 4x 4xxx − 1 2 2xx 2xxx 80b b b b3 + 320b b3b b 40b b b3b + 320b b3b b 40b b b3b 1 2xx 2xxx 4 1 2 2xx 4xxx − 1 2xx 4 4xxx 1 2 2xxx 4xx − 1 2xxx 4 4xx − 160b b3b b + 20b b3b b 320b b3b b + 40b b b b3 + 160b b3b b 1 2 4xx 4xxx 1 4 4xx 4xxx − 1 2 2x 2xxxx 1 2x 2xxxx 4 1 2 2x 4xxxx − 20b b b3b + 160b b3b b 20b b b3b 80b b3b b + 10b b3b b + 1 2x 4 4xxxx 1 2 2xxxx 4x − 1 2xxxx 4 4x − 1 2 4x 4xxxx 1 4 4x 4xxxx 5760b b2b2 b 2880b b2b2 b + 1440b b2b b2 720b b2b2 b + 1440b b2 b b2 1x 2 2x 2xx − 1x 2 2x 4xx 1x 2 2xx 4x − 1x 2 4x 4xx 1x 2x 2xx 4 − 720b b2 b2b + 360b b b2b2 180b b2b2 b 1280b b3b b + 640b b3b b + 1x 2x 4 4xx 1x 2xx 4 4x − 1x 4 4x 4xx − 1x 2 2x 2xxx 1x 2 2x 4xxx 640b b3b b 320b b3b b + 160b b b b3 80b b b3b 80b b b3b + 1x 2 2xxx 4x − 1x 2 4x 4xxx 1x 2x 2xxx 4 − 1x 2x 4 4xxx − 1x 2xxx 4 4x 40b b3b b 1920b b3b b + 960b b3b b + 960b b3b b 480b b3b b + 1x 4 4x 4xxx − 1xx 2 2x 2xx 1xx 2 2x 4xx 1xx 2 2xx 4x − 1xx 2 4x 4xx 240b b b b3 120b b b3b 120b b b3b + 60b b3b b 320b b3b b + 1xx 2x 2xx 4 − 1xx 2x 4 4xx − 1xx 2xx 4 4x 1xx 4 4x 4xx − 1xxxx 2 2x 4 34 240b b2b b2 80b b b b3 + 160b b3b b 120b b2b2b + 40b b b3b 1xxxx 2 2x 4 − 1xxxx 2 2x 4 1xxxx 2 4 4x − 1xxxx 2 4 4x 1xxxx 2 4 4x − 7680b b b3 b + 3840b b3 b b + 3840b b b3 b 1920b b3 b b + 960b b b b3 1 2 2x 2xx 1 2x 2xx 4 1 2 2x 4xx − 1 2x 4 4xx 1 2 2xx 4x − 480b b b b3 480b b b3 b + 240b b b3 b + 2880b b2b b2 + 720b b b2 b2 + 1 2xx 4 4x − 1 2 4x 4xx 1 4 4x 4xx 1 2 2x 2xx 1 2x 2xx 4 720b b2b b2 + 180b b b2b2 1440b b2b2 b 360b b2 b2b 360b b2b b2 1 2 2x 4xx 1 2x 4 4xx − 1 2 2xx 4x − 1 2xx 4 4x − 1 2 4x 4xx − 90b b2b b2 + 1920b b2b2 b + 480b b2 b b2 960b b2b2 b 240b b2 b2b + 1 4 4x 4xx 1 2 2x 2xxx 1 2x 2xxx 4 − 1 2 2x 4xxx − 1 2x 4 4xxx 480b b2b b2 + 120b b b2b2 240b b2b2 b 60b b2b2 b 64b b3b b + 1 2 2xxx 4x 1 2xxx 4 4x − 1 2 4x 4xxx − 1 4 4x 4xxx − 1 2 2xxxxx 4 48b b2b b2 16b b b b3 + 32b b3b b 24b b2b2b +8b b b3b + 1 2 2xxxxx 4 − 1 2 2xxxxx 4 1 2 4 4xxxxx − 1 2 4 4xxxxx 1 2 4 4xxxxx 960b b3b b + 1440b b2b2 b + 360b b2b b2 720b b b2 b2 180b b b2b2 1x 2 2xx 4xx 1x 2 2xx 4 1x 2 4 4xx − 1x 2 2xx 4 − 1x 2 4 4xx − 120b b b3b 320b b3b b + 160b b3b b + 240b b2b2b 120b b2b2b 1x 2xx 4 4xx − 1x 2 2xxxx 4 1x 2 4 4xxxx 1x 2 4 2xxxx − 1x 2 4 4xxxx − 80b b b b3 + 40b b b3b 2880b b2b2 b + 1440b b2b b2 1920b b b3 b + 1x 2 2xxxx 4 1x 2 4 4xxxx − 1xx 2 2x 4x 1xx 2 2x 4x − 1xx 2 2x 4 240b b b b3 720b b2 b2b + 360b b b2b2 640b b3b b + 320b b3b b + 1xx 2 4 4x − 1xx 2x 4 4x 1xx 2x 4 4x − 1xx 2 2xxx 4 1xx 2 4 4xxx 480b b2b b2 240b b2b2b 160b b b b3 + 80b b b3b + 640b b3b b + 1xx 2 2xxx 4 − 1xx 2 4 4xxx − 1xx 2 2xxx 4 1xx 2 4 4xxx 1xxx 2 2x 4x 960b b2b2 b + 240b b2b b2 480b b b2 b2 120b b b2b2 80b b b3b 1xxx 2 2x 4 1xxx 2 4 4x − 1xxx 2 2x 4 − 1xxx 2 4 4x − 1xxx 2x 4 4x − 640b b3b b + 320b b3b b + 480b b2b b2 240b b2b2b 160b b b b3 + 1xxx 2 2xx 4 1xxx 2 4 4xx 1xxx 2 2xx 4 − 1xxx 2 4 4xx − 1xxx 2 2xx 4 3 4 2 3 80b1xxxb2b4b4xx + 160b1xxxxb2b2x + 120b1xb2xxb4 + 2880b1b2b2xb2xxb4b4xx =0,

where H(t) and C(t) are arbitrary functions of t.

Determining Equations for the Third Equation

Requiring the compatability condition yield a multiple of F3[v] in the off-diagonal yields the following large coupled system of algebraic and partial differential equations

f = f = 0,g = 1 p a ,g = 1 p a ,f = p ,g = p a ,g = p a ,g = 2 8 34 − 7 1 10 20 − 3 3 9 4 3 30 − 3 1 21 − 3 7 19 p a ,g = q a ,g = 1 q a ,g = 1 q a ,g = q a ,g = q a ,f = q ,g = g = − 3 2 54 3 1 58 7 3 10 44 3 3 9 45 3 7 43 3 2 6 − 3 63 − 67 g = g = G (t),g = g = G (t),g = 1 g = g = G (t),f = f ,f = f ,q = p , − 8 4 4 66 7 7 68 − 2 3 9 9 5 3 7 1 3 − 3

35 f (g g )=0, (i = 12 18, 22, 23, 25 29, 31 33) 3 i+24 − i − − − f (g g )=0, (i =2, 3, 5, 6, 8, 10, 11) 3 i+59 − i 1 (p c ) + p (g g )=0, ±7 3 10 x 3 70 − 11 g + g =0, (i = 24 29, 48 52) i i+1,x − −

36 2 p c +4p a = p c , g p (2c c )=0, g +2g = p c , 3 3 9 3 7 3 8 47 − 3 2 − 3 18 17 − 3 6 1 g p (2c c )=0, G = G , g +2g = p c , 23 − 3 2 − 3 7 −2 4 42 41 − 3 6 1 1 G = G , G = G , g p (g g )=0, 66 −2 63 68 −2 62 3 − 3 51 − 27 1 g + g = p c , g +2g = p c , g p (g g )=0, 23 17 − 3 5 23 18 − 3 4 2 − 2 3 49 − 25 1 g + g = p c , g +2g = p c , g p (g g )=0, 47 41 − 3 5 47 42 − 3 4 5 − 2 3 55 − 31 1 g p (g g )=0, g p (g g )=0, g p (g g )=0, 6 − 3 36 − 16 8 − 3 53 − 29 10 − 4 3 39 − 15 1 1 g p (g g )=0, g + p (g g )=0, g + p (g g )=0, 11 − 6 3 40 − 16 61 2 3 49 − 25 62 3 51 − 27 1 g + p (g g )=0, g + p (g g )=0, g + p (g g )=0, 65 3 36 − 12 64 2 3 55 − 31 67 3 53 − 29 1 1 g + p (g g )=0, g + p (g g )=0, g p (g g )=0, 69 4 3 39 − 15 70 6 3 40 − 16 2x − 3 48 − 24 g p (g g )=0, g p (g g )=0, g p (g g )=0, 5x − 3 37 − 13 3x − 3 50 − 26 6x − 3 38 − 14 g p (g g )=0, g p (g g )=0, g p (g g )=0, 8x − 3 52 − 28 10x − 3 56 − 32 11x − 3 57 − 33 g + p (g g )=0, g + p (g g )=0, g + p (g g )=0, 62x 3 50 − 26 61x 3 48 − 24 64x 3 37 − 13 g + p (g g )=0, g + p (g g )=0, g + p (g g )=0, 65x 3 38 − 14 67x 3 52 − 28 69x 3 56 − 32 g (p c ) p (g g )=0, 2g + g + g =0, g + p (g g )=0, 12 − 3 2 x − 3 67 − 8 12 23x 22 70x 3 57 − 33

37 g + g =0, g + g =0, g + g + p (g g )=0, 13 31x 14 12x 15 14x 3 3 − 62 1 g (p c ) p (g g )=0, 2g + g =0, 4g (p c ) p (g g )=0, 16 − 3 7 x − 3 65 − 6 15 13x 16 − 3 3 9 x − 3 64 − 5 1 4g (p c ) + p (g g )=0, g + g =0, 3g + g =0, 40 − 3 3 9 x 3 64 − 5 22 18x 32 15x g + g =0, g + g =0, g + g p (g g )=0, 37 55x 38 36x 39 38x − 3 3 − 62 g (p c ) + p (g g )=0, g + g =0, g + g +2g =0, 40 − 3 7 x 3 65 − 6 46 42x 46 47x 36 3g + g =0, 5g + g =0, p (g g )+ f (g g )=0, 56 39x 57 40x 3 59 − 35 3 48 − 24 g16 + g16x +5g33 =0, 2g13 + g22x +2g14 =0, g22 + g17x +3g31 =0,

2g + g +2g =0, g + g +3g =0, 2g +3g p (g g )=0, 37 46x 38 46 41x 55 5 6 − 3 46 − 22 f g f (g + g )=0, 2g +3g + p (g g )=0, f g + f (g g )=0, 1t − 1x − 3 35 59 64 65 3 46 − 22 1t − 60x 3 35 − 59 f g + f (g g )=0, f g f (g g )=0, g p (g g )=0, 3t − 35x 3 60 − 1 3t − 59x − 3 60 − 1 56x − 3 2 − 61 g + p (g g )=0, g + p (g g )=0, g p (g g )=0, 32x 3 2 − 61 33x 3 10 − 69 57x − 3 10 − 69 g (p c ) =0, g (p c ) + p (g g )=0, p g p (g g )=0, 29 − 3 1 x 36 − 3 2 x 3 67 − 8 1t − 24x − 3 1 − 60 g (p c ) =0, 2g + g =0, p g + p (g g )=0. 53 − 3 1 x 39 37x 1t − 48x 3 1 − 60

Deriving a relation between the ci

In this section we reduce the previous system down to equations which depend solely on the ci’s. In doing so we ﬁnd that

g64 = g5 = g65 = g6 = g61 = g2 = g10 = g69 = g70 = g11 = G4(t)= G9(t)=0,g59 = g35,g60 = g1,g42 = g18,g40 = g16,g52 = g28,g53 = g29,g55 = g31,g36 = g12,g39 = g15,g56 = g32,g41 =

H(t) 1 g17,g57 = g33,g37 = g13,g47 = g23,p3 = ,g29 = (p3c1)x,g12 = (p3c2)x,g15 = g13x,g32 = c10 − 2

38 1 g ,g = (p c ) ,g = G (t),g = g , (i = 24 27, 48 51) − 3 15x 28 − 3 1 xx 33 33 i − ix − −

1 g = p c g , g = (p (2c c 6c )) , 23 − 3 5 − 17 13 3 3 3 − 5 − 2 xx g = p c 2g , g = g 2g +(p c ) , 18 − 3 6 − 17 22 17x − 12 3 5 x g =(p c ) , g = g 2g +(p c ) , 16 3 7 x 46 17x − 12 3 5 x 1 1 1 g = g g g (p c ) , g = (2g 2g (p c ) ), 14 12x − 2 17xx − 13 − 2 3 5 xx 31 3 12 − 17x − 3 5 x 1 1 g = g + g g (p c ) , g = p (c 2c c ), 38 − 13 12x − 2 17xx − 2 3 5 xx 17 3 3 − 2 − 5

which leads to the following conditions on the ci,

c = 10c +5c +2c 4c , (2.91) 4 − 2 3 6 − 5 2 c =4c + c , (2.92) 8 7 3 9 c c c c 2c c +2c c + c c c c =0, (2.93) 10x 6 − 6 10x − 5 10x 5x 10 10x 3 − 3x 10 12c c + 12c c + c c c c =0, (2.94) − 10x 7 7x 10 10x 9 − 9x 10 5Gc3 2Hc c c + Hc2 c +2Hc2 c Hc c c =0, (2.95) 10 − 10x 7x 10 10 7xx 10x 7 − 7 10 10xx

39 14c c2 7c c c 60c c2 + 30c c c + 60c c c 30c c2 +4c c c 3 10x − 3 10 10xx − 2 10x 2 10 10xx 2x 10 10x − 2xx 10 5x 10 10x 14c c c +7c c2 4c c2 +2c c c 2c c2 =0, (2.96) − 3x 10 10x 3xx 10 − 5 10x 5 10 10xx − 5xx 10 18c c2 18c c c c +3a c3 c 18c c c2 +9c c2 c 60c c3 3 10x − 3 10 10x 10xx 3 10 10xxx − 3x 10 10x 3x 10 10xx − 2 10x +60c c c c 10c c2 c + 60c c c2 30c c2 c 30c c2 c 2 10 10x 10xx − 2 10 10xxx 2x 10 10x − 2x 10 10xx − 2xx 10 10x +10c c3 +6c c c2 3c c2 c 3c c2 c +9c c2 c 3c c3 2xxx 10 5x 10 10x − 5x 10 10xx − 5xx 10 10x 3xx 10 10x − 3xxx 10 6c c3 + c c c c c c2 c + c c3 =0, (2.97) − 5 10x 5 10 10x 10xx − 5 10 10xxx 5xxx 10 c c7 + 5040c c c6 7c c6 c c c6 c 5040c c7 1xxxxxxx 10 1x 10 10x − 1xxxxxx 10 10x − 1 10 10xxxxxxx − 1 10x H 7c c6 c 2520c c2 c5 + t c7 + 210c c5 c2 c6 c − 1x 10 10xxxxxx − 1xx 10 10x H 10 1xxx 10 10xx − 10 10t 35c c6 c 210c c4 c3 21c c6 c + 140c c5 c2 − 1xxx 10 10xxxx − 1xxxx 10 10x − 1xxxxx 10 10xx 1x 10 10xxx 630c c4 c c c 630c c4 c3 + 5040c c3 c2 c c − 1 10 10x 10xx 10xxxx − 1x 10 10xx 1 10 10x 10xx 10xxx 630c c4 c2 c 126c c4 c2 c + 70c c5 c c − 1 10 10xx 10xxx − 1 10 10x 10xxxxx 1 10 10xxx 10xxxx +14c c5 c c 12600c c2 c3 c2 + 2520c c3 c c3 1 10 10x 10xxxxxx − 1 10 10x 10xx 1 10 10x 10xx 12600c c2 c4 c + 7560c c3 c2 c2 + 3360c c3 c3 c − 1x 10 10x 10xx 1x 10 10x 10xx 1x 10 10x 10xxx 5 5 3 3 +210c1xc10c10xxc10xxxx + 84c1xc10c10xc10xxxxx + 5040c1xxc10c10xc10xx

1260c c4 c2 c + 420c c5 c c + 210c c5 c c − 1xx 10 10x 10xxx 1xx 10 10xx 10xxx 1xx 10 10x 10xxxx 2520c c4 c c c + 840c c3 c3 c 1260c c4 c2 c − 1x 10 10x 10xx 10xxx 1 10 10x 10xxxx − 1xxx 10 10x 10xx +42c c5 c c 420c c4 c c2 630c c4 c2 c 1 10 10xx 10xxxxx − 1 10 10x 10xxx − 1x 10 10x 10xxxx 1890c c4 c c2 + 840c c3 c4 4200c c2 c4 c − 1xx 10 10x 10xx 1xxx 10 10x − 1 10 10x 10xxx 5 2 5 5 +42c1xxxxxc10c10x + 280c1xxxc10c10xc10xxx + 210c1xxxxc10c10xc10xx

21c c6 c 35c c6 c + 15210c c c5 c =0, (2.98) − 1xx 10 10xxxxx − 1xxxx 10 10xxx 1 10 10x 10xx

40 and

240c c5 + 720c c5 + 120c c5 + 40c c3 c2 20c c4 c − 3 10x 2 10x 5 10x 3xxx 10 10x − 3xxx 10 10xx 180c c3 c2 + 30c c4 c + 360c c2 c3 + 60c c4 c − 2x 10 10xx 2x 10 10xxxx 2xx 10 10x 2xx 10 10xxx +60c c4 c 30c c3 c2 +5c c4 c + 60c c2 c3 2xxx 10 10xx − 5x 10 10xx 5x 10 10xxxx 5xx 10 10x +10c c4 c + c c4 c 120c c2 c3 20c c4 c 5xxx 10 10xx 5 10 10xxxxx − 3xx 10 10x − 3xx 10 10xxx 10c c4 c + 30c c4 c +5c c4 c 10c c4 c − 3x 10 10xxxx 2xxxx 10 10x 5xxxx 10 10x − 3xxxx 10 10x c c5 720c c c4 120c c c4 + 240c c c4 − 5xxxxx 10 − 2x 10 10x − 5x 10 10x 3x 10 10x 120c c2 c2 c + 40c c3 c c + 20c c3 c c − 3 10 10x 10xxx 3 10 10xx 10xxx 3 10 10x 10xxxx +120c c3 c c 1440c c c3 c + 540c c2 c c2 3xx 10 10x 10xx − 2 10 10x 10xx 2 10 10x 10xx 60c c3 c c + 1080c c2 c2 c 240c c3 c c − 2 10 10x 10xxxx 2x 10 10x 10xx − 2x 10 10x 10xxx 40c c3 c c 60c c3 c c 240c c c3 c − 5x 10 10x 10xxx − 5xx 10 10x 10xx − 5 10 10x 10xx 20c c3 c2 + 60c c3 c2 +2c c5 180c c2 c c2 − 5xxx 10 10x 3x 10 10xx 3xxxxx 10 − 3 10 10x 10xx +180c c2 c2 c 120c c3 c c + 80c c3 c c 5x 10 10x 10xx − 2 10 10xx 10xxx 3x 10 10x 10xxx +10c c4 c 2c c4 c 6c c5 360c c2 c2 c 5xx 10 10xxx − 3 10 10xxxxx − 2xxxxx 10 − 3x 10 10x 10xx 20c c3 c c 10c c3 c c + 60c c2 c2 c +6c c4 c − 5 10 10xx 10xxx − 5 10 10x 10xxxx 5 10 10x 10xxx 2 10 10xxxxx +480c c c3 c + 360c c2 c2 c 360c c3 c c 3 10 10x 10xx 2 10 10x 10xxx − 2xx 10 10x 10xx +90c c2 c c2 120c c3 c2 =0, (2.99) 5 10 10x 10xx − 2xxx 10 10x where G(t) and H(t) are arbitrary functions of t.

41 CHAPTER 3: THE EXTENDED ESTABROOK-WAHLQUIST METHOD

In this chapter we present the extended Estabrook-Wahlquist method for deriving the differential constraints necessary for compatibility of the Lax pair associated with a variable-coefﬁcient nlpde.

We then illustrate the effectiveness of this method by deriving variable-coefficient generalizations to the nonlinear Schrodinger¨ (NLS) equation, derivative NLS equation, PT-symmetric NLS, fifth- order KdV, and the first equation in the MKdV hierarchy. As a final example, we consider a variable-coefficient extension to the nonintegrable cubic-quintic NLS and show that the extended Estabrook-Wahlquist method correctly breaks down upon attempting to satisfy the consistency conditions.

Outline of the Extended Estabrook-Wahlquist Method

In the standard Estabrook-Wahlquist method one begins with a constant-coefﬁcient nlpde and as- sumes an implicit dependence on the unknown function(s) and its (their) partial derivatives for the

spatial and time evolution matrices (F, G) involved in the linear scattering problem

ψx = Fψ, ψt = Gψ

The evolution matrices F and G are connected via a zero-curvature condition (independence of

path in spatial and time evolution) derived by requiring ψxt = ψtx. That is, it requires

F G +[F, G]=0 (3.1) t − x

42 upon satisfaction of the nlpde.

For simplicity in outlining the details of the method we will consider a system with only one unknown function, namely u(x,t). Generalization of the procedure to follow to systems with n 2 unknown functions is straightforward. Let ≥

F = F(u,ux,ut,...,umx,nt), G = G(u,ux,ut,...,ukx,jt)

∂p+qu represent the space and time evolution matrices of a Lax pair, respectively, where upx,qt = ∂xp∂tq . Plugging these into (3.1) we obtain the equivalent condition

F u G u +[F, G]=0 umx,nt mx,(n+1)t − ujx,kt (j+1)x,kt m,n j,k

As we are now letting F and G have explicit dependence on x and t and for notational clarity, it will be more convenient to consider the following version of the zero-curvature condition

D F D G +[F, G]=0 (3.2) t − x

where Dt and Dx are the total derivative operators on time and space, respectively. Recall the deﬁnition of the total derivative

∂f ∂f ∂u1 ∂f ∂u2 ∂f ∂un Dxf(x,t,u1(x,t),u2(x,t),...,un(x,t)) = + + + + ∂x ∂u1 ∂x ∂u2 ∂x ∂un ∂x

43 Thus we can write the compatibility condition as

F + F u G G u +[F, G]=0 t umx,nt mx,(n+1)t − x − ujx,kt (j+1)x,kt m,n j,k

It is important to note that the subscripted x and t denotes the partial derivative with respect to only the x and t elements, respectively. That is, although u and it’s derivatives depend on x and t this will not invoke use of the chain rule as they are treated as independent variables. This will become more clear in the examples of the next section.

From here there is often a systematic approach [26]- [29] to determining the explicit dependence of F and G on u and its derivatives which is outlined in [28] and will be utilized in the examples to follow.

Typically one takes F to depend on all terms in the nlpde for which there is a partial time derivative present. Similarly one may take G to depend on all terms for which there is a partial space

derivative present. For example, given the Camassa-Holm equation,

u +2ku u +3uu 2u u uu =0, t x − xxt x − x xx − xxx

one would consider F = F(u,uxx) and G = G(u,ux,uxx). Imposing compatibility allows one to determine the explicit form of F and G in a very algorithmic way. Additionally the compatibility condition induces a set of constraints on the coefficient matrices in F and G. These coefficient matrices subject to the constraints generate a finite dimensional matrix Lie algebra.

In the extended Estabrook-Wahlquist method we once again take F and G to be functions of u and its partial derivatives but now additionally allow dependence on x and t. Although the details

44 change, the general procedure will remain essentially the same. This typically entails equating

the coefﬁcient of the highest partial derivative in space of the unknown function(s) to zero and working our way down until we have eliminated all partial derivatives of the unknown function(s).

This is simply because the spatial derivative of the time evolution matrix G will introduce a term which contains a spatial derivative of the unknown function of degree one greater than that which G depends on. For example, in the case of the Camassa-Holm equation with G as it was given

above one would have

G G G G G Dx = x + uux + ux uxx + uxx uxxx.

The ﬁnal term in the above expression involves uxxx (i.e. a spatial derivative of u one degree higher than G depends on.) This result, coupled with the terms resulting from the elimination of the ut

using the evolution equation, yield a uxxx term whose coefﬁcient must necessarily vanish as F and

G do not depend on uxxx. Before considering some examples we make the following observations.

This extended method usually results in a large partial differential equation (in the standard Estabrook-Wahlquist method, this is an algebraic equation) which can be solved by equating the coefficients of the different powers of the unknown function(s) to zero. This final step induces a set of constraints on the coefficient matrices in F and G. Another big difference which we will see in the examples comes in the final and, arguably, the hardest step. In the standard Estabrook-

Wahlquist method the final step involves finding explicit forms for the set of coefficient matrices such that they satisfy the contraints derived in the procedure and depend on a spectral parame-

ter. Note these constraints are nothing more than a system of algebraic matrix equations. In the extended Estabrook-Wahlquist method these constraints will be in the form of matrix partial dif-

ferential equations which can be used to derive an integrability condition on the coefﬁcients in the nlpde.

45 Note that compatibility of the time and space evolution matrices will yield a set of constraints

which contain the constant coefﬁcient constraints as a subset. In fact, taking the variable coef- ﬁcients to be the appropriate constants will yield exactly the Estabrook-Wahlquist results for the

constant coefficient version of the nlpde. That is, the constraints given by the Estabrook-Wahlquist method for a constant coefficient nlpde are always a proper subset of the constraints given by a variable-coefficient version of the nlpde. This can easily be shown. Letting F and G not depend explicitly on x and t and taking the coefficients in the NLPDE to be constant the zero-curvature condition as it is written above becomes

F u G u +[F, G]=0, umx,nt mx,(n+1)t − ujx,kt (j+1)x,kt m,n j,k which is exactly the standard Estabrook-Wahlquist method.

The conditions derived via mandating (3.2) be satisfied upon solutions of the variable-coefficient nlpde may be used to determine conditions on the coefficient matrices and variable coefficients

(present in the NLPDE). Successful closure of these conditions is equivalent to the system being S-integrable. A major advantage to using the Estabrook-Wahlquist method that carries forward with the extension is the fact that it requires little guesswork and yields quite general results.

PT-Symmetric and Standard Nonlinear Schrodinger¨ Equation

We begin with the derivation of the Lax pair and differential constraints for the variable-coefﬁcient standard NLS equation. Following with the procedure outlined above we choose

F = F(x,t,q,r), and G = G(x,t,r,q,rx,qx).

46 Compatibility requires these matrices satisfy the zero-curvature conditions given by (3.2). Plug- ging F and G into (3.2) we have

F + F r + F q G G r G q G r G q +[F, G]=0. (3.3) t r t q t − x − r x − q x − rx xx − qx xx

Now requiring this be satisﬁed upon solutions of (2.3) we follow the standard technique of eliminating rt and ut via (2.3) from which we obtain

F iF (fr + gr2q +(υ iγ)r)+ iF (fq + gq2r +(υ + iγ)q) t − r xx − q xx G G r G q G r G q +[F, G]=0. (3.4) − x − r x − q x − rx xx − qx xx

Since F and G do not depend on qxx or rxx we collect the coefﬁcients of qxx and rxx and equate them to zero. This requires

ifF G =0, and ifF G =0. (3.5) − r − rx q − qx

Solving this linear system yields

G = if(F q F r )+ K0(x,t,q,r). (3.6) q x − r x

Plugging this expression for G into equation (3.4) gives us the updated requirement

47 F iF (gr2q +(υ iγ)r)+ iF (gq2r +(υ + iγ)q) if (F q F r ) t − r − q − x q x − r x K0q K0r if(F q F r ) if(F q2 F r2) K0 + ifq [F, F ] − q x − r x − qx x − rx x − qq x − rr x − x x q ifr [F, F ]+ F, K0 =0. (3.7) − x r

F K0 2 2 Now since and do not depend on qx and rx we collect the coefﬁcients of the qx and rx and equate them to zero. This now requires

ifF =0= ifF . rr − qq

From this, it follows via simple integration that F depends on q and r explicitly as follows,

F = X1(x,t)+ X2(x,t)q + X3(x,t)r + X4(x,t)rq,

where the Xi in this expression are arbitrary matrices whose elements are functions of x and t but do not depend on q, r, or their partial derivatives. Plugging this expression for F into equation (3.7) we obtain

48 X + X q + X r + X rq i(X + X q)(gr2q +(υ iγ)r)+ i(X + X r)(gq2r +(υ + iγ)q) 1,t 2,t 3,t 4,t − 3 4 − 2 4 if ((X + X r)q (X + X q)r ) K0q K0r if((X + X r)q (X + X q)r ) − x 2 4 x − 3 4 x − q x − r x − 2,x 4,x x − 3,x 4,x x K0 + if[X , X ]q + if[X , X ]rq + if[X , X ]rqq + if[X , X ]rq + if[X , X ]rqq − x 1 2 x 3 2 x 4 2 x 1 4 x 2 4 x +if[X , X ]r2q if[X , X ]r if[X , X ]qr if[X , X ]qrr if[X , X ]qr 3 4 x − 1 3 x − 2 3 x − 4 3 x − 1 4 x if[X , X ]q2r if[X , X ]qrr +[X , K0]+[X , K0]q +[X , K0]r +[X , K0]qr =0. (3.8) − 2 4 x − 3 4 x 1 2 3 4

Noting the antisymmetry of the commutator, that is [A, B]= [B,A] (which is required if the X − i are elements of a Lie algebra), we can further simplify the previous expression to obtain

X + X q + X r + X rq i(X + X q)(gr2q +(υ iγ)r)+ i(X + X r)(gq2r +(υ + iγ)q) 1,t 2,t 3,t 4,t − 3 4 − 2 4 if ((X + X r)q (X + X q)r ) K0q K0r if((X + X r)q (X + X q)r ) − x 2 4 x − 3 4 x − q x − r x − 2,x 4,x x − 3,x 4,x x K0 + if[X , X ]q + if[X , X ]rq + if[X , X ]rq + if[X , X ]r2q if[X , X ]r − x 1 2 x 3 2 x 1 4 x 3 4 x − 1 3 x if[X , X ]qr if[X , X ]qr if[X , X ]q2r =0. (3.9) − 2 3 x − 1 4 x − 2 4 x

0 As before, since the Xi and K do not depend on rx or qx we equate the coefﬁcients of the qx and rx terms to zero. We therefore require the following equations are satisﬁed,

49 if (X + X r) K0 if(X + X r)+ if[X , X ]+ ifr[X , X ] − x 2 4 − q − 2,x 4,x 1 2 1 4 ifr[X , X ]+ ifr2[X , X ]=0, (3.10) − 2 3 3 4 if (X + X q) K0 + if(X + X q) if[X , X ] ifq[X , X ] x 3 4 − r 3,x 4,x − 1 3 − 1 4 ifq[X , X ] ifq2[X , X ]=0. (3.11) − 2 3 − 2 4

Upon trying to integrate this system one ﬁnds that the system is in its current state inconsistent unless a consistency condition is satisﬁed. Recall that given a system of PDEs

Ψq = ξ(q,r) and Ψr = η(q,r),

if we are to recover Ψ we must satisfy a consistency condition. That is, we must have ξr =Ψqr =

Ψrq = ηq. In terms of equations (3.10) and (3.11) we have

ξ(q,r) = if (X + X r) if(X + X r)+ if[X , X ]+ ifr[X , X ] − x 2 4 − 2,x 4,x 1 2 1 4 ifr[X , X ]+ ifr2[X , X ]=0 (3.12) − 2 3 3 4 η(q,r) = if (X + X q)+ if(X + X q) if[X , X ] ifq[X , X ] x 3 4 3,x 4,x − 1 3 − 1 4 ifq[X , X ] ifq2[X , X ]=0 (3.13) − 2 3 − 2 4

Thus the consistency condition (ξr = ηq) requires that

50 if X ifX + if[X , X ] if[X , X ]+2if[X , X ]r = if X + ifX if[X , X ] − x 4 − 4,x 1 4 − 2 3 3 4 x 4 4,x − 1 4 if[X , X ] 2if[X , X ]q − 2 3 − 2 4 hold. But this means we must have

2if X +2ifX 2if[X , X ] 2if[X , X ](r + q)=0. (3.14) x 4 4,x − 1 4 − 3 4

One easy choice to make the system consistent, and for the purpose of demonstrating how this method can reproduce results previously obtained in the literature, is to set X4 = 0. Thus the system becomes

K0 = if X ifX + if[X , X ] ifr[X , X ], (3.15) q − x 2 − 2,x 1 2 − 2 3 K0 = if X + ifX if[X , X ] ifq[X , X ]. (3.16) r x 3 3,x − 1 3 − 2 3

Integrating the ﬁrst equation with respect to q we obtain

0 K = if X q ifX q + if[X , X ]q if[X , X ]rq + K∗(x,t,r). − x 2 − 2,x 1 2 − 2 3

0 K Now differentiating this and requiring that it equal our previous expression for Kr we ﬁnd that ∗ must satisfy

51 K∗ = if X + ifX if[X , X ]. r x 3 3,x − 1 3

Integrating this expression with respect to r we easily ﬁnd that

K∗ = if X r + ifX r if[X , X ]r + X (x,t), x 3 3,x − 1 3 0

where X0 is an arbitrary matrix whose elements are functions of x and t and does not depend on q, r, or their partial derivatives. Finally, plugging this expression for K∗ into our previous expression for K0 we have

K0 = if (X r X q)+if(X r X q)+if[X , X ]q if[X , X ]r if[X , X ]qr+X . (3.17) x 3 − 2 3,x − 2,x 1 2 − 1 3 − 2 3 0

Plugging this into equation (3.9) we obtain

52 X + X q + X r iX (gr2q +(υ iγ)r)+ iX (gq2r +(υ + iγ)q) if (X r X q) 1,t 2,t 3,t − 3 − 2 − xx 3 − 2 2if (X r X q) if(X r X q) i(f[X , X ]) q + i(f[X , X ]) r − x 3,x − 2,x − 3,xx − 2,xx − 1 2 x 1 3 x X + if ([X , X ]r [X , X ]q)+ if([X , X ]r [X , X ]q)+ if[X , [X , X ]]q − 0,x x 1 3 − 1 2 1 3,x − 1 2,x 1 1 2 if[X , [X , X ]]r if[X , [X , X ]]qr +[X , X ]+ if [X , X ]qr + i(f[X , X ]) qr − 1 1 3 − 1 2 3 1 0 x 2 3 2 3 x +if[X , [X , X ]]q2 if[X , [X , X ]]qr if[X , [X , X ]]q2r +[X , X ]q if [X , X ]rq 2 1 2 − 2 1 3 − 2 2 3 2 0 − x 3 2 +if([X , X ]r2 [X , X ]rq)+ if[X , [X , X ]]rq if[X , [X , X ]]r2 if[X , [X , X ]]qr2 3 3,x − 3 2,x 3 1 2 − 3 1 3 − 3 2 3 +if([X , X ]qr [X , X ]q2)+[X , X ]r =0. (3.18) 2 3,x − 2 2,x 3 0

Since the Xi are independent of r and q we equate the coefﬁcients of the different powers of r and q to zero and thus obtain the following constraints at each order in r and q:

53 O(1) : X X +[X , X ]=0, (3.19) 1,t − 0,x 1 0 O(q) : X + iX (υ + iγ)+ i(fX ) i(f[X , X ]) if [X , X ] if[X , X ] 2,t 2 2 xx − 1 2 x − x 1 2 − 1 2,x

+if[X1, [X1, X2]]+[X2, X0]=0, (3.20)

O(r) : X iX (υ iγ) i(fX ) + i(f[X , X ]) + if [X , X ]+ if[X , X ] 3,t − 3 − − 3 xx 1 3 x x 1 3 1 3,x if[X , [X , X ]]+[X , X ]=0, (3.21) − 1 1 3 3 0 O(qr) : 2i(f[X , X ]) if[X , [X , X ]] + if [X , X ] if[X , [X , X ]] 2 3 x − 1 2 3 x 2 3 − 2 1 3

+if[X3, [X1, X2]]=0, (3.22)

O(q2) : if[X , X ] if[X , [X , X ]]=0, (3.23) 2 2,x − 2 1 2 O(r2) : if[X , X ] if[X , [X , X ]]=0, (3.24) 3 3,x − 3 1 3 O(q2r) : igX if[X , [X , X ]]=0, (3.25) 2 − 2 2 3 2 O(r q) : igX3 + if[X3, [X2, X3]]=0. (3.26)

These equations collectively determine the conditions for the Lax-integrability of the system (2.3). Note that in general, as with the standard Estabrook-Wahlquist method, the solution to the above system is not unique. Moreover, as we will show, one finds that resolution of the system of equations derived in the process of requiring compatibility under the zero-curvature equation (such as the system above) cannot be done in general. Rather it will require additional compatibility conditions be satisfied between the coefficients in system (2.3). This result is not unique to this system but in fact typical of most systems (if not all). Provided we can find representations for the Xi, and in doing so derive the necessary compatibility conditions between the coefficient functions in

(2.3), we will obtain our Lax pair F and G. We will now show how to reproduce the results given

54 in Khawaja’s paper. Let us consider Khawaja’s choices for the Xi matrices. That is, we take

g 0 f 0 0 ip 0 0 X 1 X 1 X 1 X 0 =   , 1 =   , 2 =   , 3 =   . (3.27) 0 g13 0 f7 0 0 ip2 0       −          Plugging these into our integrability conditions yields

O(1) : f g =0, (3.28) 1t − 1x O(1) : f g =0, (3.29) 7t − 13x O(q) : ip ip (g g iυ + γ) (fp ) + 2(f f )(p f) 1t − 1 1 − 13 − − 1 xx 1 − 7 1 x fp (f f )2 + fp (f f ) =0, (3.30) − 1 1 − 7 1 1 − 7 x O(r) : ip + ip (g g iυ γ)+(fp ) + 2(f f )(fp ) 2t 2 1 − 13 − − 2 xx 1 − 7 2 x +(f f )2fp + fp (f f ) =0, (3.31) 1 − 7 2 2 1 − 7 x

O(qr) : fxp1p2 + 2(fp1p2)x =0, (3.32)

2 2 O(q r) and O(r q) : g +2fp1p2 =0. (3.33)

Note that equations (3.23) and (3.24) were identically satisﬁed and thus omitted here. In Khawaja’s paper we see equations (3.28), (3.29), (3.32) and (3.33) given exactly as they are above. To see that the other conditions are equivalent we note that in his paper he had the additional determining equations

55 (fp ) fp (f f ) g =0, (3.34) 1 x − 1 1 − 7 − 6 (fp ) + fp (f f ) g =0, (3.35) 2 x 2 1 − 7 − 10 g (f f ) ip (g g iυ + γ) g + ip =0, (3.36) 6 1 − 7 − 1 1 − 13 − − 6x 1t g (f f )+ ip (g g iυ γ)+ g + ip =0. (3.37) 10 1 − 7 2 1 − 13 − − 10x 2t

We begin by solving equations (3.34) and (3.35) for g6 and g10, respectively. Now plugging g6 into equation (3.36) and g10 into equation (3.37) we obtain the system

2(fp ) (f f ) fp (f f )2 ip (g g iυ + γ) (fp ) 1 x 1 − 7 − 1 1 − 7 − 1 1 − 13 − − 1 xx +fp (f f ) + ip =0, (3.38) 1 1 − 7 x 1t (fp ) (f f )+ fp (f f )2 + ip (g g iυ γ)+(fp ) 2 x 1 − 7 2 1 − 7 2 1 − 13 − − 2 xx +fp (f f ) + ip =0, (3.39) 2 1 − 7 x 2t which are exactly equations (3.30) and (3.31). At this point the derivation of the ﬁnal conditions on the coefﬁcient functions is exactly as it was given in the previous chapter and thus will be omitted here. The Lax pair for this system is given by

56 F = X1 + X2q + X3r (3.40)

G = if(X q X r )+ if (X r X q)+ if(X r X q)+ if[X , X ]q if[X , X ]r 2 x − 3 x x 3 − 2 3,x − 2,x 1 2 − 1 3 if[X , X ]qr + X (3.41) − 2 3 0

As the PT-symmetric NLS is a special case of the system considered above we may obtain the results through the necessary reductions. Letting γ = v = 0, f = a , g = a and taking − 1 − 2 r(x,t)= q∗( x,t) where ∗ denotes the complex conjugate we obtain −

O(1) : X X +[X , X ]=0, (3.42) 1,t − 0,x 1 0 O(q) : X i(a X ) + i(a [X , X ]) + ia [X , X ]+ ia [X , X ] 2,t − 1 2 xx 1 1 2 x 1x 1 2 1 1 2,x

+if[X1, [X1, X2]]+[X2, X0]=0, (3.43)

O(r) : X + i(a X ) i(a [X , X ]) ia [X , X ] ia [X , X ] 3,t 1 3 xx − 1 1 3 x − 1x 1 3 − 1 1 3,x

ia1[X1, [X1, X3]]+[X3, X0]=0, (3.44)

O(qr) : 2i(a [X , X ]) + ia [X , [X , X ]] ia [X , X ]+ ia [X , [X , X ]] − 1 2 3 x 1 1 2 3 − 1x 2 3 1 2 1 3 ia [X , [X , X ]]=0, (3.45) − 1 3 1 2 O(q2) : ia [X , X ]+ ia [X , [X , X ]]=0, (3.46) − 1 2 2,x 1 2 1 2 O(r2) : ia [X , X ]+ ia [X , [X , X ]]=0, (3.47) − 1 3 3,x 1 3 1 3 O(q2r) : ia X + ia [X , [X , X ]]=0, (3.48) − 2 2 1 2 2 3 O(r2q) : ia X ia [X , [X , X ]]=0. (3.49) − 2 3 − 1 3 2 3

57 Utilizing the same set of generators

g 0 f 0 0 ip 0 0 X 1 X 1 X 1 X 0 =   , 1 =   , 2 =   , 3 =   , (3.50) 0 g13 0 f7 0 0 ip2 0       −          we have the following set of conditions

O(1) : f g =0, (3.51) 1t − 1x O(1) : f g =0, (3.52) 7t − 13x O(q) : ip ip (g g )+(a p ) 2(f f )(p a ) 1t − 1 1 − 13 1 1 xx − 1 − 7 1 1 x +a p (f f )2 a p (f f ) =0, (3.53) 1 1 1 − 7 − 1 1 1 − 7 x O(r) : ip + ip (g g ) (a p ) 2(f f )(a p ) 2t 2 1 − 13 − 1 2 xx − 1 − 7 1 2 x (f f )2a p a p (f f ) =0, (3.54) − 1 − 7 1 2 − 1 2 1 − 7 x O(qr) : a p p 2(a p p ) =0, (3.55) − 1x 1 2 − 1 1 2 x 2 2 O(q r) and O(r q) : a2 +2a1p1p2 =0. (3.56)

As with the standard NLS, the ﬁnal conditions on the coefﬁcient functions can be found in the previous chapter. The Lax pair for this system is then given by

F = X1 + X2q + X3r, (3.57)

G = ia (X q X r ) ia (X r X q) ia (X r X q) ia [X , X ]q + ia [X , X ]r − 1 2 x − 3 x − 1x 3 − 2 − 1 3,x − 2,x − 1 1 2 1 1 3

+ia1[X2, X3]qr + X0. (3.58)

58 Derivative Nonlinear Schrodinger Equation

In this section we derive the Lax pair and differential constraints for the variable-coefﬁcient derivative nonlinear Schrodinger¨ equation. The details of this example will be similar to that of the standard NLS and PT-symmetric NLS. Following the extended Estabrook-Wahlquist procedure as outlined at the beginning of the chapter we let

F = F(x,t,r,q), and G = G(x,t,r,q,rx,qx).

Plugging F and G as given above into equation (3.2) we obtain

F + F q + F r G G q G q G r G r +[F, G]=0. (3.59) t q t r t − x − q x − qx xx − r x − rx xx

Now using substituting for qt and rt using equations (2.42) and (2.43), respectively, we obtain the following equation,

F +(ia F G )q (ia F + G )r F (2a rqq + a q2r ) t 1 q − qx xx − 1 r rx xx − q 2 x 2 x F (2a qrr + a r2q ) G G q G r +[F, G]=0. (3.60) − r 2 x 2 x − x − q x − r x

Since F and G do not depend on qxx or rxx we can set the coefﬁcients of the qxx and rxx terms to

59 zero. This requires F and G satisfy the equations

ia F G =0 and ia F + G =0. (3.61) 1 q − qx 1 r rx

Solving this in the same manner as in the NLS example we obtain

G = ia (F q F r )+ K0(x,t,r,q). (3.62) 1 q x − r x

Plugging this expression for G into equation (3.60) we obtain

F F (2a rqq + a q2r ) F (2a qrr + a r2q ) i(a F ) q + i(a F ) r K0 t − q 2 x 2 x − r 2 x 2 x − 1 q x x 1 r x x − x +ia F r2 K0r ia F q2 K0q + ia [F, F ]q ia [F, F ]r +[F, K0]=0. (3.63) 1 rr x − r x − 1 qq x − q x 1 q x − 1 r x

0 Now since F and K do not depend on qx or rx we can set the coefﬁcients of the different powers

2 2 of rx and qx to zero. Setting the coefﬁcients of the qx and rx terms to zero we obtain the conditions

ia F = ia F =0. (3.64) − 1 qq 1 rr

From these conditions it follows that F must be of the form F = X1(x,t)+ X2(x,t)r + X3(x,t)q +

X4(x,t)qr where the Xi are matrices whose elements are functions of x and t. As with the previous examples these matrices are independent of q, r, and their partial derivatives. Now setting the coefﬁcients of the qx and rx terms to zero we obtain the following conditions,

60 a X q2 a X q2r + i(a X ) + i(a X ) q K0 ia [X , X ] ia [X , X ]q − 2 3 − 2 4 1 2 x 1 4 x − r − 1 1 2 − 1 1 4 ia [X , X ]q ia [X , X ]q2 2a X rq 2a X q2r =0, (3.65) − 1 3 2 − 1 3 4 − 2 2 − 2 4 a X r2 a X r2q i(a X ) i(a X ) r K0 + ia [X , X ]+ ia [X , X ]r − 2 2 − 2 4 − 1 3 x − 1 4 x − q 1 1 3 1 1 4 +ia [X , X ]r + ia [X , X ]r2 2a X rq 2a X r2q =0. (3.66) 1 2 3 1 2 4 − 2 3 − 2 4

In much the same way as for the NLS we denote the left-hand side of (3.65) as ξ(r, q) and the

0 left-hand side of (3.66) as η(r, q). For recovery of K we require that ξq = ηr. Computing ξq and

ηr we ﬁnd

ξ = 2a X q 2a X qr + i(a X ) ia [X , X ] ia [X , X ] 2ia [X , X ]q q − 2 3 − 2 4 1 4 x − 1 1 4 − 1 3 2 − 1 3 4 2a X r 4a X qr, (3.67) − 2 2 − 2 4 η = 2a X r 2a X rq i(a X ) + ia [X , X ]+ ia [X , X ]+2ia [X , X ]r r − 2 2 − 2 4 − 1 4 x 1 1 4 1 2 3 1 2 4 2a X q 4a X rq. (3.68) − 2 3 − 2 4

As with the NLS, consistency requires ξq and ηr be equal. This is equivalent to the condition

2i(a X ) 2ia [X , X ] 2ia [X , X ]q 2ia [X , X ]r =0. (3.69) 1 4 x − 1 1 4 − 1 3 4 − 1 2 4

Since the Xi do not depend on q or r this previous condition requires

61 2i(a X ) 2ia [X , X ]=0, (3.70) 1 4 x − 1 1 4 2ia [X , X ]=0, (3.71) − 1 3 4 2ia [X , X ]=0. (3.72) − 1 2 4

Once again in following with the standard NLS we will take X4 =0 in order to simplify computa- tions. Therefore we have

K0 = a X r2 i(a X ) + ia [X , X ]+ ia [X , X ]r 2a X rq, (3.73) q − 2 2 − 1 3 x 1 1 3 1 2 3 − 2 3 K0 = a X q2 + i(a X ) ia [X , X ] ia [X , X ]q 2a X rq. (3.74) r − 2 3 1 2 x − 1 1 2 − 1 3 2 − 2 2

Integrating the ﬁrst equation with respect to q yields

0 2 2 K = a X r q i(a X ) q + ia [X , X ]q + ia [X , X ]rq a X q r + K∗(x,t,r). − 2 2 − 1 3 x 1 1 3 1 2 3 − 2 3

Differentiating this equation with respect to r and requiring that it equal our previous expression

K0 K for r we ﬁnd that ∗ must satisfy

K∗ = i(a X ) ia [X , X ]. r 1 2 x − 1 1 2

From this it follows

62 K∗ = i(a X ) r ia [X , X ]r + X (x,t), 1 2 x − 1 1 2 0

where X0 a matrix whose elements are functions of x and t and which does not depend on q, r, or

0 their partial derivatives. Plugging this expression for K∗ into our previous expression for K we obtain the following ﬁnal expression for K0,

K0 = i(a X ) r i(a X ) q ia [X , X ]r + ia [X , X ]q + ia [X , X ]rq a X r2q 1 2 x − 1 3 x − 1 1 2 1 1 3 1 2 3 − 2 2 a X q2r + X (x,t). (3.75) − 2 3 0

Now plugging this and our expression for F into equation (3.63) we get

X + X r + X q i(a X ) r + i(a X ) q + i(a [X , X ]) r i(a [X , X ]) q 1,t 2,t 3,t − 1 2 xx 1 3 xx 1 1 2 x − 1 1 3 x i(a [X , X ]) rq +(a X ) r2q +(a X ) q2r X + i[X , (a X ) ]r [X , (a X ) ]q − 1 2 3 x 2 2 x 2 3 x − 0,x 1 1 2 x − 1 1 3 x ia [X , [X , X ]]r + ia [X , [X , X ]]q + ia [X , [X , X ]]rq a [X , X ]r2q a [X , X ]q2r − 1 1 1 2 1 1 1 3 1 1 2 3 − 2 1 2 − 2 1 3 +[X , X ]+ i[X , (a X ) ]r2 i[X , (a X ) ]rq ia [X , [X , X ]]r2 + ia [X , [X , X ]]rq 1 0 2 1 2 x − 2 1 3 x − 1 2 1 2 1 2 1 3 +ia [X , [X , X ]]r2q +[X , X ]r + i[X , (a X ) ]rq i[X , (a X ) ]q2 ia [X , [X , X ]]rq 1 2 2 3 2 0 3 1 2 x − 3 1 3 x − 1 3 1 2 2 2 +ia1[X3, [X1, X3]]q + ia1[X3, [X2, X3]]q r +[X3, X0]q =0. (3.76)

Since the Xi are independent of r and q we equate the coefﬁcients of the different powers of r and q to zero thereby obtaining the following ﬁnal constraints:

63 O(1) : X X +[X , X ]=0, (3.77) 1,t − 0,x 1 0 O(q) : X + i(a X ) i(a [X , X ]) i[X , (a X ) ]+ ia [X , [X , X ]] 3,t 1 3 xx − 1 1 3 x − 1 1 3 x 1 1 1 3

+[X3, X0]=0, (3.78)

O(r) : X i(a X ) + i(a [X , X ]) + i[X , (a X ) ] ia [X , [X , X ]] 2,t − 1 2 xx 1 1 2 x 1 1 2 x − 1 1 1 2

+[X2, X0]=0, (3.79)

O(rq) : (a [X , X ]) + a [X , [X , X ]] [X , (a X ) ]+ a [X , [X , X ]]+[X , (a X ) ] − 1 2 3 x 1 1 2 3 − 2 1 3 x 1 2 1 3 3 1 2 x a [X , [X , X ]]=0, (3.80) − 1 3 1 2 O(q2) : [X , (a X ) ]+ a [X , [X , X ]]=0, (3.81) − 3 1 3 x 1 3 1 3 O(r2) : [X , (a X ) ] a [X , [X , X ]]=0, (3.82) 2 1 2 x − 1 2 1 2 O(r2q) : (a X ) a [X , X ]+ ia [X , [X , X ]]=0, (3.83) 2 2 x − 2 1 2 1 2 2 3 O(q2r) : (a X ) a [X , X ]+ ia [X , [X , X ]]=0. (3.84) 2 3 x − 2 1 3 1 3 2 3

We take the following forms for the generators

g g f 0 0 f 0 0 X 1 2 X 1 X 3 X 0 =   , 1 =   , 2 =   , 3 =   , (3.85) g3 g4 0 f2 0 0 f4 0                

where the fi and gj are yet to be determined functions of x and t. Note that with this choice the (3.81) and (3.82) equations are immediately satisfed. From equations (3.78) and (3.79) we obtain the following conditions,

64 g3f4 = g2f3 =0, (3.86)

f + i(a f ) i(a f (f f )) +(f f )(a f ) + ia f (f f )2 4t 1 4 xx − 1 4 1 − 2 x 2 − 1 1 4 x 1 4 1 − 2 +f (g g )=0, (3.87) 4 4 − 1 f i(a f ) i(a f (f f )) +(f f )(a f ) ia f (f f )2 3t − 1 3 xx − 1 3 1 − 2 x 2 − 1 1 3 x − 1 3 1 − 2 f (g g )=0. (3.88) − 3 4 − 1

To keep X2 and X3 nonzero (and thus obtain nontrivial results) we force g2 = g3 = 0. The condition given by equation (3.80) reduces to the single equation

(a1f3f4)x + f3(a1f4)x + f4(a1f3)x =0. (3.89)

The ﬁnal two conditions now yield the system

(a f ) a f (f f ) 2ia f 2f =0, (3.90) 2 3 x − 2 3 2 − 1 − 1 3 4 (a f ) + a f (f f )+2ia f 2f =0. (3.91) 2 4 x 2 4 2 − 1 1 4 3

At this point the resolution of the system given by equations (3.77) and (3.87) - (3.91) such that the ai are real-valued functions requires either f3 = 0 or f4 = 0. Without loss of generality we choose f3 =0 from which we obtain the new system of equations

65 f g =0, (3.92) 1t − 1x f g =0, (3.93) 2t − 4x f + i(a f ) i(a f (f f )) +(f f )(a f ) + ia f (f f )2 4t 1 4 xx − 1 4 1 − 2 x 2 − 1 1 4 x 1 4 1 − 2 +f (g g )=0, (3.94) 4 4 − 1 (a f ) + a f (f f )=0. (3.95) 2 4 x 2 4 2 − 1

Solving equations (3.92), (3.94) and (3.95) for f1, g4, and f2, respectively, we obtain

f1 = g1xdt + F1(x), (3.96) (a2f4)x f2 = + g1xdt + F1(x), (3.97) − a2f4 2 2 2 ia2f4a1xx + ia1a2f4a2xx 2ia1a2xf4 +2ia2a2xa1xf4 f4ta2 g4 = − − 2 − + g1, (3.98) a2f4

where F1(x) is an arbitrary function of x. Plugging these expressions into equation (3.93) yields the integrability condition

a3a ia a a + ia a2 3a2a a 4a3 a +5a a a a +4a2 a a 2 1xxx − 2t 2x 2 2xt 2 − 2 2xx 1x − 2x 1 1 2 2x 2xx 2x 2 1x a2a a 2a a2a =0. (3.99) − 2 1 2xxx − 2x 2 1xx

Since we require that the ai be real this equation splits into the conditions

66 a a a a =0, (3.100) 2t 2x − 2xt 2 a3a 3a2a a 4a3 a +5a a a a +4a2 a a 2 1xxx − 2 2xx 1x − 2x 1 1 2 2x 2xx 2x 2 1x a2a a 2a a2a =0. (3.101) − 2 1 2xxx − 2x 2 1xx

With the aid of MAPLE we ﬁnd that the previous system is exactly solvable with solution given by

x dx dx a (x,t) = F (t)F (x) c + c x c + c x , (3.102) 1 4 2 1 2 − 1 F (x) 1 F (x) 2 2 a2(x,t) = F2(x)F3(t), (3.103)

where F2, F3, and F4 are arbitrary functions in their respective variables and c1 and c2 are arbitrary constants. The Lax pair for this system is then found to be

F = X1 + X3q, (3.104)

G = ia X q i(a X ) q + ia [X , X ]q a X q2r + X . (3.105) 1 3 x − 1 3 x 1 1 3 − 2 3 0

Fifth-Order Korteweg-de-Vries Equation

In this section we derive the Lax pair and differential constraints for the generalized ﬁfth-order

variable-coefﬁcient KdV. Following the procedure outlined earlier in the chapter we let

67 F = F(x,t,u) and G = G(x,t,u,ux,uxx,uxxx,uxxxx).

Plugging F and G into equation (3.2) we obtain

F +F u G G u G u G u G u G u +[F, G]=0. (3.106) t u t − x − u x − ux xx − uxx xxx − uxxx xxxx − uxxxx xxxxx

Next, substituting for ut in the previous expression using equation (2.66) we obtain the equation

F F a uu + a u u + a u2u + a uu + a u + a u + a u + a u G t − u 1 xxx 2 x xx 3 x 4 x 5 xxx 6 xxxxx 7 8 x − x G u G u G u G u G u +[F, G]=0. (3.107) − u x − ux xx − uxx xxx − uxxx xxxx − uxxxx xxxxx

Since F and G do not depend on uxxxxx we can equate the coefﬁcient of the uxxxxx term to zero. This requires that F and G satisfy

G + a F =0 G = a F u + K0(x,t,u,u ,u ,u ). uxxxx 6 u ⇒ − 6 u xxxx x xx xxx

Updating equation (3.107) using the above expression for G we obtain the equation

68 F F a uu + a u u + a u2u + a uu + a u + a u + a u + a F u t − u 1 xxx 2 x xx 3 x 4 x 5 xxx 7 8 x 6x u xxxx +a F u K0 K0 u K0 u K0 u K0 u + a F u u 6 xu xxxx − x − u x − ux xx − uxx xxx − uxxx xxxx 6 uu x xxxx [F, F ]a u +[F, K0]=0. (3.108) − u 6 xxxx

0 Since F and K do not depend on uxxxx we can equate the coefﬁcient of the uxxxx term to zero. This requires that K0 satisﬁes

a F + a F + a F u K0 [F, F ]a =0. (3.109) 6x u 6 xu 6 uu x − uxxx − u 6

0 Integrating with respect to uxxx and solving for K we obtain

K0 = a F u + a F u + a F u u [F, F ]a u + K1(x,t,u,u ,u ). (3.110) 6x u xxx 6 xu xxx 6 uu x xxx − u 6 xxx x xx

Updating equation (3.108) by plugging in our expression for K0 we obtain the equation

69 F F a uu + a u u + a u2u + a uu + a u + a u + a u a F u t − u 1 xxx 2 x xx 3 x 4 x 5 xxx 7 8 x − 6xx u xxx 2a F u a F u a F u u a F u u +[F , F ]a u − 6x xu xxx − 6 xxu xxx − 6x uu x xxx − 6 xuu x xxx x u 6 xxx +[F, F ]a u +[F, F ]a u K1 a F u u a F u u K1 u xu 6 xxx u 6x xxx − x − 6x uu x xxx − 6 xuu x xxx − u x a F u2 u +[F, F ]a u u a F u u K1 u K1 u +[F, K1] − 6 uuu x xxx uu 6 x xxx − 6 uu xx xxx − ux xx − uxx xxx +a [F, F ]u + a [F, F ]u + a [F, F ]u u [F, [F, F ]]a u =0. (3.111) 6x u xxx 6 xu xxx 6 uu x xxx − u 6 xxx

1 Since F and K do not depend on uxxx we can equate the coefﬁcient of the uxxx term to zero. This requires that K1 satisﬁes the equation

F (a u + a ) a F 2a F a F a F u a F u − u 1 5 − 6xx u − 6x xu − 6 xxu − 6x uu x − 6 xuu x +[F , F ]a +[F, F ]a +[F, F ]a a F u a F u a F u2 x u 6 xu 6 u 6x − 6x uu x − 6 xuu x − 6 uuu x +[F, F ]a u a F u K1 + a [F, F ]+ a [F, F ]+ a [F, F ]u uu 6 x − 6 uu xx − uxx 6x u 6 xu 6 uu x [F, [F, F ]]a =0. (3.112) − u 6

1 Integrating with respect to uxx and solving for K and collecting like terms we have

K1 = F (a u + a )u (a F ) u 2(a F ) u u + 2(a [F, F ]) u − u 1 5 xx − 6 u xx xx − 6 uu x x xx 6 u x xx 1 a F u2u +2a [F, F ]u u a F u2 a [F , F ]u − 6 uuu x xx 6 uu x xx − 2 6 uu xx − 6 x u xx a [F, [F, F ]]u + K2(x,t,u,u ). (3.113) − 6 u xx x

70 Plugging the expression for K1 into equation (3.111) and simplifying a little bit we obtain the large equation

F F (a u u + a u2u + a uu + a u + a u )+(a F ) uu +(a F ) u t − u 2 x xx 3 x 4 x 7 8 x 1 u x xx 5 u x xx +(a F ) u + 2(a F ) u u (a [F, F ]) u +(a F ) u2u 6 u xxx xx 6 uu xx x xx − 6 u xx xx 6 uuu x x xx 1 + (a F ) u2 ([F, (a F ) ]) u +(a [F, [F, F ]]) u K2 + F a uu u 2 6 uu x xx − 6 u x x xx 6 u x xx − x uu 1 x xx F F F F 2 F 3 + ua1uxuxx + uua5uxuxx +(a6 uu)xxuxuxx + 2(a6 uuu)xuxuxx + a6 uuuuuxuxx 5 a [F , F ]u2 u a [F, F ]u2u + a F u2 u [F , (a F ) ]u u − 6 u uu x xx − 6 uuu x xx 2 6 uuu xx x − u 6 u x x xx 2[F, (a F ) ]u u a [F , F ]u2 u a [F, F ]u2u + a [F , [F, F ]]u u − 6 uu x x xx − 6 u uu x xx − 6 uuu x xx 6 u u x xx 3 +a [F, [F, F ]]u u K2 u + 2(a F ) u2 a [F, F ]u2 K2 u 6 uu x xx − u x 6 uu x xx − 2 6 uu xx − ux xx a [F, F ]u [F, (a F ) ]u [F, (a F ) ]u u +[F, (a [F, F ]) ]u − 5 u xx − 6 u xx xx − 6 uu x x xx 6 u x xx a [F, F ]u2u +2a [F, [F, F ]]u u +[F, [F, (a F ) ]]u a [F, F ]uu − 6 uuu x xx 6 uu x xx 6 u x xx − 1 u xx a [F, [F, [F, F ]]]u 3(a [F, F ]) u u +[F, K2]=0. (3.114) − 6 u xx − 6 uu x x xx

K2 F 2 Since and do not depend on uxx we can start by setting the coefﬁcients of the uxx and the uxx terms to zero. Note the difference here to that of the previous steps. Here we have multiple powers

2 of uxx present in equation (3.114). Setting the O(uxx) term to zero we obtain the condition

3 5 3 (a F ) + a F u a [F, F ]=0. (3.115) 2 6 uu x 2 6 uuu x − 2 6 uu

Since F does not depend on ux we must additionally require that the coefﬁcient of the ux term in this previous expression is zero. This is equivalent to F satisfying

71 1 F =0 F = X (x,t)+ X (x,t)u + X (x,t)u2, uuu ⇒ 1 2 2 3

where the Xi are matrices whose elements are functions of x and t and do not depend on u or its partial derivatives. Plugging this expression for F into equation (3.115) we obtain the condition

3(a X ) 3a ([X , X ]+[X , X ]u)=0. (3.116) 6 3 x − 6 1 3 2 3

Since the Xi do not depend on u we can set the coefﬁcient of the u term to zero. That is, we require

that X2 and X3 commute. We ﬁnd now that equation (3.116) reduces to

(a X ) a [X , X ]=0. (3.117) 6 3 x − 6 1 3

For ease of computation and in order to immediately satisfy equation (3.117) we set X3 = 0. Plugging our expression for F into equation (3.114) we obtain the large equation

72 X + X u X (a u u + a u2u + a uu + a u + a u )+(a X ) uu +(a X ) u 1,t 2,t − 2 2 x xx 3 x 4 x 7 8 x 1 2 x xx 5 2 x xx +(a X ) u (a [X , X ]) u K2 + X a u u [X , (a X ) ]u u 6 2 xxx xx − 6 1 2 xx xx − x 2 1 x xx − 2 6 2 x x xx ([X , (a X ) ]) u ([X , (a X ) ]) uu +(a [X , [X , X ]]) u +(a [X , [X , X ]]) uu − 1 6 2 x x xx − 2 6 2 x x xx 6 1 1 2 x xx 6 2 1 2 x xx K2 u K2 u a [X , X ]uu + a [X , [X , X ]]u u +[X , (a [X , X ]) ]uu − u x − ux xx − 1 1 2 xx 6 2 1 2 x xx 2 6 1 2 x xx a [X , X ]u [X , (a X ) ]u [X , (a X ) ]uu +[X , (a [X , X ]) ]u − 5 1 2 xx − 1 6 2 xx xx − 2 6 2 xx xx 1 6 1 2 x xx 2 +[X1, [X1, (a6X2)x]]uxx +[X1, [X2, (a6X2)x]]uuxx +[X2, [X1, (a6X2)x]]uuxx +[X1, K ]

a [X , [X , [X , X ]]]u a [X , [X , [X , X ]]]uu a [X , [X , [X , X ]]]uu − 6 1 1 1 2 xx − 6 1 2 1 2 xx − 6 2 1 1 2 xx +[X , [X , (a X ) ]]u2u a [X , [X , [X , X ]]]u2u +[X , K2]u =0. (3.118) 2 2 6 2 x xx − 6 2 2 1 2 xx 2

2 Again using the fact that the Xi and K do not depend on uxx we can set the coefﬁcient of the uxx term in equation (3.118) to zero. This requires

(a X ) (a [X , X ]) + a X u [X , (a X ) ]u a X u 6 2 xxx − 6 1 2 xx 1 2 x − 2 6 2 x x − 2 2 x ([X , (a X ) ]) ([X , (a X ) ]) u +(a [X , [X , X ]]) +(a [X , [X , X ]]) u − 1 6 2 x x − 2 6 2 x x 6 1 1 2 x 6 2 1 2 x K2 a [X , X ]u + a [X , [X , X ]]u +[X , (a [X , X ]) ]u +(a X ) − ux − 1 1 2 6 2 1 2 x 2 6 1 2 x 5 2 x a [X , X ] [X , (a X ) ] [X , (a X ) ]u +[X , (a [X , X ]) ] − 5 1 2 − 1 6 2 xx − 2 6 2 xx 1 6 1 2 x

+[X1, [X1, (a6X2)x]]+[X1, [X2, (a6X2)x]]u +[X2, [X1, (a6X2)x]]u +(a1X2)xu

a [X , [X , [X , X ]]] a [X , [X , [X , X ]]]u a [X , [X , [X , X ]]]u − 6 1 1 1 2 − 6 1 2 1 2 − 6 2 1 1 2 +[X , [X , (a X ) ]]u2 a [X , [X , [X , X ]]]u2 =0. (3.119) 2 2 6 2 x − 6 2 2 1 2

73 2 Integrating equation (3.119) with respect to ux and solving for K we ﬁnd

1 1 1 K2 = (a X ) u + a X u2 [X , (a X ) ]u2 a X u2 +(a [X , [X , X ]]) uu 6 2 xxx x 2 1 2 x − 2 2 6 2 x x − 2 2 2 x 6 2 1 2 x x (a [X , X ]) u ([X , (a X ) ]) u ([X , (a X ) ]) uu +(a [X , [X , X ]]) u − 6 1 2 xx x − 1 6 2 x x x − 2 6 2 x x x 6 1 1 2 x x 1 a [X , X ]uu + a [X , [X , X ]]u2 +[X , (a [X , X ]) ]uu +(a X ) uu − 1 1 2 x 2 6 2 1 2 x 2 6 1 2 x x 1 2 x x a [X , X ]u [X , (a X ) ]u [X , (a X ) ]uu +[X , (a [X , X ]) ]u − 5 1 2 x − 1 6 2 xx x − 2 6 2 xx x 1 6 1 2 x x

+[X1, [X1, (a6X2)x]]ux +[X1, [X2, (a6X2)x]]uux +[X2, [X1, (a6X2)x]]uux +(a5X2)xux

a [X , [X , [X , X ]]]u a [X , [X , [X , X ]]]uu a [X , [X , [X , X ]]]uu − 6 1 1 1 2 x − 6 1 2 1 2 x − 6 2 1 1 2 x +[X , [X , (a X ) ]]u2u a [X , [X , [X , X ]]]u2u + K3(x,t,u). (3.120) 2 2 6 2 x x − 6 2 2 1 2 x

It is helpful at this stage to deﬁne the following new matrices

X4 =[X1, X2], X5 =[X1, X4], X6 =[X2, X4] (3.121)

X7 =[X1, X5], X8 =[X2, X5], X9 =[X1, X6], X10 =[X2, X6] (3.122)

for clarity and in order to reduce the size of the equations to follow. Plugging the expression for K2 into equation (3.118) we obtain the updated equation

74 X + X u X (a u2u + a uu + a u + a u ) (a X ) u2 + a X u2 1,t 2,t − 2 3 x 4 x 7 8 x − 6 6 x x 1 4 x 1 1 1 (a X ) u (a X ) u2 + ([X , (a X ) ]) u2 + (a X ) u2 − 6 2 xxxx x − 2 1 2 x x 2 2 6 2 x x x 2 2 2 x x +(a X ) u + ([X , (a X ) ]) u + ([X , (a X ) ]) uu (a X ) u 6 4 xxx x 1 6 2 x xx x 2 6 2 x xx x − 6 5 xx x 1 +(a X ) uu (a X ) u2 ([X , (a X ) ]) uu (a X ) u (a X ) uu 1 4 x x − 2 6 6 x x − 2 6 4 x x x − 5 2 xx x − 1 2 xx x +(a X ) u + ([X , (a X ) ]) u + ([X , (a X ) ]) uu ([X , (a X ) ]) u 5 4 x x 1 6 2 xx x x 2 6 2 xx x x − 1 6 4 x x x ([X , [X , (a X ) ]]) u ([X , [X , (a X ) ]]) uu ([X , [X , (a X ) ]]) uu − 1 1 6 2 x x x − 1 2 6 2 x x x − 2 1 6 2 x x x ([X , [X , (a X ) ]]) u2u +(a X ) u2u K3 + ([X , (a X ) ]) u2 + a X u2 − 2 2 6 2 x x x 6 10 x x − x 2 6 2 x x x 6 9 x [X , (a X ) ]u2 +[X , (a X ) ]u2 + a X u2 +(a X ) uu +(a X ) uu − 2 6 4 x x 2 6 2 xx x 6 8 x 6 9 x x 6 8 x x [X , [X , (a X ) ]]u2 [X , [X , (a X ) ]]u2 (a X ) u2 (a X ) uu − 1 2 6 2 x x − 2 1 6 2 x x − 1 2 x x − 6 6 xx x 1 2[X , [X , (a X ) ]]uu2 +2a X uu2 K3 u +[X , (a X ) ]u a X u2 − 2 2 6 2 x x 6 10 x − u x 1 6 5 x x − 2 2 4 x 1 1 +[X , (a X ) ]u + a X u2 [X , [X , (a X ) ]]u2 +[X , (a X ) ]uu 1 6 2 xxx x 2 1 4 x − 2 1 2 6 2 x x 1 6 6 x x [X , (a X ) ]u [X , ([X , (a X ) ]) ]u [X , ([X , (a X ) ]) ]uu − 1 6 4 xx x − 1 1 6 2 x x x − 1 2 6 2 x x x 1 a X uu + a X u2 +[X , [X , (a X ) ]]uu +[X , (a X ) ]u +(a X ) u − 1 5 x 2 6 9 x 1 2 6 4 x x 1 5 2 x x 6 7 x x a X u [X , [X , (a X ) ]]u [X , [X , (a X ) ]]uu +[X , [X , (a X ) ]]u − 5 5 x − 1 1 6 2 xx x − 1 2 6 2 xx x 1 1 6 4 x x

+[X1, [X1, [X1, (a6X2)x]]]ux +[X1, [X1, [X2, (a6X2)x]]]uux +[X1, [X2, [X1, (a6X2)x]]]uux

a [X , X ]u a [X , X ]uu a [X , X ]uu +[X , [X , (a X ) ]]uu − 6 1 7 x − 6 1 9 x − 6 1 8 x 2 1 6 4 x x +[X , [X , [X , (a X ) ]]]u2u a [X , X ]u2u +[X , K3]+[X , (a X ) ]uu 1 2 2 6 2 x x − 6 1 10 x 1 1 1 2 x x 1 +[X , (a X ) ]uu [X , [X , (a X ) ]]uu2 +[X , (a X ) ]u2u +[X , (a X ) ]uu 2 6 2 xxx x − 2 2 2 6 2 x x 2 6 6 x x 2 6 5 x x [X , (a X ) ]uu [X , ([X , (a X ) ]) ]uu [X , ([X , (a X ) ]) ]u2u − 2 6 4 xx x − 2 1 6 2 x x x − 2 2 6 2 x x x 1 a X u2u + a X uu2 +[X , [X , (a X ) ]]u2u +[X , (a X ) ]uu − 1 6 x 2 6 10 x 2 2 6 4 x x 2 5 2 x x a X uu [X , [X , (a X ) ]]uu [X , [X , (a X ) ]]u2u a [X , X ]u2u − 5 6 x − 2 1 6 2 xx x − 2 2 6 2 xx x − 6 2 8 x

75 2 2 +[X2, [X1, [X1, (a6X2)x]]]uux +[X2, [X1, [X2, (a6X2)x]]]u ux +[X2, (a1X2)x]u ux

+[X , [X , [X , (a X ) ]]]u2u a [X , X ]uu a [X , X ]u2u a [X , X ]u3u 2 2 1 6 2 x x − 6 2 7 x − 6 2 9 x − 6 2 10 x 3 3 +[X2, [X2, [X2, (a6X2)x]]]u ux +[X2, K ]u =0. (3.123)

K3 X 2 Now since and the i do not depend on ux we can set the coefﬁcient of the ux term to zero in equation (3.123). This is equivalent to the requirement that

3([X , (a X ) ]) 3(a X ) +(a X ) 3(a X ) +3a X +3a X 2 6 2 x x − 1 2 x 2 2 x − 6 6 x 1 4 6 9 3[X , [X , (a X ) ]] 2[X , [X , (a X ) ]] 2[X , (a X ) ]+2[X , (a X ) ] − 1 2 6 2 x − 2 1 6 2 x − 2 6 4 x 2 6 2 xx a X +2a X +5a X 5[X , [X , (a X ) ]]=0. (3.124) − 2 4 6 8 6 10 − 2 2 6 2 x

Further since we know that the Xi do not depend on u we can decouple this condition into the system

3([X , (a X ) ]) 3(a X ) +(a X ) 3(a X ) +3a X +3a X 2 6 2 x x − 1 2 x 2 2 x − 6 6 x 1 4 6 9 3[X , [X , (a X ) ]] 2[X , [X , (a X ) ]] 2[X , (a X ) ]+2[X , (a X ) ] − 1 2 6 2 x − 2 1 6 2 x − 2 6 4 x 2 6 2 xx a X +2a X =0, (3.125) − 2 4 6 8 a X [X , [X , (a X ) ]]=0. (3.126) 6 10 − 2 2 6 2 x

76 3 Taking these conditions into account and once again noting the fact that K and the Xi do not depend on ux we can simplify and equate the coefﬁcient of the ux in equation (3.123) to zero. This results in the condition

X (a u2 + a u + a ) (a X ) K3 +[X , (a X ) ] − 2 3 4 8 − 6 2 xxxx − u 1 6 5 x +(a X ) + ([X , (a X ) ]) + ([X , (a X ) ]) u (a X ) 6 4 xxx 1 6 2 x xx 2 6 2 x xx − 6 5 xx +(a X ) u ([X , (a X ) ]) u (a X ) (a X ) u a X u2 1 4 x − 2 6 4 x x − 5 2 xx − 1 2 xx − 1 6 +(a X ) + ([X , (a X ) ]) + ([X , (a X ) ]) u ([X , (a X ) ]) 5 4 x 1 6 2 xx x 2 6 2 xx x − 1 6 4 x x ([X , [X , (a X ) ]]) ([X , [X , (a X ) ]]) u ([X , [X , (a X ) ]]) u − 1 1 6 2 x x − 1 2 6 2 x x − 2 1 6 2 x x +(a X ) u +(a X ) u (a X ) u +[X , (a X ) ]+[X , (a X ) ]u 6 9 x 6 8 x − 6 6 xx 1 6 2 xxx 1 6 6 x [X , (a X ) ] [X , ([X , (a X ) ]) ] [X , ([X , (a X ) ]) ]u − 1 6 4 xx − 1 1 6 2 x x − 1 2 6 2 x x a X u +[X , [X , (a X ) ]]u +[X , (a X ) ]+(a X ) +[X , [X , (a X ) ]]u2 − 1 5 1 2 6 4 x 1 5 2 x 6 7 x 2 2 6 4 x a X [X , [X , (a X ) ]] [X , [X , (a X ) ]]u +[X , [X , (a X ) ]] − 5 5 − 1 1 6 2 xx − 1 2 6 2 xx 1 1 6 4 x

+[X1, [X1, [X1, (a6X2)x]]]+[X1, [X1, [X2, (a6X2)x]]]u +[X1, [X2, [X1, (a6X2)x]]]u

a [X , X ] a [X , X ]u a [X , X ]u +[X , [X , (a X ) ]]u − 6 1 7 − 6 1 9 − 6 1 8 2 1 6 4 x 2 +[X2, (a6X2)xxx]u +[X2, (a6X6)x]u +[X2, (a6X5)x]u +[X1, (a1X2)x]u

[X , (a X ) ]u [X , ([X , (a X ) ]) ]u [X , ([X , (a X ) ]) ]u2 − 2 6 4 xx − 2 1 6 2 x x − 2 2 6 2 x x a X u [X , [X , (a X ) ]]u [X , [X , (a X ) ]]u2 +[X , (a X ) ]u − 5 6 − 2 1 6 2 xx − 2 2 6 2 xx 2 5 2 x 2 2 +[X2, [X1, [X1, (a6X2)x]]]u +[X2, [X1, [X2, (a6X2)x]]]u +[X2, (a1X2)x]u

+[X , [X , [X , (a X ) ]]]u2 a [X , X ]u a [X , X ]u2 a [X , X ]u2 =0. (3.127) 2 2 1 6 2 x − 6 2 7 − 6 2 9 − 6 2 8

Integrating with respect to u and solving for K3 in equation (3.127) we ﬁnd

77 1 1 1 K3 = a X u3 a X u2 a X u (a X ) u +[X , (a X ) ]u a [X , X ]u2 −3 3 2 − 2 4 2 − 8 2 − 6 2 xxxx 1 6 5 x − 2 6 2 7 1 +(a X ) u + ([X , (a X ) ]) u + ([X , (a X ) ]) u2 (a X ) u 6 4 xxx 1 6 2 x xx 2 2 6 2 x xx − 6 5 xx 1 1 1 1 + (a X ) u2 ([X , (a X ) ]) u2 (a X ) u (a X ) u2 a X u3 2 1 4 x − 2 2 6 4 x x − 5 2 xx − 2 1 2 xx − 3 1 6 1 +(a X ) u + ([X , (a X ) ]) u + ([X , (a X ) ]) u2 ([X , (a X ) ]) u 5 4 x 1 6 2 xx x 2 2 6 2 xx x − 1 6 4 x x 1 1 ([X , [X , (a X ) ]]) u ([X , [X , (a X ) ]]) u2 ([X , [X , (a X ) ]]) u2 − 1 1 6 2 x x − 2 1 2 6 2 x x − 2 2 1 6 2 x x 1 1 1 1 + (a X ) u2 + (a X ) u2 (a X ) u2 +[X , (a X ) ]u + [X , (a X ) ]u2 2 6 9 x 2 6 8 x − 2 6 6 xx 1 6 2 xxx 2 1 6 6 x 1 1 [X , (a X ) ]u [X , ([X , (a X ) ]) ]u [X , ([X , (a X ) ]) ]u2 a X u2 − 1 6 4 xx − 1 1 6 2 x x − 2 1 2 6 2 x x − 2 1 5 1 1 + [X , [X , (a X ) ]]u2 +[X , (a X ) ]u +(a X ) u + [X , [X , (a X ) ]]u3 2 1 2 6 4 x 1 5 2 x 6 7 x 3 2 2 6 4 x 1 a X u [X , [X , (a X ) ]]u [X , [X , (a X ) ]]u2 +[X , [X , (a X ) ]]u − 5 5 − 1 1 6 2 xx − 2 1 2 6 2 xx 1 1 6 4 x 1 1 +[X , [X , [X , (a X ) ]]]u + [X , [X , [X , (a X ) ]]]u2 + [X , [X , [X , (a X ) ]]]u2 1 1 1 6 2 x 2 1 1 2 6 2 x 2 1 2 1 6 2 x 1 1 1 a [X , X ]u a [X , X ]u2 a [X , X ]u2 + [X , [X , (a X ) ]]u2 − 6 1 7 − 2 6 1 9 − 2 6 1 8 2 2 1 6 4 x 1 1 1 1 + [X , (a X ) ]u2 + [X , (a X ) ]u3 + [X , (a X ) ]u2 + [X , (a X ) ]u2 2 2 6 2 xxx 3 2 6 6 x 2 2 6 5 x 2 1 1 2 x 1 1 1 [X , (a X ) ]u2 [X , ([X , (a X ) ]) ]u2 [X , ([X , (a X ) ]) ]u3 −2 2 6 4 xx − 2 2 1 6 2 x x − 3 2 2 6 2 x x 1 1 1 1 a X u2 [X , [X , (a X ) ]]u2 [X , [X , (a X ) ]]u3 + [X , (a X ) ]u2 −2 5 6 − 2 2 1 6 2 xx − 3 2 2 6 2 xx 2 2 5 2 x 1 1 1 + [X , [X , [X , (a X ) ]]]u2 + [X , [X , [X , (a X ) ]]]u3 + [X , (a X ) ]u3 2 2 1 1 6 2 x 3 2 1 2 6 2 x 3 2 1 2 x 1 1 1 + [X , [X , [X , (a X ) ]]]u3 a [X , X ]u3 a [X , X ]u3 + X (x,t). (3.128) 3 2 2 1 6 2 x − 3 6 2 9 − 3 6 2 8 0

where X0 is a matrix whose elements are functions of x and t and does not depend on u or its partial derivatives. Next, we update equation (3.123) by plugging in the expression for K3. Upon doing this we will have a rather large expression which is nothing more than an algebraic equation in u. In this ﬁnal step, as the Xi do not depend on u, we set the coefﬁcients of the different powers

78 of u in this last, lengthy expression to zero. At the different orders of u we ﬁnd the following conditions

O(1) : X X +[X , X ]=0, (3.129) 1,t − 0,x 1 0 O(u) : [X , X ] a [X , [X , X ]]+[X , [X , [X , (a X ) ]]]+[X , [X , [X , [X , (a X ) ]]]] 2 0 − 6 1 1 7 1 1 1 6 4 x 1 1 1 1 6 2 x +[X , [X , (a X ) ]] [X , [X , [X , (a X ) ]]]+[X , (a X ) ] [X , [X , (a X ) ]] 1 1 5 2 x − 1 1 1 6 2 xx 1 6 7 x − 1 1 6 4 xx a X [X , [X , ([X , (a X ) ]) ]]+[X , [X , (a X ) ]] [X , ([X , (a X ) ]) ] − 5 7 − 1 1 1 6 2 x x 1 1 6 2 xxx − 1 1 6 4 x x [X , ([X , [X , (a X ) ]]) ]+[X , (a X ) ]+[X , ([X , (a X ) ]) ] [X , (a X ) ] − 1 1 1 6 2 x x 1 5 4 x 1 1 6 2 xx x − 1 5 2 xx [X , (a X ) ]+[X , (a X ) ]+[X , ([X , (a X ) ]) ] a X [X , (a X ) ] − 1 6 5 xx 1 6 4 xxx 1 1 6 2 x xx − 8 4 − 1 6 2 xxxx +[X , [X , (a X ) ]]+(a [X , X ]) ([X , [X , [X , (a X ) ]]]) ([X , [X , (a X ) ]]) 1 1 6 5 x 6 1 7 x − 1 1 1 6 2 x x − 1 1 6 4 x x +(a X ) + ([X , [X , (a X ) ]]) ([X , (a X ) ]) (a X ) + ([X , (a X ) ]) 8 2 x 1 1 6 2 xx x − 1 5 2 x x − 6 7 xx 1 6 4 xx x +([X , ([X , (a X ) ]) ]) ([X , (a X ) ]) + ([X , [X , (a X ) ]]) +(a X ) 1 1 6 2 x x x − 1 6 2 xxx x 1 1 6 2 x xx 5 5 x +X ([X , (a X ) ]) (a X ) +(a X ) +(a X ) ([X , (a X ) ]) 2,t − 1 6 2 xx xx − 5 4 xx 5 2 xxx 6 5 xxx − 1 6 2 x xxx ([X , (a X ) ]) +(a X ) (a X ) a X + ([X , (a X ) ]) =0, (3.130) − 1 6 5 x x 6 2 xxxxx − 6 4 xxxx − 7 2 1 6 4 x xx

79 O(u2) : 2a X 2a [X , [X , X ]]+2[X , [X , [X , [X , (a X ) ]]]] + 2[X , [X , [X , (a X ) ]]] − 5 8 − 6 2 1 7 2 1 1 1 6 2 x 2 1 1 6 4 x 2[X , [X , [X , (a X ) ]]] + 2[X , [X , (a X ) ]] 2[X , [X , ([X , (a X ) ]) ]] a X − 2 1 1 6 2 xx 2 1 5 2 x − 2 1 1 6 2 x x − 4 4 +2[X , (a X ) ] 2[X , [X , (a X ) ]]+2[X , [X , (a X ) ]] 2[X , ([X , (a X ) ]) ] 2 6 7 x − 2 1 6 4 xx 2 1 6 2 xxx − 2 1 6 4 x x 2[X , ([X , [X , (a X ) ]]) ]+2[X , (a X ) ]+2[X , ([X , (a X ) ]) ] 2[X , (a X ) ] − 2 1 1 6 2 x x 2 5 4 x 2 1 6 2 xx x − 2 5 2 xx +2[X , (a X ) ]+2[X , ([X , (a X ) ]) ] 2[X , (a X ) ]+2[X , [X , (a X ) ]] 2 6 4 xxx 2 1 6 2 x xx − 2 6 2 xxxx 2 1 6 5 x 2[X , (a X ) ] a X +[X , [X , (a X ) ]]+[X , [X , [X , [X , (a X ) ]]]] − 2 6 5 xx − 5 9 1 2 5 2 x 1 2 1 1 6 2 x [X , [X , [X , (a X ) ]]]+[X , [X , (a X ) ]] [X , [X , ([X , (a X ) ]) ]] − 1 2 1 6 2 xx 1 1 1 2 x − 1 2 1 6 2 x x +[X , [X , [X , [X , (a X ) ]]]] [X , [X , (a X ) ]]+[X , [X , [X , (a X ) ]]] 1 1 1 2 6 2 x − 1 2 6 4 xx 1 2 1 6 4 x a [X , [X , X ]]+[X , [X , (a X ) ]]+[X , [X , [X , [X , (a X ) ]]]] − 6 1 1 9 1 2 6 2 xxx 1 1 2 1 6 2 x +[X , [X , (a X ) ]] [X , [X , [X , (a X ) ]]] [X , [X , ([X , (a X ) ]) ]] 1 2 6 5 x − 1 1 2 6 2 xx − 1 1 2 6 2 x x a [X , [X , X ]]+[X , [X , [X , (a X ) ]]] a X +[X , [X , (a X ) ]] − 6 1 1 8 1 1 2 6 4 x − 1 7 1 1 6 6 x [X , ([X , [X , (a X ) ]]) ]+[X , (a X ) ]+[X , (a X ) ] [X , (a X ) ] − 1 2 1 6 2 x x 1 6 9 x 1 6 8 x − 1 6 6 xx [X , ([X , [X , (a X ) ]]) ]+[X , ([X , (a X ) ]) ] [X , ([X , (a X ) ]) ] − 1 1 2 6 2 x x 1 2 6 2 xx x − 1 2 6 4 x x [X , (a X ) ]+[X , (a X ) ]+[X , ([X , (a X ) ]) ] a [X , [X , X ]] − 1 1 2 xx 1 1 4 x 1 2 6 2 x xx − 6 1 2 7 ([X , (a X ) ]) ([X , [X , [X , (a X ) ]]]) + ([X , [X , (a X ) ]]) − 2 5 2 x x − 2 1 1 6 2 x x 2 1 6 2 xx x +(a X ) + ([X , ([X , (a X ) ]) ]) ([X , (a X ) ]) ([X , (a X ) ]) 5 6 x 2 1 6 2 x x x − 2 6 5 x x − 1 1 2 x x +([X , (a X ) ]) +(a [X , X ]) +(a [X , X ]) ([X , [X , (a X ) ]]) 2 6 4 xx x 6 1 9 x 6 1 8 x − 2 1 6 4 x x ([X , (a X ) ]) ([X , [X , [X , (a X ) ]]]) ([X , [X , [X , (a X ) ]]]) − 2 6 2 xxx x − 1 1 2 6 2 x x − 1 2 1 6 2 x x +([X , [X , (a X ) ]]) ([X , [X , (a X ) ]]) + ([X , ([X , (a X ) ]) ]) 1 2 6 2 xx x − 1 2 6 4 x x 1 2 6 2 x x x +(a X ) ([X , (a X ) ]) (a X ) (a X ) + ([X , [X , (a X ) ]]) 1 5 x − 1 6 6 x x − 6 9 xx − 6 8 xx 1 2 6 2 x xx +(a X ) + ([X , [X , (a X ) ]]) ([X , (a X ) ]) + ([X , (a X ) ]) 6 6 xxx 2 1 6 2 x xx − 2 6 2 xx xx 2 6 4 x xx +(a X ) (a X ) ([X , (a X ) ]) +(a [X , X ]) +(a X ) =0, (3.131) 1 2 xxx − 1 4 xx − 2 6 2 x xxx 6 2 7 x 4 2 x 80 O(u3) : 2([X , (a X ) ]) + 3[X , [X , [X , [X , (a X ) ]]]] 3[X , [X , [X , (a X ) ]]] − 2 6 6 x x 2 2 1 1 6 2 x − 2 2 1 6 2 xx +2(a X ) 3[X , [X , (a X ) ]]+3[X , [X , (a X ) ]]+3[X , [X , (a X ) ]] 3 2 x − 2 2 6 4 xx 2 2 6 5 x 2 1 1 2 x 3[X , [X , ([X , (a X ) ]) ]] 3a [X , [X , X ]]+3[X , [X , [X , [X , (a X ) ]]]] − 2 2 1 6 2 x x − 6 2 1 9 2 1 2 1 6 2 x 3a [X , [X , X ]]+3[X , [X , [X , (a X ) ]]] + 3[X , [X , [X , [X , (a X ) ]]]] − 6 2 1 8 2 2 1 6 4 x 2 1 1 2 6 2 x +3[X , [X , (a X ) ]] 3[X , [X , [X , (a X ) ]]] 3[X , [X , ([X , (a X ) ]) ]] 2 2 6 2 xxx − 2 1 2 6 2 xx − 2 1 2 6 2 x x 3a X + 3[X , [X , [X , (a X ) ]]] + 3[X , [X , (a X ) ]] 3[X , (a X ) ] − 1 8 2 1 2 6 4 x 2 1 6 6 x − 2 6 6 xx 3[X , ([X , [X , (a X ) ]]) ]+3[X , (a X ) ] 3[X , ([X , [X , (a X ) ]]) ] − 2 2 1 6 2 x x 2 6 9 x − 2 1 2 6 2 x x +3[X , ([X , (a X ) ]) ] 3[X , (a X ) ]+3[X , ([X , (a X ) ]) ]+3[X , (a X ) ] 2 2 6 2 xx x − 2 1 2 xx 2 2 6 2 x xx 2 1 4 x +3[X , (a X ) ] 3[X , ([X , (a X ) ]) ] 3a [X , [X , X ]] 2a [X , [X , X ]] 2 6 8 x − 2 2 6 4 x x − 6 2 2 7 − 6 1 2 9 +2[X , [X , (a X ) ]]+2[X , [X , [X , [X , (a X ) ]]]] 2[X , [X , ([X , (a X ) ]) ]] 1 2 1 2 x 1 2 2 1 6 2 x − 1 2 2 6 2 x x +2[X , [X , [X , [X , (a X ) ]]]] 2[X , [X , [X , (a X ) ]]] 2a [X , [X , X ]] 1 2 1 2 6 2 x − 1 2 2 6 2 xx − 6 1 2 8 +2[X , [X , (a X ) ]]+2[X , [X , [X , (a X ) ]]] 2a X 2([X , (a X ) ]) 1 2 6 6 x 1 2 2 6 4 x − 1 9 − 2 1 2 x x 2a X + 2(a [X , X ]) + 2(a [X , X ]) 2([X , [X , [X , (a X ) ]]]) − 3 4 6 2 9 x 6 2 8 x − 2 1 2 6 2 x x 2([X , [X , [X , (a X ) ]]]) + 2([X , [X , (a X ) ]]) + 2([X , ([X , (a X ) ]) ]) − 2 2 1 6 2 x x 2 2 6 2 xx x 2 2 6 2 x x x 2([X , [X , (a X ) ]]) + 2(a X ) =0, (3.132) − 2 2 6 4 x x 1 6 x

O(u4) : [X , [X , [X , [X , (a X ) ]]]] a [X , [X , X ]]+[X , [X , [X , [X , (a X ) ]]]] 2 2 2 1 6 2 x − 6 2 2 8 2 2 1 2 6 2 x a [X , [X , X ]] [X , [X , [X , (a X ) ]]] [X , [X , ([X , (a X ) ]) ]] − 6 2 2 9 − 2 2 2 6 2 xx − 2 2 2 6 2 x x

+[X2, [X2, (a6X6)x]]+[X2, [X2, [X2, (a6X4)x]]] = 0. (3.133)

81 Note that if we decouple equation (3.125) into the following three conditions

([X , (a X ) ]) (a X ) + a X [X , [X , (a X ) ]]=0, (3.134) 2 6 2 x x − 6 6 x 6 9 − 1 2 6 2 x [X , [X , (a X ) ]]+[X , (a X ) ] [X , (a X ) ] a X =0, (3.135) 2 1 6 2 x 2 6 4 x − 2 6 2 xx − 6 8 ((a 3a )X ) (a 3a )X =0, (3.136) 2 − 1 2 x − 2 − 1 4 then the O(u4) equation is identically satisﬁed. To reduce the complexity of the O(u3) equation we can decouple it into the following two equations

[X , [X , [X , (a X ) ]]] [X , [X , (a X ) ]] [X , (a X ) ]+[X , (a X ) ] 2 1 1 6 2 x − 2 1 6 2 xx − 2 6 4 xx 2 6 5 x +[X , (a X ) ] [X , ([X , (a X ) ]) ]+[X , [X , (a X ) ]]+[X , (a X ) ] 1 1 2 x − 2 1 6 2 x x 2 1 6 4 x 2 6 2 xxx a X (a X ) +(a X ) a [X , X ]=0, (3.137) − 1 5 − 1 2 xx 1 4 x − 6 2 7 (a X ) +[X , [X , (a X ) ]] a X ([X , (a X ) ]) a X +(a X ) =0(3.138). 3 2 x 1 2 1 2 x − 1 9 − 2 1 2 x x − 3 4 1 6 x

From this last condition, we can use equations (3.134)-(3.138) to reduce the O(u2) condition to the following equation

82 a X a [X , [X , X ]]+[X , [X , [X , [X , (a X ) ]]]] + [X , [X , [X , (a X ) ]]] − 5 8 − 6 2 1 7 2 1 1 1 6 2 x 2 1 1 6 4 x [X , [X , [X , (a X ) ]]]+[X , [X , (a X ) ]] [X , [X , ([X , (a X ) ]) ]] − 2 1 1 6 2 xx 2 1 5 2 x − 2 1 1 6 2 x x +[X , (a X ) ] [X , [X , (a X ) ]]+[X , [X , (a X ) ]] [X , ([X , (a X ) ]) ] 2 6 7 x − 2 1 6 4 xx 2 1 6 2 xxx − 2 1 6 4 x x [X , ([X , [X , (a X ) ]]) ]+[X , (a X ) ]+[X , ([X , (a X ) ]) ] [X , (a X ) ] − 2 1 1 6 2 x x 2 5 4 x 2 1 6 2 xx x − 2 5 2 xx +[X , (a X ) ]+[X , ([X , (a X ) ]) ] [X , (a X ) ]+[X , [X , (a X ) ]] 2 6 4 xxx 2 1 6 2 x xx − 2 6 2 xxxx 2 1 6 5 x 1 1 1 1 [X , (a X ) ] a X + [X , [X , (a X ) ]] ([X , (a X ) ]) + (a X ) − 2 6 5 xx − 2 5 9 2 1 2 5 2 x − 2 2 5 2 x x 2 5 6 x 1 1 a X + (a X ) =0. (3.139) −2 4 4 2 4 2 x

Decoupling this equation allows for the simpliﬁcation of the O(u) equation. Thus we write the previous condition as the following system of equations

[X , (a X ) ] [X , [X , (a X ) ]] + a X +[X , (a X ) ] (a X ) − 1 6 4 x − 1 1 6 2 x 5 4 1 6 2 xx − 5 2 x +(a X ) + ([X , (a X ) ]) (a X ) (a X ) =0, (3.140) 6 4 xx 1 6 2 x x − 6 2 xxx − 6 5 x 1 1 a X +[X , [X , (a X ) ]] ([X , (a X ) ]) +(a X ) a X + (a X ) =0(3.141). − 5 9 1 2 5 2 x − 2 5 2 x x 5 6 x − 2 4 4 2 4 2 x

Using the previous system the O(u) equation is reduced to the following, simpler equation

X +[X , X ] a X +(a X ) a X =0. (3.142) 2,t 2 0 − 8 4 8 2 x − 7 2

We therefore ﬁnd that the ﬁnal, reduced constraints are given by equations (3.126), (3.134)-(3.138)

83 and (3.140)-(3.142). In order to satisfy these constraints we begin with the following forms for our

generators,

g (x,t) g (x,t) 0 f (x,t) 0 f (x,t) X 1 2 X 1 X 3 0 =   , 1 =   , 2 =   . g3(x,t) g4(x,t) f2(x,t) 0 f4(x,t) 0             To get more general results we will assume a =3a . Note that had we instead opted for the forms 2 1

g (x,t) g (x,t) f (x,t) f (x,t) f (x,t) f (x,t) X 1 12 X 1 3 X 2 4 0 =   , 1 =   , 2 =   , g23(x,t) g34(x,t) f5(x,t) f7(x,t) f6(x,t) f8(x,t)             we would obtain an equivalent system to that obtained in [21]. The additional unknown functions which appear in Khawaja’s method [21] can be introduced with the proper substitutions via their functional dependence on the twelve unknown functions given above.

Taking the naive approach of beginning with the smaller conditions ﬁrst we begin with equation

(3.136) which, utilizing the given forms for X0, X1, and X2, becomes

(a 3a )(f f f f )=0, (3.143) 2 − 1 1 4 − 2 3 ((a 3a )f ) =0, j =3, 4. (3.144) 2 − 1 j x

Solving this system for f2, f3 and f4 in this previous system yields

84 F (t) f (x,t) = 3 , (3.145) 3 a (x,t) 3a (x,t) 2 − 1 F (t) f (x,t) = 4 , (3.146) 4 a (x,t) 3a (x,t) 2 − 1 f1(x,t)F4(t) f2(x,t) = , (3.147) F3(t) (3.148)

where F3,4(t) are arbitrary functions of t. With these choices we’ve elected to satisfy X4 =0 rather than a2 =3a1. This will greatly reduce the complexity of the remaining conditions while allowing for the possibility of a less trivial relation between a1 and a2. Looking next at equation (3.142) with X4 =0 we obtain the system

F F a F a 1 F a j j 7 + j 8 + j 4 a 3a − a 3a a 3a 2 a 3a 2 − 1 t 2 − 1 2 − 1 x 2 − 1 x F (g g ) +( 1)j j 4 − 1 =0 j =3, 4, (3.149) − a2 3a1 F g −F g 3 3 4 2 =0. (3.150) a 3a − a 3a 2 − 1 2 − 1

Solving the second equation for g3 yields

F4(t)g2(x,t) g3 = F3(t)

Considering next the O(1) equation, we obtain the following system of equations

85 g1x = g4x =0, (3.151)

f g + f (g g )=0, (3.152) 1t − 2x 1 4 − 1 F (F f ) f F F g F F + F F f (g g )=0. (3.153) 3 4 1 t − 1 4 3t − 2x 4 3 3 4 1 1 − 4

It follows that we must have g1(x,t)= G1(t) and g4(x,t)= G4(t) where G1 and G4 are arbitrary functions of t. Since equations (3.152) and (3.153) do not depend on the ai they will not affect

the conditions on the ai required for Lax-integrability of (2.66). On the other hand the remaining

conditions have been reduced to conditions involving soley the ai and the previously introduced arbitrary functions of t. The remaining conditions are given by

a a 5 + 6 =0, (3.154) a 3a a 3a 2 − 1 x 2 − 1 xxx a 3 =0, (3.155) a 3a 2 − 1 x a 1 =0, (3.156) a 3a 2 − 1 xx a 4 =0. (3.157) a 3a 2 − 1 x

One can easily solve the system of equations given by equations (3.149), (3.154)-(3.157) yielding

86 2 R (G4 G1)dt F4 = c1F3e − (3.158)

g = (f + f (G G ))dx + F (3.159) 2 1t 1 4 − 1 10 (3F 1 3F x)a a = 1 − − 2 1 (3.160) 2 − F x F 2 − 1 F5a1 a3 = (3.161) F2x F1 F −a a = 6 1 (3.162) 4 F x F 2 − 1 (F + F x + F x2)a x y a (z,t) dz dy a = 7 8 9 1 5 (3.163) 6 F x F − a (z,t) 3a (z,t) 2 − 1 2 − 1 a 3a F a a = 2 − 1 3 +(a 3a ) 8 + G G (3.164) 7 F a 3a 2 − 1 a 3a 4 − 1 3 2 − 1 t 2 − 1 x

where F5 10 are arbitrary functions of t. Note that a1,a5 and a8 have no restrictions beyond the − appropriate differentiability and integrability conditions. The Lax pair for the generalized variable- coefﬁcient KdV equation with the previous integrability conditions on the variable coefﬁcients is therefore given by

F = X1 + X2u, (3.165)

G = a X u +(a X ) u X (a u + a )u (a X ) u a X u − 6 2 xxxx 6 2 x xxx − 2 1 5 xx − 6 2 xx xx − 8 2 1 1 1 1 + a X u2 a X u2 +(a X ) uu a X u3 a X u2 + X . (3.166) 2 1 2 x − 2 2 2 x 1 2 x x − 3 3 2 − 2 4 2 0

87 Modiﬁed Korteweg-de-Vries Equation

In this section we derive the Lax pair and differential constraints for the ﬁrst equation in the variable-coefﬁcient mKdV hierarchy. Following the procedure outlined at the beginning of the

chapter we let

F = F(x,t,u) and G = G(x,t,u,ux,uxx).

Plugging F and G into equation (3.2) we obtain

F + F v G G v G v G v +[F, G]=0. (3.167) t v t − x − v x − vx xx − vxx xxx

Using equation (2.74) to substitute for vt in the equation given above we have

F G (G + b F v2)v G v (G + b F )v +[F, G]=0. (3.168) t − x − v 2 v x − vx xx − vxx 1 v xxx

Since F and G do not depend on vxxx we must set the coefﬁcient of the vxxx term to zero from which we ﬁnd that F and G must satisfy

G + b F =0 G = b F v + K0(x,t,v,v ). vxx 1 v ⇒ − 1 v xx x

88 Substituting this expression for G into equation (3.168) we obtain the updated equation

F +(b F ) v K0 + b F v v K0v b F v2v K0 v b [F, F ]v +[F, K0]=0. t 1 v x xx − x 1 vv x xx − v x − 2 v x − vx xx − 1 v xx (3.169)

0 Since F and K do not depend on vxx we can equate the coefﬁcient of the vxx term to zero which is equivalent to the requirement

(b F ) + b F v K0 b [F, F ]=0. (3.170) 1 v x 1 vv x − vx − 1 v

0 Integrating with respect to vx and solving for K we have

1 K0 =(b F ) v + b F v2 b [F, F ]v + K1(x,t,v). 1 v x x 2 1 vv x − 1 v x

Substituting this expression for K0 into equation (3.169) we have the following updated equation

1 1 F (b F ) v (b F ) v2 +(b [F, F ]) v K1 (b F ) v2 b F v3 t − 1 v xx x − 2 1 vv x x 1 v x x − x − 1 vv x x − 2 1 vvv x 1 K1v + b [F, F ]v2 b F v2v +[F, (b F ) ]v + b [F, F ]v2 b [F, [F, F ]]v − v x 1 vv x − 2 v x 1 v x x 2 1 vv x − 1 v x +[F, K1]=0. (3.171)

F K1 2 3 Since and do not depend on vx we can equate the coefﬁcients of the vx, vx and vx terms to zero from which we obtain the following system,

89 3 F O(vx) : vvv =0, (3.172) 1 1 O(v2) : (b F ) +(b F ) b [F, F ] b [F, F ]=0, (3.173) x 2 1 vv x 1 vv x − 1 vv − 2 1 vv O(v ) : (b F ) (b [F, F ]) + K1 + b F v2 [F, (b F ) ]+ b [F, [F, F ]]=0(3.174). x 1 v xx − 1 v x v 2 v − 1 v x 1 v

Since the MKdV equation does not contain a vvt term and for ease of computation we take Fvv =0

from which we have F = X1(x,t)+ X2(x,t)v where X1(x,t) and X2(x,t) are matrices whose elements are functions of x and t and do not depend on v or its partial derivatives. With this

F 3 2 3 requirement on the O(vx) and O(vx) equations are immediately satisﬁed. Integrating the O(vx) equations with respect to v and solving for K1 we ﬁnd

1 K1 = (b X ) v +(b [X , X ]) v +[X , (b X ) ]v + [X , (b X ) ]v2 b [X , [X , X ]]v − 1 2 xx 1 1 2 x 1 1 2 x 2 2 1 2 x − 1 1 1 2 1 1 b X v3 b [X , [X , X ]]v2 + X (x,t), (3.175) −3 2 2 − 2 1 2 1 2 0

where X0 is a matrix whose elements are functions of x and t and does not depend on v or its partial derivatives. Substituting the expression for K1 into equation (3.171) we obtain

90 1 1 X +(b X ) v (b [X , X ]) v + (b X ) v3 ([X , (b X ) ]) v ([X , (b X ) ]) v2 1,t 1 2 xxx − 1 1 2 xx 3 2 2 x − 1 1 2 x x − 2 2 1 2 x x 1 +(b [X , [X , X ]]) v + (b [X , [X , X ]]) v2 X [X , (b X ) ]v +[X , (b [X , X ]) ]v 1 1 1 2 x 2 1 2 1 2 x − 0,x − 1 1 2 xx 1 1 1 2 x 1 1 +X v b [X , X ]v3 +[X , [X , (b X ) ]]v + [X , [X , (b X ) ]]v2 b [X , [X , [X , X ]]]v 2,t − 3 2 1 2 1 1 1 2 x 2 1 2 1 2 x − 1 1 1 1 2 1 b [X , [X , [X , X ]]]v2 [X , (b X ) ]v2 +[X , (b [X , X ]) ]v2 +[X , [X , (b X ) ]]v2 −2 1 1 2 1 2 − 2 1 2 xx 2 1 1 2 x 2 1 1 2 x 1 1 +[X , X ]+ [X , [X , (b X ) ]]v3 b [X , [X , [X , X ]]]v2 b [X , [X , [X , X ]]]v3 1 0 2 2 2 1 2 x − 1 2 1 1 2 − 2 1 2 2 1 2

+[X2, X0]v =0 (3.176)

Since the Xi do not depend on v we can equate the coefﬁcients of the different powers of v to zero. In doing so we obtain the following system of equations

O(1) : X X +[X , X ]=0, (3.177) 1,t − 0,x 1 0 O(v) : X ([X , (b X ) ]) +(b [X , [X , X ]]) [X , (b X ) ]+[X , (b [X , X ]) ] 2,t − 1 1 2 x x 1 1 1 2 x − 1 1 2 xx 1 1 1 2 x (b [X , X ]) +(b X ) +[X , [X , (b X ) ]] b [X , [X , [X , X ]]] − 1 1 2 xx 1 2 xxx 1 1 1 2 x − 1 1 1 1 2

+[X2, X0]=0, (3.178) 1 1 1 O(v2) : ([X , (b X ) ]) + (b [X , [X , X ]]) + [X , [X , (b X ) ]] [X , (b X ) ] −2 2 1 2 x x 2 1 2 1 2 x 2 1 2 1 2 x − 2 1 2 xx 1 b [X , [X , [X , X ]]]+[X , (b [X , X ]) ] b [X , [X , [X , X ]]] −2 1 1 2 1 2 2 1 1 2 x − 1 2 1 1 2

+[X2, [X1, (b1X2)x]]=0, (3.179) 1 1 1 1 O(v3) : (b X ) b [X , X ]+ [X , [X , (b X ) ]] b [X , [X , [X , X ]]] = 0.(3.180) 3 2 2 x − 3 2 1 2 2 2 2 1 2 x − 2 1 2 2 1 2

Note that if we decouple the O(v3) equation into the following system of equations

91 b [X , X ] (b X ) =0, (3.181) 1 1 2 − 1 2 x b [X , X ] (b X ) =0, (3.182) 2 1 2 − 2 2 x we ﬁnd that the O(v2) equation is immediately satisﬁed and the O(v) equation reduces to

[X2, X0]+ X2,t =0. (3.183)

Again at this point we should note that should we opt for the forms

g (x,t) g (x,t) f (x,t) f (x,t) f (x,t) f (x,t) X 1 4 X 1 3 X 2 4 0 =   , 1 =   , 2 =   g10(x,t) g16(x,t) f5(x,t) f7(x,t) f6(x,t) f8(x,t)             we would obtain an equivalent system of equations to that obtained in [21] for the mKdV. The additional unknown functions which appear in Khawaja’s method [21] can be introduced (as in the case of the ﬁfth-order KdV equation) with the proper substitutions via their functional dependence on the twelve unknown functions given above. Therefore, utilizing the same generators as in the generalized KdV equation of the previous section we obtain the system of equations

92 f f f f =0, (3.184) 1 4 − 2 3

(b1fj)x =(b2fj)x =0 j =3, 4, (3.185)

f g f g =0, (3.186) 3 3 − 4 2 f +( 1)jf (g g )=0 j =3, 4, (3.187) jt − j 1 − 4 g +( 1)j(f g f g )=0 j =1, 4, (3.188) jx − 1 3 − 2 2 f g +( 1)jf (g g )=0 j =2, 3. (3.189) jt − (j+1)x − j 4 − 1

Solving this system with the aid of MAPLE yields

Fj(t) fj(x,t) = j =3, 4, (3.190) b1(x,t) F4(t)g2(x,t) g3(x,t) = , (3.191) F3(t) F4(t)f1(x,t) f2(x,t) = , (3.192) F3(t)

g1(x,t) = G1(t), (3.193)

g4(x,t) = G4(t), (3.194)

such that F3, F4, G1, G4, and the bi are subject to the constraints

93 F j +( 1)j(G G )=0 j =3, 4, (3.195) b − 1 − 4 1 t b F 2 j =0 j =3, 4, (3.196) b 1 x f g +( 1)jf (G G )=0 j =2, 3. (3.197) jt − (j+1)x − j 4 − 1

Solving (3.195) and (3.196) for F4, b1, b2 and G4 we obtain

2 c1F2(t) F4(t) = , (3.198) F3(t)

b1(x,t) = F1(x)F2(t), (3.199)

b2(x,t) = F1(x)F5(t), (3.200)

F3(t)F2′(t) F2(t)F3′(t)+ G1(t)F2(t)F3(t) G4(t) = − , (3.201) F2(t)F3(t) [f (x,t)] F (t)F (t) f (x,t)F ′(t)F (t)+ f (x,t)F (t)F ′(t) g (x,t) = 1 t 3 2 − 1 3 2 1 3 2 dx + F (3.202)(t), 2 F (t)F (t) 6 3 2

where F1 and F2 are arbitrary functions in their respective variables and c1 is an arbitrary constant.

The Lax pair for the variable-coefﬁcient MKdV equation with the previous integrability conditions is then given by

F = X1 + X2v (3.203) 1 G = b X v b X v3 + X (3.204) − 1 2 xx − 3 2 2 0

94 Cubic-Quintic Nonlinear Schrodinger¨ Equation

In this section we consider an extended version of a standard non-integrable nlpde, namely the cubic-quintic nonlinear Schrodinger¨ equation (CQNLS). Since the constant-coefﬁcient systems can be obtained as reductions of the extended systems in which the coefﬁcient functions are taken to be the appropriate constants, one would expect this method to breakdown in the case of the

CQNLS. As we will show in this section, the extended Estabrook-Wahlquist method indeed breaks down for the CQNLS by requiring that the quintic term be removed or that the Lax pair be trivial (i.e. F and G are both the zero matrix). Consider the following variable-coefﬁcient generalization to the nonintegrable CQNLS,

iψ + fψ + hψ + g ψ 2ψ + g ψ 4ψ =0, (3.205) t xx 1| | 2| | where f,h,g , and g are real functions of x and t. It is imperative that the condition g =0 hold. 1 2 2 Otherwise equation (3.205) reduces to the well-known cubic-NLS for which the results are given in the PT-symmetric and standard nonlinear Schrodinger¨ section. As with the cubic NLS, it will be notationally cleaner to work with the following equivalent system,

2 3 2 iqt + fqxx + hq + g1q r + g2q r =0, (3.206a)

ir + fr + hr + g r2q + g r3q2 =0. (3.206b) − t xx 1 2

Following the procedure outlined earlier in the chapter we make an initial assumption only on the implicit dependence of the Lax pair on the unknown function and its derivatives augmented by al-

95 lowing dependence on x and t as well. That is, we take F = F(q,r,x,t) and G = G(q,r,qx,rx,x,t). Plugging these into the zero curvature condition yields

F q + F r + F G q G r G q G r G +[F, G]=0. (3.207) q t r t t − q x − r x − qx xx − rx xx − x

Utilizing equations (3.206) to eliminate the qt and rt terms in the equation above we obtain

ifF q + ihF q + ig F q2r + iF q3r2 ifF r ihF r ig F r2q ig F r3q2 q xx q 1 q q − r xx − r − 1 r − 2 r +F G q G r G q G r G +[F, G]=0. (3.208) t − q x − r x − qx xx − rx xx − x

Since F and G do not depend on qxx or rxx the coefﬁcients of the qxx and rxx must be zero. This is equivalent to the conditions

ifF G =0, (3.209a) q − qx F G if r + rx =0. (3.209b)

Solving this system for G in a method analogous to that discussed in the case of the NLS we ﬁnd

G = ifF q ifF r + K0(q,r,x,t). Plugging this expression for G into equation (3.208) we q x − r x obtain the updated zero-curvature condition

96 ihF q + ig F q2r + ig F q3r2 ihF r ig F r2q ig F r3q2 + F i(fF ) q q 1 q 2 q − r − 1 r − 2 r t − q x x ifF q2 + ifF r2 + i(fF ) r + if[F, F ]q if[F, F ]r K0q K0r − qq x rr x r x x q x − r x − q x − r x K0 +[F, K0]=0. (3.210) − x

0 Since F and K do not depend on qx or rx the coefﬁcients of the different powers of qx and rx must be zero. This is equivalent to the system

Fqq = Frr =0, (3.211a)

K0 + i(fF ) if[F, F ]=0, (3.211b) q q x − q K0 i(fF ) + if[F, F ]=0. (3.211c) r − q x r

From the former two conditions it is clear that F must be of the form F = X1 + X2q + X3r + X4qr where the Xi are matrices whose elements are functions of x and t and do not depend on q, r, or their partial derivatives. The inclusion of the X4qr term will not lead to any speciﬁc terms present in equation (3.206) and thus we will take X4 to be zero in order to satisfy the consistency conditions

(fX ) f[X , X ]=0, [X , X ]=[X , X ]=0, (3.212) 4 x − 1 4 2 4 3 4 which arise in the process of recovering K0 from equations (3.211b) and (3.211c). Using the derived explicit form for F we ﬁnd that K0 takes the form

97 K0 = i(fX ) q + if[X , X ]q + if[X , X ]rq + i(fX ) r if[X , X ]r (3.213) − 2 x 1 2 3 2 2 x − 1 3

+X0, (3.214)

where X0 is a matrix whose elements are functions of x and t and does not depend on q, r, or their

0 partial derivatives. Using X4 = 0 and the expressions for F and K to update the zero-curvature condition (3.210) we obtain a rather long expression which is nothing more than a polynomial in q and r. In order for the zero-curvature condition to be satisﬁed upon solutions to equations

(3.206) we must require that the coefﬁcients of the different powers of q and r be zero in much the same fashion as we have done previously. Decoupling this large expression results in the following system,

98 O(1) : X X +[X , X ]=0, (3.215) 1,t − 0,x 1 0 O(q) : X + ihX + i(fX ) i(f[X , X ]) i[X , (fX ) ]+ if[X , [X , X ]] 2,t 2 2 xx − 1 2 x − 1 2 x 1 1 2

+[X2, X0]=0, (3.216)

O(r) : X ihX i(fX ) + i(f[X , X ]) + i[X , (fX ) ] if[X , [X , X ]] 3,t − 3 − 2 xx 1 3 x 1 2 x − 1 1 2

+[X3, X0]=0, (3.217)

O(rq) : i(f[X , X ]) + if[X , [X , X ]] + if[X , X ] if[X , [X , X ]] − 3 2 x 1 3 2 2 2,x − 2 1 3 i[X , (fX ) ]+ if[X , [X , X ]]=0, (3.218) − 3 2 x 3 1 2 O(r2) : [X , (fX ) ] f[X , [X , X ]]=0, (3.219) 3 2 x − 3 1 3 O(q2) : [X , X ] [X , [X , X ]]=0, (3.220) 2 2,x − 2 1 2 O(r2q) : f[X , [X , X ]] g X =0, (3.221) 3 3 2 − 1 3 2 O(q r) : f[X2, [X3, X2]] + g1X2 =0, (3.222)

3 2 O(q r ) : X2g2 =0, (3.223)

3 2 O(r q ) : X3g2 =0. (3.224)

At this point from the ﬁnal two conditions it is clear that we must either have X2 = X3 = 0 or g = 0. As we required g = 0 this means we must take X = X = 0. But this completely 2 2 2 3 removes the q and r dependence of both F and G rendering the Lax pair trivial. Therefore no nontrivial Lax pair exists to equation (3.205).

99 PART II: SOLUTIONS OF CONSTANT COEFFICIENT INTEGRABLE

SYSTEMS

100 CHAPTER 4: EXACT SOLUTIONS OF NONLINEAR PDES

Introduction

In this chapter we employ three types of singular manifold methods, namely the truncated Painleve,´ invariant truncated Painleve,´ and generalized Hirota expansions to derive exact traveling and non- traveling wave solutions to various PDEs which occur in mathematical physics. Truncated Painleve,´ invariant truncated Painleve,´ and the generalized Hirota expansion methods have been extensively used over the past 30 years to derive solutions to a wide variety of nlpdes. In this chapter we will give a brief introduction to each method and subsequently demonstrate each method on a classic example for which the results are well known. Among the nlpdes considered will be the KdV equation, Kadomtsev-Petviashvili II (KP-II) equation, a microstructure PDE which arises within the context of one-dimensional wave propagation in microstructured solids [34] - [38]

v bv (v2) δ(βv γv ) =0 (4.1) tt − xx − 2 xx − tt − xx xx

where b, , δ, β, and γ are dimensionless parameters and v denotes the macro-deformation, and two versions

(u u ) (a u + a u2 + a u3) =0 (4.2) − xx tt − 1 2 3 xx and

(u u ) (a u + a u3 + a u5) =0 (4.3) − xx tt − 1 3 5 xx

101 of a Pochhammer-Chree equation which describe the propagation of longitudinal deformation

waves [39] - [44] in elastic rods. While the results for the KdV and KP equations are known, the solutions derived by the aforementioned methods for equations (4.1), (4.2), and (4.3) are new.

Singular Manifold Methods

In this section we will give a brief description of the truncated Painleve,´ invariant Painleve,´ and generalized Hirota expansion methods which will be subsequently utilized to derive exact solutions. Each of these methods stem from a broader class of solution methods known as singular manifold methods and thus will begin quite similarly. Given a nonlinear PDE we seek a Laurent series solution centered about a movable singular manifold φ(x,t)=0. That is, for a nonlinear PDE

∂i+lu N u, =0, l + l = l (4.4) l1 1 n ∂ti∂x ∂xln 1 n in (n + 1), we seek solution of the form

α α n u(x,t)= φ(x,t)− un(x,t)φ(x,t) (4.5) n=0 where the un are functions to be determined and α is the singularity degree. This truncated expansion of course only makes sense for α N thus implying that the function u(x,t) is single- ∈ valued about the movable, singularity manifolds. This condition was given by Weiss, Tabor, and Carnevale [45] as an extension of the well-known Painleve´ property from ODEs to PDEs. How-

ever, this may prove to be too strict a condition while endeavoring to obtain exact solutions. To

102 circumvent this issue and thus extend the applicability of the truncated singular manifold method

for the cases when α / N one may deﬁne a substitution v(x,t) = F (u(x,t)) for some suitably ∈ well-deﬁned (invertible) function F ( ) such that for the transformed equation

∂i+lu N˜ u, =0, l + l = l (4.6) l1 1 n ∂ti∂x ∂xln 1 n we may seek a solution of the form

β β n v(x,t)= φ(x,t)− vn(x,t)φ(x,t) (4.7) n=0

where the v are functions to be determined and β N is the singularity degree. The solution to n ∈ 1 the original nonlinear PDE is then given by u(x,t)= F − (v(x,t)).

Plugging the truncated series expansion for u(x,t) into the nonlinear PDE and reconciling the powers of the unknown function φ will yield a recurrence relation from which we will determine the un and α.

Truncated Painleve´ Analysis Method

In this section we will brieﬂy outline the truncated Painleve´ analysis method for nonlinear PDEs as it was introduced by Weiss, Tabor, and Carnevale [45] in 1983. Following their procedure and continuing from where we left off in the previous section, we may reduce the order and complexity of the recurrence relations with the introduction of the additional functions

103 φt C0(x0,...,xn,t) = (4.8) φx0

φx1 C1(x0,...,xn,t) = (4.9) φx0 . . (4.10)

φxn Cn(x0,...,xn,t) = (4.11) φx0

φx0x0 V (x0,...,xn,t) = (4.12) φx0

It is clear that after making use of these additonal functions we will have eliminated all partial

derivatives of φ excepting φx. For simplicity it is common to allow the Ci(x,t) and V (x,t) to be constants, thereby reducing a system of PDEs (more than likely nonlinear) in C (x,t),V (x,t) to { i } an algebraic system in C ,V for (i = 0,...,n). It is straightforward to see that this simpliﬁca- { i } tion is equivalent assuming a traveling-wave form for φ.

Once we have determined the unknown coefﬁcient functions un in the truncated series solution as well as the unknown Ci(x,t) and V (x,t) we will have the required information to recover the singularity manifold φ. Upon determining φ we may plug it into the truncated expansion, thus obtaining an exact solution to the original PDE.

Example: KP-II Equation

Before presenting the main results for the microstructure PDE and Pochhammer-Chree equations we will demonstrate the effectiveness of the truncated Painleve´ analysis method on a classic example in (2+1), the KP-II equation. The KP-II equation [51]- [54], a two-dimensional generalization of the well-known KdV equation describing weakly transverse water waves in a long-wave regime

104 with small surface tension is given by

2 (ut + uux + ǫ uxxx)x + λuyy =0. (4.13)

α Plugging φ− into (4.13) and balancing the (uux)x and uxxxx terms we ﬁnd that the degree of the singularity, α, is equal to 2. Therefore we seek a solution of the form

u (x,y,t) u (x,y,t) u(x,y,t)= 0 + 1 + u (x,y,t) (4.14) φ(x,y,t)2 φ(x,y,t) 2 for equation (4.13). Plugging the truncated expansion (4.14) into (4.13) and resolving the powers of φ yields the recurrence relation

un 4,xt +(n 5)un 3,xφt +(n 5)un 3φxt +(n 4)(n 5)un 2φxφt − − − − − − − − 2 +(n 5)un 3,tφx + ǫ (un 4,xxxx + 4(n 5)un 3,xxxφx + 6(n 5)un 3,xxφxx − − − − − − − 2 + 6(n 4)(n 5)un 2,xxφx + 4(n 5)un 3,xφxxx + 12(n 4)(n 5)un 2,xφxφxx − − − − − − − − 3 + 4(n 3)(n 4)(n 5)un 1,xφx +(n 5)un 3φxxxx + 4(n 4)(n 5)un 2φxφxxx − − − − − − − − − 2 2 + 3(n 4)(n 5)un 2φxx + 6(n 3)(n 4)(n 5)un 1φxφxx − − − − − − − 4 +(n 2)(n 3)(n 4)(n 5)unφx)+ λ(un 4,yy + 2(n 5)un 3,yφy +(n 5)un 3φyy − − − − − − −n − − 2 +(n 4)(n 5)un 2φy)+ (uk,xun k 2,x +(n k 3)uk,xun k 1φx +(k 2)ukun k 1,xφx − − − − − − − − − − − − k=0 2 +(k 2)(n k 2)ukun kφx + uk(un k 2,xx + 2(n k 3)un k 1,xφx − − − − − − − − − − 2 +(n k 3)un k 1φxx +(n k 2)(n k 3)un kφx))=0 (4.15) − − − − − − − − −

105 where u = 0 if n < 0 and u ∂ (u ). Upon satisfaction of the recurrence relation at the n n,x ≡ ∂x n lowest powers of φ we ﬁnd

n =0 : u = 12ǫ2φ2 (4.16) 0 − x 2 n =1 : u1 = 12ǫ φxx (4.17)

2 2 2 2 2 n =2 : u = φ− (3ǫ φ φ φ 4ǫ φ φ λφ ) (4.18) 2 x xx − x t − x xxx − y which we may write in the cleaner form

u(x,y,t)= 2[log(φ)] + u (x,y,t) (4.19) − xx 2

For simplicity we allow C(x,y,t) (our C0(x,y,t) function), V (x,y,t) and Q(x,y,t) (our C1(x,y,t) function) to be the constants C, V , and Q, respectively. Upon making this simpliﬁcation we easily reconcile the remainder of the coefﬁcients of φ thus obtaining the solution

u (x,y,t) = 12ǫ2φ2 (4.20) 0 − x 2 u1(x,y,t) = 12ǫ Vφx (4.21)

u (x,y,t) = 3ǫ2V 2 C 4ǫ2V 2 λQ2 (4.22) 2 − − − V (x+Qy+Ct) φ(x,y,t) = c1 + c2e (4.23)

106 where c1 and c2 are arbitrary constants. Our solution can be written in the more compact form

u(x,y,t)= 12ǫ2 [log(φ(x,y,t))] +3ǫ2V 2 C 4ǫ2V 2 λQ2 (4.24) − xx − − −

From this form of the solution it follows that a nontrivial (in this context non-constant) solution

requires c =0. 1

We will now plot the solution derived above for two differents choices for the set of parameters (see

ﬁgure 4.1). For the ﬁrst choice we take (ǫ,λ,V,Q,C,c ,c ) = (1, 1, 1, 1,i, 1, 1). The solution 1 2 − (4.24) at t =1 then becomes

12e x y i − − − (4.25) u(x,y, 1) = x y i 2 2 i. −(1 + e− − − ) − −

It is clear that for this set of parameters the solution is complex-valued. Therefore to better visualize the solution we plot the real and imaginary parts separately.

107 Figure 4.1: Plots of the real (left) and imaginary (right) parts of the solution to the KP-II equation at t =1 on the rectangle [ 3π, 3π] [ 3π, 3π] with the choice of parameters (ǫ,λ,V,Q,C,c1,c2)= (1, 1, 1, 1,i, 1, 1). − × − −

As a second choice for the set of parameters we will take (ǫ,λ,V,Q,C,c ,c )=(1, 1, 1, 1, 2, 1, 1) 1 2 − − − for which the solution (4.24) at t =1 becomes

12e x+y+2 − sech2 (4.26) u(x,y, 1) = x+y+2 2 = 6 ((1/2)( x + y + 2)) . −(1 + e− ) − −

The plot for this solution is given in ﬁgure 4.2.

108 Figure 4.2: Plot of the solution to the KP-II equation at t =1 on the rectangle [ 10, 10] [ 10, 10] with the choice of parameters (ǫ,λ,V,Q,C,c ,c )=(1, 1, 1, 1, 2, 1, 1). − × − 1 2 − − −

Microstructure PDE

In this section we use the truncated Painleve´ analysis method to construct an exact solution to the

α microstructure PDE given earlier in the section by equation (4.1). Plugging φ− into (4.1) and

2 balancing the (v )xx and vxxxx terms we ﬁnd that the degree of the singularity, α, is equal to 2. In fact, this amounted to balancing the same terms that we did for the KP-II equation so we could

have borrowed the results from the leading order analysis in the previous section. Since α =2, we seek a solution of the form

u (x,t) u (x,t) u(x,t)= 0 + 1 + u (x,t) (4.27) φ(x,t)2 φ(x,t) 2

109 for equation (4.1). Plugging the truncated expansion (4.27) into (4.1) and resolving the powers of

φ yields the large recurrence relation

um 4,tt bum 4,xx +(m 5)(2um 3,tφt 2bum 3,xφx + um 3(φtt bφxx) − − − − − − − m − − 2 2 +(m 4)um 2(φt bφx)) uk,xum k 2,x + ukum k 2,xx − − − − { − − − − k=0 2 +(m k 3)(uk,xum k 1φx +2ukum k 1,xφx +(m k 2)ukum kφx − − − − − − − − − 2 +ukum k 1φxx)+(k 2)(ukum k 1,xφx +(m k 2)ukum kφx) δβ(um 4,xxtt − − − − − − − − − − +(m 5)(2um 3,xxtφt + um 3,xxφtt +2um 3,xttφx +4um 3,xtφxt +2um 3,xφxtt − − − − − − 2 2 +um 3,ttφxx +2um 3,tφxxt + um 3φxxtt +(m 4)(um 2,xxφt + um 2,ttφx − − − − − − 2 +4um 2,xtφxφt +4um 2,xφtφxt +2um 2,xφxφtt +4um 2,tφxφxt +2um 2φxt − − − − −

+2um 2φxφxtt +2um 2,tφtφxx + um 2φtφxxt + um 2φxxφtt + um 2φxxtφt − − − − − 2 2 2 2 +(m 3)(2um 1,xφxφt +2um 1,tφxφt +4um 1φxφtφxt + um 1φxφtt + um 1φxxφt − − − − − − 2 2 +(m 2)umφxφt )))) + δγ(um 4,xxxx +(m 5)(um 3φxxxx +4um 3,xφxxx − − − − − 2 +6um 3,xxφxx +4um 3,xxxφx +(m 4)(um 2(4φxφxxx +3φxx)+12um 2,xφxφxx − − − − − 2 3 2 4 +6um 2,xxφx +(m 3)(4um 1,xφx +6um 1φxφxx +(m 2)umφx)))) = 0 (4.28) − − − − −

where once again u =0 if n< 0 and u ∂ (u ). Upon satisfaction of the recurrence relation n n,x ≡ ∂x n at the lowest powers of φ we ﬁnd

110 12δ m =0 : u = (γφ2 βφ2) (4.29) 0 − t − x 12δ m =1 : u = 5γ2φ2φ +5β2φ2φ βγ(φ2φ + φ2φ ) 8βγφ φ φ (4.30) 1 5(γφ2 βφ2) x xx t tt − t xx x tt − t x xt x − t 1 m =2 : u = (25β3φ φ8 25γ3φ7φ2 + 25γ3bφ9 25β3bφ3φ9 75β2γφ3φ6 2 25φ3 (βφ2 γφ2)3 x t − x t x − x t − x t x t − x + 75βγ2φ5φ4 + 171δγ4φ7φ2 120δγ4φ8φ + 75β2γbφ5φ4 75βγ2bφ2φ2 x t x xx − x xxx x t − x t 120δβ3γφ2φ φ φ5 + 292δβ2γ2φ4φ φ φ3 856δβγ3φ6φ φ φ3 128δβ3γφ4φ φ3φ − x xx xt t x xx xt t − x xx xt t − x xt t tt 10δβ3γφ3φ φ4φ + 340δβ2γ2φ5φ φ2φ 356δβ2γ2φ6φ φ φ + 160δβ3γφ5φ3φ − x xx t tt x xx t tt − x xt t tt x t ttt 60δβ2γ2φ7φ φ 476δβ3γφ3φ2 φ4 + 584δβ2γ2φ5φ2 φ2 + 215δβ2γ2φ3φ2 φ4 − x t ttt − x xt t x xt t x xx t 110δβγ3φ5φ2 φ2 100δβ3γφ φ2 φ6 100δβ3γφ2φ φ6 60δβ3γφ3φ φ5 − x xx t − x xx t − x xxx t − x xxt t + 120δβ2γ2φ4φ φ4 360δβ2γ2φ5φ φ3 + 100δβγ3φ6φ φ2 + 300δβγ3φ7φ φ x xxx t − x xxt t x xxx t x xxt t 220δβ4φ2φ φ5φ + 22δβγ3φ7φ φ + 480δβ3γφ4φ φ4 420δβ2γ2φ6φ φ2 − x xt t tt x xx tt x xtt t − x xtt t + 150δβ3γφ5φ2φ2 100δβ4φ3φ5φ + 336δβ4φ φ2 φ6 + 120δβ4φ φ φ7 + 260δβγ3φ7φ2 x t tt − x t ttt x xt t x xxt t x xt 20δβ4φ φ φ7 140δβ4φ2φ φ6 + 80δβγ3φ8φ + 75δβ4φ3φ4φ2 49δβ2γ2φ7φ2 (4.31)) − xx xt t − x xtt t x xtt x t tt − x tt

Once again we will ﬁnd that without the assumption that C(x,t) and V (x,t) be constants the calculations will become unnecessarily complicated. Under this reduction we ﬁnd that elimina-

tion of the remaining coefﬁcients of φ requires that C = 1 while V can remain arbitrary. Using C(x,t)=1 and V (x,t)= V (constant) the previous equations reduce to

111 12δ(γ β)φ2 u (x,t) = − x (4.32) 0 12δ(γ β)Vφ u (x,t) = − x (4.33) 1 − V 2δ(γ β)+1 b u (x,t) = − − (4.34) 2 and we obtain the solution for the singularity manifold

V (x+t) φ(x,t)= c1 + c2e (4.35)

where c1 and c2 are arbitrary constants. As with the KP-II equation we may write the the equation in a more convenient form

12δ(β γ) V 2δ(γ β)+1 b u(x,t)= − (log(φ(x,t))) + − − (4.36) xx

From this form of the solution we make the observation that a nontrivial (once again taken to mean non-constant) solution will require that c =0 but also δ =0 and β = γ. 1

We will now plot the solution derived above for the set of parameters (c1,c2,V,γ,,δ,β,b) = (1, 1, 1, 1, 1, 1, 1, 1). The solution (4.36) then becomes − −

(e 2x 2t 10e x t + 1) − − − − (4.37) u(x,t)=2 − x t 2 . (1 + e− − )

The plot for this solution is given in ﬁgure 4.3.

112 Figure 4.3: Plot of the solution (4.37) for t = 10, 0, 10 on the interval [ 20, 20] with the choice of parameters (c ,c ,V,γ,,δ,β,b)=(1, 1, −1, 1, 1, 1, 1, 1). − 1 2 − −

Generalized Pochhammer-Chree Equations

In this section we use the truncated Painleve´ analysis method to construct an exact solution to the generalized Pochhammer-Chree equations given earlier in the section by equations (4.2) and (4.3).

α 3 Plugging φ− into (4.2) and balancing the (u )xx and uxxtt terms we ﬁnd that the degree of the singularity, α, is equal to 1. Following the procedure outlined earlier in the chapter using α = 1 and thus we seek a solution of the form

u (x,t) u(x,t)= 0 + u (x,t). (4.38) φ(x,t) 1 for equation (4.2). Now consider the following generalized higher-order Pochhammer-Chree equa-

113 tion

n (u u ) a u2i+1 =0, (4.39) − xx tt − 2i+1 i=0 xx

α 2n+1 where a = 0. Plugging φ− into (4.39) and balancing the (u ) and u terms leads to 2n+1 xx xxtt α = 1 . For n 2 (e.g. polynomial nonlinearity of degree 5 or higher), α will be noninteger. We n ≥ 1 may circumvent this by substituting u(x,t)=[v(x,t)] n . for which α becomes 1 for all n 1. ≥ 1 4 4 1 Plugging u(x,t)=[v(x,t)] n into equation (4.39) and multiplying through by n [v(x,t)] − n we obtain the following complicated new NLPDE

0 = (1 n)(1 2n)(1 3n)v2v2 n(1 n)(1 2n)v(v2v +4v v v + v2v ) − − − − t x − − − x tt x t xt t xx 2 2 2 2 3 3 +n (1 n)v (vt 2vxvxtt vxxvtt 2vtvxxt 2vxt)+ n v (vtt vxxtt) −n − − − − − 2 2 2i 2 v n a (2i + 1)v n (2i +1 n)v + nvv . (4.40) − 2i+1 − x xx i=0

Note that if n = 1 (corresponding to equation (4.2)) then most of the terms in equation (4.40)

vanish yielding fewer (and smaller) determining equations for the u , C, and V . For n = 1, i

however, these terms do not vanish. We then ﬁnd that the resulting system of equations for the ui, C, and V are overdetermined and in fact inconsistent. For this reason we will proceed with only the derivation of the solution to equation (4.2) here. Plugging (4.38) into (4.2) with α =1 gives us the following recurrence relation for the um’s

114 2 0 = 2(m 2)(m 3)(m 4)φxφt um 1,x (m 3)(m 4)(2φtφxxt + φttφxx +2φxφxtt)um 2 − − − − − − − − −

2(m 3)(m 4)(φtφxx +2φxφxt)um 2,t 4(m 3)(m 4)φtφxum 2,xt − − − − − − − − 2 2 2(m 3)(m 4)(φxφtt +2φtφxt)um 2,x + um 4,tt (m 1)(m 2)(m 3)(m 4)φt φxum − − − − − − − − − − 2 2 um 4,xxtt 2(m 2)(m 3)(m 4)φtφxum 1,t 2(m 3)(m 4)φxtum 2 − − − − − − − − − − − 2 2 2 (m 3)(m 4)φt um 2,xx (m 2)(m 3)(m 4)φt φxxum 1 +(m 3)(m 4)φt um 2 − − − − − − − − − − − − 2 2 2 (m 2)(m 3)(m 4)φxφttum 1 (m 3)(m 4)φxum 2,tt a1((m 3)(m 4)φxum 2 − − − − − − − − − − − − −

+2(m 4)φxum 3,x +(m 4)φxxum 3 + um 4,xx) 4(m 2)(m 3)(m 4)φxφtφxtum 1 − − − − − − − − − −

+2(m 4)φtum 3,t 2(m 4)φtum 3,xxt 2(m 4)φxum 3,xtt (m 4)φxxum 3,tt − − − − − − − − − − −

2(m 4)φxttum 3,x (m 4)φxxttum 3 2(m 4)φxxtum 3,t (m 4)φttum 3,xx − − − − − − − − − m − − − 2 4(m 4)φxtum 3,xt +(m 4)φttum 3 2a2 [(m j 2)(m j 3)φxujum j 1 − − − − − − − − − − − − j=0 +2(m j 3)φxujum j 2,x +(m j 3)φxxujum j 2 + ujum j 3,xx] − − − − − − − − − − m m j − 2 3a3 [(m k j 1)(m k j 2)φxujukum k j + ujukum k j 2,xx − − − − − − − − − − − − j=0 k=0 +(m k j 2)φxxujukum k j 1 + 2(m k j 2)φxujukum k j 1,x] − − − − − − − − − − − − m m j − 2 6a3 [(m k j 1)(k 1)φxujukum k j +(m k j 2)φxujuk,xum k j 1 − − − − − − − − − − − − − j=0 k=0 m 2 +ujuk,xum k j 2,x +(k 1)φxujukum j k 1,x] 2a2 [(j 1)(m j 2)φxujum j 1 − − − − − − − − − − − − − j=0 +(j 1)φxujum j 2,x +(m j 3)φxuj,xum j 2 + uj,xum j 3,x] − − − − − − − − − where u =0 if m< 0 and u ∂ (u ). m m,x ≡ ∂x m

Iterating through m values we obtain the coefﬁcients of the different powers of φ. Solving the ﬁrst two for u0 and u1 we have

115 2φ m =0 : u = t (4.41) 0 −√ 2a − 3 1 a φ √ 2a 3a φ m =1 : u = 2 t − 3 − 3 tt (4.42) 1 −3 a φ √ 2a 3 t − 3 (4.43)

Substituting in C and V and letting both be constants the solution to the remaining system of equations is easily found to be

2Cφ u (x,t) = x (4.44) 0 √ 2a − 3 √2(3CVa + a √ 2a ) u (x,t) = 3 2 − 3 (4.45) 1 6( a )3/2 − 3 C = sgn 3a a a2 (4.46) − 1 3 − 2 6a (3a a a2 3a ) V = 3 1 3 − 2− 3 (4.47) 3a3 V (x+Ct) φ(x,t) = c1 + c2e (4.48)

1 , x> 0 where sgn(x)=  . This again lends itself to a rather nice expression for u(x,t)  1 , x< 0  − given by 

2C √2(3CVa + a √ 2a ) 3 2 3 (4.49) u(x,t)= (log(φ(x,t)))x + 3/2 − √ 2a 6( a3) − 3 −

for equation (4.2). We will now plot the solution derived above for the set of parameters (c1,c2,a1,a2,a3)=

116 (1, 1, 1, 1, 1). The solution (4.49) then becomes −

√ 2 (x+t) 2 e− 3 1+ √3 u(x,t)= + . (4.50) √ √ 2 (x+t) 3 − 3 1+ e− 3

The plot for this solution is given in ﬁgure 4.4.

Figure 4.4: Plot of the solution (4.50) for t = 10, 0, 10 on the interval [ 20, 20] with the choice of parameters (c ,c ,a ,a ,a )=(1, 1, 1, 1, −1). − 1 2 1 2 3 −

117 Invariant Painleve´ Analysis Method

In 1989 Conte [46] showed that the Painleve´ analysis of PDEs was in fact invariant under an arbitrary homographic transformation of the singularity manifold φ. That is, the analysis was invariant under any transformation of φ of the form

aφ + b φ s.t. ad bc =1. (4.51) → cφ + d −

He found the ”best“ choice of new expansion function to be the function

1 ψ φ φ − χ = x1 x1x1 (4.52) ≡ ψ φ φ − 2φ x1 − 0 x1

1/2 ψ =(φ φ )φ− (4.53) − 0 x1

From this deﬁnition it can then be shown that the new expansion variable χ satisﬁes the following Ricatti equations

1 χ = 1+ Sχ2 x1 2 1 χ = C + C χ (C S + C )χ2 t − 1 1,x1 − 2 1 1,x1x1 1 χ = C + C χ (C S + C )χ2 x2 − 2 2,x1 − 2 2 2,x1x1 . . 1 χ = C + C χ (C S + C )χ2 (4.54) xn − n n,x1 − 2 n n,x1x1

118 and thus ψ satisﬁes the following linear equations

1 ψ = Sψ x1x1 −2 1 ψ = C ψ C ψ t 2 1,x1 − 1 x1 1 ψ = C ψ C ψ x2 2 1,x1 − 1 x1 . . 1 ψ = C ψ C ψ (4.55) xn 2 n,x1 − n x1

where S (the Schwarzian derivative) and the Ci are deﬁned by

φ 3 φ 2 S = x1x1x1 x1x1 (4.56) φ − 2 φ x1 x1 φt C1 = (4.57) −φx1

φxi Ci = (2 i n) (4.58) −φx1 ≥ ≤

and are also invariant under the group of homographic transformations. It is important to note that

the systems (4.54) and (4.55) are equivalent to each other. The Ci and S are linked by the cross derivative conditions

119 φx1x1x1t = φtx1x1x1 (4.59)

φx1x1x1x2 = φx2x1x1x1 (4.60) . . (4.61)

φx1x1x1xn = φxnx1x1x1 (4.62)

which are equivalent to the conditions

St + C1,x1x1x1 +2C1,x1 S + C1Sx1 = 0 (4.63)

Sx2 + C2,x1x1x1 +2C2,x1 S + C2Sx1 = 0 (4.64) . . (4.65)

Sxn + Cn,x1x1x1 +2Cn,x1 S + CnSx1 = 0 (4.66)

Upon determining the unknown coefﬁcients in our truncated expansion and resolving these condi-

tions, thereby determining our expansion function χ we will have obtained an exact solution to the original PDE.

Example: KdV Equation

Before presenting the main results for the microstructure PDE and Pochhammer-Chree equations we would like to demonstrate the effectiveness of the truncated invariant Painleve´ analysis method

on the KdV equation. The KdV equation enjoys a wide variety of applications accross multiple

120 disciplines. The KdV equation [55] - [57], which describes the propagation of weakly dispersive

and weakly nonlinear water waves, is often given by

u + u 6uu =0 (4.67) t xxx − x

α Plugging χ− into (4.67) and balancing the uux and uxxx terms we ﬁnd that the degree of the singularity, α, is equal to 2. Therefore we seek a solution of the form

u (x,t) u (x,t) u(x,t)= 0 + 1 + u (x,t) (4.68) χ(x,t)2 χ(x,t) 2

for equation (4.67). Plugging the truncated expansion (4.68) into (4.67), eliminating the partial

derivatives of χ, and collecting in powers of χ yields the system of equations

121 5 2 O(χ− ) : 24u + 12u =0 (4.69) − 0 0 4 O(χ− ) : 6u (u u )+12u u + 18u 6u =0 (4.70) − 0 0,x − 1 0 1 0,x − 1 3 O(χ− ) : 2u C 20u S 6u (u u S) 6u (u u )+12u u 6u +6u (4.71)=0 0 − 0 − 0 1,x − 0 − 1 0,x − 1 2 0 − 0,xx 1,x 2 1 O(χ− ) : u u 4u S + 21u S 6u u u S 6u (u u S) 0,t − 1,xx − 1 0,x − 0 2,x − 2 1 − 1 1,x − 0 6u (u u )+5u S 2u C + u C + u =0 (4.72) − 2 0,x − 1 0 x − 0 x 1 0,xxx 1 1 O(χ− ) : u +3u S 3u S 6u u u S 6u (u u S) 1,t 1,x − 0,x x − 1 2,x − 2 1 − 2 1,x − 0 1 +u (CS + C )+ u S 3u S 2u S +2S2 u C + u =0(4.73) 0 xx 1 x − 0,xx − 0 2 xx − 1 x 1,xxx 3 1 3 3 3 3 O(χ0) : u S2 u S +2S2 + u S2 + u SS + u u S u S 2 0,x − 1 2 xx 2 1 2 0 x 2,t − 2 1,x x − 2 1,xx 1 1 6u u u S + u + u (CS + C )=0 (4.74) − 2 2,x − 2 1 2,xxx 2 1 xx

We now have a system of equations for the unknown functions u0,u1,u2,C and S. It is often useful to impose certain conditions (such as C,S constant) to reduce computational complexity, however a major drawback as one may deduce is that our solutions will become more trivial.

Instead, leaving S and C as functions of x and t and solving this system with the aid of MAPLE yields the following results

122 u0(x,t) = 2, (4.75)

u1(x,t) = 0, (4.76) 2 1 u (x,t) = S(x,t) C(x,t), (4.77) 2 3 − 6 x C(x,t) = , (4.78) 3t + C1 x S(x,t) = . (4.79) 3t + C1

Using our equations for ψ and χ we are then able to ﬁnd the following rather complicated solutions

3 3 (3t + C )1/3 C x F (, 4 ; x )+ C (3t + C )1/3 F (, 2 ; x ) 1 3 0 1 3 54t+18C1 2 1 0 1 3 54t+18C1 ψ = − − , (4.80) √9t +3C1 3 3 C x F (, 4 ; x )+ C (3t + C )1/3 F (, 2 ; x ) 3 0 1 3 54t+18C1 2 1 0 1 3 54t+18C1 χ = − 3 − 3 (4.81), 3 7 x 1/3 2 5 x 3 C3x 0F1(, ; ) C2(3t+C1) x 0F1(, ; ) 4 x 9 3 − 54t+18C1 9 3 − 54t+18C1 C3 0F1(, ; ) 3 − 54t+18C1 − 4 54t+18C1 − 2 54t+18C1 from which our equation for u(x,t) gives us the following solution

3 3 2 3 7 x 1/3 2 5 x 3 C3x 0F1(, ; ) C2(3t+C1) x 0F1(, ; ) 4 x 9 3 − 54t+18C1 9 3 − 54t+18C1 C3 0F1(, ; ) u(x,t) = 2 3 − 54t+18C1 − 4 54t+18C1 − 2 54t+18C1  4 x3 1/3 2 x3  C3x 0F1(, ; )+ C2(3t + C1) 0F1(, ; ) 3 − 54t+18C1 3 − 54t+18C1  x  + , (4.82) 6t +2C1

123 where 0F1(,b; z) is the hypergeometric function deﬁned by

∞ Γ(b)zk F (,b; z)= , (4.83) 0 1 Γ(b + k)k! k=0 and Γ(a) is the gamma function.

We will now plot the solution derived above for the set of parameters (C1,C2,C3) = (1, 1, 0) for which the solution (4.82) then becomes

2 √2 x3 x I2/3 3 3t+1 u(x,t)= − . (4.84) −3t +1  √2 x3  I 1/3 3 3t+1  − −    The plot for this solution is given in ﬁgure 4.5.

124 Figure 4.5: Plot of the solution (4.84) for t = 2 and 2 on the interval [ 35, 35] with the choice − − of parameters (C1,C2,C3)=(1, 1, 0).

We remark that u(x,t) is not of traveling wave form. The inclusion of both u(x, 2) and u(x, 2) − in the same plot was to show near symmetry of the solution (4.84) in t for t 1. | |≥

MicroStructure PDE

In this section we use the truncated invariant Painleve´ analysis method to construct an exact solu-

α tion to the microstructure PDE given earlier in the chapter by equation (4.1). Plugging χ− into

2 (4.1) and balancing the (v )xx and vxxxx terms we ﬁnd that the degree of the singularity, α, is equal

125 to 2. Since α =2, we seek a solution of the form

v (x,t) v (x,t) v(x,t)= 0 + 1 + v (x,t) (4.85) χ(x,t)2 χ(x,t) 2 for equation (4.1). Plugging the truncated expansion (4.85) into (4.1) and and eliminating all derivatives of χ yields the Painleve-Backlund¨ equations order by order in χ. Due to the complex-

ity of the Painleve-B´ acklund¨ equations we shall make a further assumption on C(x,t) that it be constant. With this assumption on the function C(x,t) the ﬁrst three Painleve-Backlund¨ equations become

12βδC2v 12γδv + v2 =0, (4.86) 0 − 0 0 (8v (v v )+2v (2v 4v )+12v v ) δ[β(24C2v 48C2v − 2 0 1 − 0,x 0 1 − 0,x 0 1 − 1 − 0,x +48Cv ) γ(24v 96v )]=0, (4.87) 0,t − 1 − 0,x 6C2v 6bv 8v v + 2(v v )2 +2v (v 2v )+2v (2v 4v ) 0 − 0 − 2 − 0 1,x 0,x − 1 0 0,xx − 1,x 1 1 − 0,x +12v v ) δ β(24v C2S 12C2v +6C2v + 12Cv 24Cv +6v ) 0 2 − 0 − 1,x 0,xx 1,t − 0,xt 0,tt γ(24v S 24 v + 36v )) 8v2S 96δv S βC2 γ =0, (4.88) − 0 − 1,x 0,xx − 0 − 0 − where it is important to keep in mind that at this point we have not made any additional assumptions on the function S(x,t). Solving the three equations above for v0, v1, and v2 with the aid of MAPLE we obtain

126 12δ(C2β γ) v = − , (4.89) 0 −

v1 = 0, (4.90) 4C2βδS 4γδS C2 + b v = − − . (4.91) 2 −

The next step is to use the remaining Painlee-B´ acklund¨ equations to determine the function S(x,t) and, if necessary, the constant C. Using the previous expressions for v0, v1, and v2 with C constant the remaining Painleve-B´ acklund¨ equations become

1 2 2 2 24− C β γ δ 18C β + γ S + 19CβS =0, (4.92) − x t 1 2 2 2 24− δ γ C β 3C β + γ S +4CβS =0, (4.93) − xx xt 1 2 2 δγ 2 17Cδβ 2 12− δ C β γ 8C δβ + (S ) + (S ) + δ (CβS + γS ) − 2 x 2 t t x xx C2S CS =0 , (4.94) − x − t 1 2 2 2 2 2− δ γ C β δ 14C β +4γ SS + 18CδβSS + δ 8C β + γ S − xx xt x 18δβS 2 9CδβS S +2S 2C2S +2δγS 2δβS =0. (4.95) − t − x t tt − xx xxxx − xxtt

It follows that a rather simple solution would be to take C = γ . Looking back at the forms ± β of u0 and u2 we see that this choice would make u0 = 0 and u2 = constant thereby giving only a constant solution. For less trivial results we will require that C = γ . Another simple, ± β yet sufﬁcient, solution to the previous system is to take S to be a constant. Using C(x,t) = C, S(x,t)= S (where C and S are constants) and the determining equations for ψ(x,t) we ﬁnd that

127 ψ(x,t) takes the form

S S ψ(x,t)= c1 sin (x Ct) + c2 cos (x Ct) , (4.96) 2 − 2 −

where c1 and c2 are arbitrary constants. Substituting this expression for ψ into the expression for χ in terms of ψ (given earlier in the section) we arrive at the following traveling wave solution to equation (4.1),

2 2 S S 6Sδ(γ C β) c1 cos 2 (x Ct) c2 sin 2 (x Ct) v(x,t) = − − − − 2 c sin S (x Ct) + c cos S (x Ct) 1 2 − 2 2 − 1 2 2 − (4Sδ (C βγ) C + b). (4.97) − − −

From this form of the solution we make the observation that a nontrivial (once again taken to mean non-constant) solution will require that c = 0 and c = 0. We make the additional remark 1 2 that the solution will have two qualitatively different forms depending on the sign of S. On one hand, for S > 0 and t = t0 ﬁxed the solution will be periodic with inﬁnitely many singularities a

2 1 c2 x = Ct0 tan− . On the other hand, for S < 0 the solution will involve hyperbolic − S − c1 sines and hyperbolic cosines and thus will be continuous on R2 for most parameter set choices.

Note that to ensure the solution is real-valued we will take c2 to be imaginary. We will now plot the solution derived above for two parameter sets such that S > 0 in the ﬁrst set and S < 0 in the second set. For the choice (c ,c ,S,C,γ,,δ,β,b) = (1, 1, 1, 1, 1, 1, 1, 1, 1) (S > 0) the 1 2 − −

128 solution (4.97) simpliﬁes to

4sin √2(x + t) +5 v(x,t)= , (4.98) − sin √2(x + t) 1 − whereas for the choice (c ,c ,S,C,γ,,δ,β,b)=(1, 2i, 1, 1, 1, 1, 1, 1, 1) (S < 0) the solution 1 2 − − (4.97) simpliﬁes to

2 t x t x t x 4 5cosh − + 4cosh − sinh − 40 √2 √2 √2 v(x,t)= − . (4.99) 2 t x t x t x 5cosh − + 4cosh − sinh − 1 √2 √2 √2 − The plots of these solutions are given in ﬁgure 4.6.

Figure 4.6: (Left): Plot of the solution (4.97) for the choice of parameters (c ,c ,S,C,γ,,δ,β,b) = (1, 1, 1, 1, 1, 1, 1, 1, 1) at t = 1 in the rectangle [ 5π, 5π]. 1 2 − − − (Right): Plot of the solution (4.97) for the choice of parameters (c1,c2,S,C,γ,,δ,β,b) = (1, 2i, 1, 1, 1, 1, 1, 1, 1) at t = 5, 0, 5 on the interval [ 15, 15]. − − − −

129 Generalized Pochhammer-Chree Equations

In this section we use the truncated invariant Painleve´ analysis method to construct an exact solution to the generalized Pochhammer-Chree equations given earlier in the chapter by equations (4.2)

α 3 and (4.3). Plugging χ− into (4.2) and balancing the (u )xx and uxxtt terms we ﬁnd that the degree

(1) 5 of the singularity, α , is equal to 1. When balancing the (u )xx and uxxtt terms in equation (4.3), however, we ﬁnd that α(2) is equal to 1/2. Due to the non-integer value of α(2) we need to ﬁnd an appropriate substitution to make α(2) a nonnegative integer. The substitution we use to accomplish 1/2 this is u(2)(x,t)= v(2)(x,t) where it is important to keep in mind that the superscripts in the parenthesis are not powers nor derivatives but rather indicate which equation the term corresponds to. That is, any term with (1) corresponds to equation (4.2) and any term with (2) corresponds to equation (4.3). Running through the leading order analysis again on the new system now yields

α(2) =1. Therefore we seek solutions of the form

u(1)(x,t) u(1)(x,t)= 0 + u(1)(x,t) (4.100) χ(1)(x,t) 1

and v(2)(x,t) v(2)(x,t)= 0 + v(2)(x,t) (4.101) χ(2)(x,t) 1

for equations (4.2) and (4.3), respectively. Plugging these truncated expansions into their respective

equations and eliminating all derivatives of χ yields the Painleve-B´ acklund¨ equations order by order in χ for equations (4.2) and (4.3). Due to the complexity of these systems of equations in

both cases we shall once again require that C(x,t) be a constant. For equation (4.2) the ﬁrst two Painleve-B´ acklund¨ equations are then

130 2 12u(1) 2C2 + a u(1) =0, (4.102) − 0 3 0 2 2 2 2Cu(1) +3a u(1) u(1) 3a u(1) u(1) + a u(1) 2C2u(1) =0, (4.103) 0,t 3 0 1 − 3 0 0,x 2 0 − 0,x and for equation (4.3) the ﬁrst two Painleve-B´ acklund¨ equations are

35 4 2 3 v(1) a v(1) + C2 =0, (4.104) − 4 0 5 0 4 5 3 2 2 v(1) 64a v(1) v(1) 20a v(1) v(1) + 36C2v(1) 3C2v(1) −8 0 5 0 1 − 5 0 0,x 1 − 0,x 2 (1) (1) +6a3 v0 +3Cv0,t =0. (4.105)

Solving the ﬁrst two equations for both cases yields the following results

2 u(1) = C (4.106) 0 − a 3 (1) a2 − u1 = (4.107) −3a3

√3C v(2) = (4.108) 0 2√ a − 5 (2) 3a3 v1 = (4.109) −8a5

At this point taking S(i)(x,t) to be a constant for i =1, 2 reduces the remaining Painleve-B´ acklund¨

131 equations to a system of algebraic equations in the C(i) and S(i). Solving these systems for the

S(i)(x,t) and C(i)(x,t) we ﬁnd

C(1) = C, (4.110) 1 a3 +3C2a 3a a (1) 2 3 1 3 (4.111) S = 2 − , 3 C a3

and

1 2(9a2 32a a ) C(2) = 3 − 1 5 , (4.112) 4 a (S(2) 4) 5 − 6a2 S(2) = 3 . (4.113) 16a a 3a2 1 5 − 3

Note that for equation (4.3) the C term must take a speciﬁc form (dependent on the ai) whereas for equation (4.2) we can take the C term to be arbitrary. In keeping with the notation of this section

we will use ψ(1)(x,t) and ψ(2)(x,t) for equations (4.2) and (4.3), respectively. From our values for the S(i),s and C(i),s and the determining equations for the ψ(i),s given earlier in the section we ﬁnd

that ψ(1) and ψ(2) take the following forms

ψ(1)(x,t)= c cos(λ(Ct x)) + c sin(λ(Ct x)) , (4.114) 1 − 2 −

132 2 2 3C a3 3a1a3+a where λ = − 2 , and 6Ca3

√ √ (2) 3a3 4x√a5 + tσ 3a3 4x√a5 + tσ ψ (x,t)= c1 cos + c2 sin , (4.115) 4√a5σ 4√a5σ where σ = 16a a 3a2. Therefore we have the following traveling wave solutions, 1 5 − 3

2 x x C λ(c1 sin(λ( t)) c2 cos(λ( t)))) a3 C C a u(1)(x,t)= − − − − 2 (4.116) c cos(λ( x t))) + c sin(λ( x t)) − 3a 1 C − 2 C − 3

and, noting that u(2)(x,t)= v(2)(x,t),

3(32a a 9a2) a ( c sin(y(x,t)) + c cos(y(x,t))) 3a 1/2 u(2)(x,t)= 1 5 − 3 3 − 1 2 3 , (4.117) 4√ a (c cos(y(x,t)) + c sin(y(x,t))) − 8a − 5 1 2 5

√ where y(x,t) = 3a3(4x√a5+tσ) . These solutions have the potential to become complex-valued 4√a5σ but may be taken to be real provided we make suitable choices for the arbitrary constants. For

example, a rather simple requirement for u(1) is a < 0 and λ,C R. The condition λ R 3 ∈ ∈ 3 is equivalent to requiring that a satisfy a < a2 + C2. Due to the nature of u(2) being that of 1 1 3a3 a rational expression involving trigonometric functions inside a radical one cannot guarantee the

solutions are real for all x and t as we did previously without eliminating the x and t dependence. However, given adequate choices it is possible to ensure the solutions are real for some spatial and time intervals.

We will now plot the solutions derived above for parameter sets which lead to qualitatively different plots. We ﬁnd that the choice of parameter sets may lead to solutions with zero, one, or inﬁnitely

133 many singularities. The case of zero or one singularity corresponds to parameters sets such that

λ iR (the set of imaginary numbers). That is, if λ iR then c1 1 corresponds to one ∈ ∈ c2 ≤ singularity whereas c1 > 1 corresponds to no singularities. On the other hand, if λ R then c2 ∈ there will be inﬁnitely many singularities for all choices of c ,c R. For the parameter set 1 2 ∈ (c ,c ,C,a ,a ,a )=(1, 1, 2, 1, 1, 1) the solution (4.116) becomes 1 2 1 2 3 −

1/2 1/2 1 (2√6 1)sin 3− (2t x) + (2√6+1)cos 3− (2t x) u(1)(x,t)= − − − , (4.118) −3 sin(3 1/2(2t x)) cos(3 1/2(2t x)) − − − − − whereas for the choice (c ,c ,C,a ,a ,a )=(3, 2i, 1, 2, 1, 1) the solution becomes 1 2 1 2 3 − −

√ 2 √ 2 1 (6 3 2) sinh 3 (t x) + (4 3 3) cosh 3 (t x) u(1)(x,t)= − − − − . (4.119) −3 3cosh 2 (t x) + 2sinh 2 (t x) 3 − 3 − The plots of these solutions are given in ﬁgure 4.7.

134 Figure 4.7: (Left): Plot of the solution (4.116) for the choice of parameters (c1,c2,C,a1,a2,a3)= (1, 1, 2, 1, 1, 1) at t = 1 in the interval [ 15, 15]. (Right): Plot of the solution (4.116) for the − − choice of parameters (c1,c2,C,a1,a2,a3) = (3, 2i, 1, 2, 1, 1) at t = 5, 0, 5 in the interval [ 12, 12]. − − − −

As mentioned above, for some choices of parameter set (c1,c2,a1,a3,a5) the term inside the square root in the solution (4.117) may oscillate between positive and negative values and thus will only be real-valued in certain regions of the xt plane. For this reason we will only consider cases − where the solution is real-valued for all (x,t) R2. In particular, we consider the parameter set ∈ (c ,c ,a ,a ,a )=(2i, 1, 1, 2, 1) for which the solution (4.117) becomes 1 2 1 3 5 −

1/2 √21(t√7+2x) √21(t√7+2x) 3 sinh 14 + cosh 14 u(2)(x,t)= (4.120) 2  √21(t√7+2x) √21(t√7+2x)  sinh 14 + 2cosh 14   The plot of this solution for t = 5, 0, 5 is given in ﬁgure 4.8. −

135 Figure 4.8: Plot of the solution (4.117) for t = 5, 0, 5 on the interval [ 20, 15] with the choice of parameters (c ,c ,a ,a ,a )=(2i, 1, 1, 2, 1)−. − 1 2 1 3 5 −

Generalized Hirota Expansion Method

In this section we will give a brief description of the generalized Hirota expansion method [50]. As previously mentioned, after plugging the truncated series solution into the PDE and reconciling the powers of φ we will obtain a recurrence relation from which we will determine the un and α. In general we ﬁnd that, after arranging the equations according to increasing order in φ, the

ﬁrst equation will determine u0, the second u1, etc. We continue this process until we have found u0,...,uα 1 and keep the remaining uα unknown. The ﬁnal term can be expanded in a power −

136 series about x1 =0

∞ i uα(x1,...,xn,t)= uα,i(x2,...,xn,t)x1 (4.121) i=0

Since α is finite and in general we will have an PDE with finite order of nonlinearity we will have a finite number of remaining conditions to satisfy. Therefore plugging (4.121) into the remaining equations will yield a heavily underdetermined system. Thus there will exist some N N such ∈ that u (x ,...,x ,t)=0 for all i N. That is, u (x ,...,x ,t) can be represented by a finite α,i 2 n ≥ α 1 n series. For ease of computation we force φ(x1,...,xn,t) to be of the form φ(x1,...,xn,t) = 1 + exp Γ(t)+ n x Ω (t) . This functional form is somewhat reminiscent of the standard { l=1 l l } ansatz one considers whilst searching for a one-soliton solution via Hirota’s bilinear method. To further reduce complexity we let Γ(t)= k +k t and Ω (t)= k where k C (l =1,...,n+2). 1 2 l l+2 l ∈ Plugging the new expansion with known u , n =0, 1,...,α 1 into the remaining equations gives n − rise to a new set of equations for each equation in the previous set of equations. From these we will determine the unknown uα,i(t) and kl. If, in theory, we can solve for these terms we will have found the last term in our truncated series expansion, as well as the form of the singularity manifold φ and therefore will have a solution to the original NLPDE.

Example: KP-II Equation

Before presenting the main results for the microstructure PDE and Pochhammer-Chree equations we will demonstrate the effectiveness of the generalized Hirota expansion method on a classic example in (2+1), the KP-II equation. As the procedure begins exactly as the standard truncated Painleve´ method we ﬁnd that the initial steps in this procedure yield the same results. That is, for the KP-II equation given earlier by equation (4.13) we ﬁnd that the degree of the singularity α,

137 found by balancing the (uux)x and uxxxx terms, is equal to 2. Therefore, we will once again seek a solution of the form

u (x,y,t) u (x,y,t) u(x,y,t)= 0 + 1 + u (x,y,t). (4.122) φ(x,y,t)2 φ(x,y,t) 2

Plugging the truncated expansion (4.122) into (4.13) yields the previously found recurrence relation

+ (uk,xun k 2,x +(n k 3)uk,xun k 1φx +(k 2)ukun k 1,xφx − − − − − − − − − k=0 2 +(k 2)(n k 2)ukun kφx + uk(un k 2,xx + 2(n k 3)un k 1,xφx − − − − − − − − − − 2 +(n k 3)un k 1φxx +(n k 2)(n k 3)un kφx))=0 (4.123) − − − − − − − − − where u = 0 if n < 0 and u ∂ (u ). The difference here is that we will solve for only the n n,x ≡ ∂x n functions u0 and u1, leaving the determination of u2 to the end. Upon satisfaction of the recurrence

138 relation at the lowest powers of φ we ﬁnd that

u (x,y,t)= 12ǫ2φ2 and u (x,y,t)=12ǫ2φ (4.124) 0 − x 1 xx

Following the procedure as outlined earlier in the section with the assumed forms for Γ(t) and Ω(t)

we ﬁnd that u2 takes the relatively simple form

2 4 2 k3k4 + ǫ k3 + λk2 u2(x,y,t)= 2 (4.125) − k3

where k1 4 are arbitrary constants. With this the remaining equations at each order of φ are iden- − tically satisfied and thus we find the final solution given by

(ǫ2k4 + k k + λk2)e2k1+2k2y+2k3x+2k4t + 2(λk2 + k k 5ǫ2k4)ek1+k2y+k3x+k4t u(x,y,t) = 3 3 4 2 2 3 4 − 3 2 k1+k2y+k3x+k4t 2 k3(1 + e ) ǫ2k4 + k k + λk2 + 3 3 4 2 (4.126) 2 k1+k2y+k3x+k4t 2 k3(1 + e )

We will now plot (see ﬁgure 4.9) the solution given by (4.126) for two different choices for the set of parameters just as we did in truncated Painleve´ analsis section. For the ﬁrst choice we take

(k , k , k , k ,ǫ,λ)=(1, 1, 1,i, 1, 1). The solution (4.126) at t =1 then becomes 1 2 3 4 − −

2+2i 2y 2x 1+i y x (2 i)(1+ e − − ) (8+2i)e − − u(x,y, 1) = − − . (4.127) 1+i y x 2 (1 + e − − )

It is clear that for this set of parameters the solution is complex-valued. Therefore, to better visu-

139 Figure 4.9: Plots of the real (left) and imaginary (right) parts of the solution to the KP-II equation at t =1 on the rectangle [ 3π, 3π] [ 3π, 3π] with the choice of parameters (k1, k2, k3, k4,ǫ,λ)= (1, 1, 1,i, 1, 1). − × − − − alize the solution we plot the real and imaginary parts separately.

As a second choice for the set of parameters we will take (k , k , k , k ,ǫ,λ)=(1, 1, 1, 1, 1, 1) 1 2 3 4 − − − for which the solution (4.24) at t =1 becomes

x y 2 3(1 e− − ) u(x,y, 1) = − = 3tanh2 ((1/2)( x y)) . (4.128) x y 2 (1 + e− − ) − −

The plot for this solution is given in ﬁgure 4.10.

140 Figure 4.10: Plot of the solution to the KP-II equation at t = 1 on the rectangle [ 10, 10] [ 10, 10] with the choice of parameters (k , k , k , k ,ǫ,λ)=(1, 1, 1, 1, 1, 1). − × − 1 2 3 4 − − −

Microstructure PDE

In this section we use the generalized Hirota expansion method to construct an exact solution to

α the microstructure PDE given earlier in the chapter by equation (4.1). Plugging φ− into equation

2 (4.1) and balancing the (v )xx and vxxxx terms we ﬁnd that the degree of the singularity, α, is equal to 2. Since α =2, we seek a solution of the form

v (x,t) v (x,t) v(x,t)= 0 + 1 + v (x,t) (4.129) φ(x,t)2 φ(x,t) 2 for equation (4.1). Plugging the truncated expansion (4.129) into equation (4.1) yields the recurrence relation given previously in the chapter in the Painleve´ analyis section,

141 vm 4,tt bvm 4,xx +(m 5)(2vm 3,tφt 2bvm 3,xφx + vm 3(φtt bφxx) − − − − − − − m − − 2 2 +(m 4)vm 2(φt bφx)) vk,xvm k 2,x + vkvm k 2,xx − − − − { − − − − k=0 2 +(m k 3)(vk,xvm k 1φx +2vkvm k 1,xφx +(m k 2)vkum kφx − − − − − − − − − 2 +vkvm k 1φxx)+(k 2)(vkvm k 1,xφx +(m k 2)vkvm kφx) δβ(vm 4,xxtt − − − − − − − − − − +(m 5)(2vm 3,xxtφt + vm 3,xxφtt +2vm 3,xttφx +4vm 3,xtφxt +2vm 3,xφxtt − − − − − − 2 2 +vm 3,ttφxx +2vm 3,tφxxt + vm 3φxxtt +(m 4)(vm 2,xxφt + vm 2,ttφx − − − − − − 2 +4vm 2,xtφxφt +4vm 2,xφtφxt +2vm 2,xφxφtt +4vm 2,tφxφxt +2vm 2φxt − − − − −

+2vm 2φxφxtt +2vm 2,tφtφxx + vm 2φtφxxt + vm 2φxxφtt + vm 2φxxtφt − − − − − 2 2 2 2 +(m 3)(2vm 1,xφxφt +2vm 1,tφxφt +4vm 1φxφtφxt + vm 1φxφtt + vm 1φxxφt − − − − − − 2 2 +(m 2)vmφxφt )))) + δγ(vm 4,xxxx +(m 5)(vm 3φxxxx +4vm 3,xφxxx − − − − − 2 +6vm 3,xxφxx +4vm 3,xxxφx +(m 4)(vm 2(4φxφxxx +3φxx)+12vm 2,xφxφxx − − − − − 2 3 2 4 +6vm 2,xxφx +(m 3)(4vm 1,xφx +6vm 1φxφxx +(m 2)vmφx)))) = 0 (4.130) − − − − −

where once again v =0 if n< 0 and v ∂ (v ). Solving the ﬁrst two equations (m =0 and n n,x ≡ ∂x n m =1) for v0 and v1 we again ﬁnd

12δ v = (γφ2 βφ2) (4.131) 0 − t − x 12δ (5γ2φ2φ +5β2φ2φ βγ(φ2φ + φ2φ ) 8βγφ φ φ ) v = x xx t tt − t xx x tt − t x xt (4.132) 1 5(γφ2 βφ2) x − t

Note that we do not solve for v2 at this time as we did previously when following the Painleve´

142 analysis method. Rather, following the procedure for the generalized Hirota expansion method as

outlined earlier in the section and expanding the v2 in x with the assumed forms for Γ(t) and Ω(t)

we ﬁnd that v2 takes the relatively simple form

2 2 2 2 2 v2(x,t) = 2 δk3(k3γ βk2) bk3 + k2 (4.133) k3 − −

where k1 3 are arbitrary constants. With this the remaining equations at each order of φ are iden- − tically satisfied and thus we find the final solution given by

2 2 12δ(βk2 γk3)exp k1 + k2t + k3x v(x,t)= − { 2 } − (1 + exp k1 + k2t + k3x ) 2 2{ 2 4 2}4 12δ(10βγk2k3 5γ k3 5β k2)exp k1 + k2t + k3x 2 − 2 − { } − (βk2 γk3)(1 + exp k1 + k2t + k3x ) 4 2 − 2 2 2 { } δγk3 βk3 δβk2k3 + k2 + − − 2 (4.134) k3

We will now plot the solution derived for the parameter set (k , k , k ,γ,,δ,β,b)=(1, 1, 1, 1, 1, 1, 1, 1). 1 2 3 − − − For this choice of parameter set the solution (4.134) simpliﬁes to

2t 2x+2 t x+1 2(e− − 10e− − + 1) v(x,t)= − . (4.135) t x+1 2 (1 + e− − )

The plot of this solution for t = 5, 0, 5 is given in ﬁgure 4.11. −

143 Figure 4.11: Plot of the solution (4.134) to the microstructure PDE at t = 5, 0, 5 on the interval [ 12, 12] with the choice of parameters (k , k , k ,γ,,δ,β,b)=(1, 1, −1, 1, 1, 1, 1, 1). − 1 2 3 − − −

Generalized Pochhammer-Chree Equations

In this section we use the generalized Hirota expansion method to construct an exact solution to the generalized Pochhammer-Chree equations given earlier in the chapter by equations (4.2) and

α 3 (4.3). Plugging φ− into equation (4.2) and balancing the (u )xx and uxxtt terms we ﬁnd that the degree of the singularity, α, is equal to 1. Following the procedure and using the proper substitution

(u(x,t) = v(x,t)) for equation (4.3) as outlined in the previous sections we ﬁnd that α(i) = 1 (i = 1, 2). From this point on we will adopt the same notation used in the previous section for differentiating between results for equations (4.2) and (4.3). That is, we will use the superscript (1) for terms corresponding to equation (4.2) and the superscript (2) for terms corresponding to

144 equation (4.3). From the leading order analysis we seek solutions of the form

u (x,t) u(x,t)= 0 + u (x,t) (4.136) φ(1)(x,t) 1

v (x,t) v(x,t)= 0 + v (x,t) (4.137) φ(2)(x,t) 1

Plugging (4.136) and (4.137) into equation (4.2) we obtain the recurrence relation found earlier in the chapter in the Painleve´ analysis section for equation (4.2),

145 2 2(m 2)(m 3)(m 4)φxφt um 1,x (m 3)(m 4)(2φtφxxt + φttφxx +2φxφxtt)um 2 − − − − − − − − −

+2(m 4)φxum 3,x +(m 4)φxxum 3 + um 4,xx) 4(m 2)(m 3)(m 4)φxφtφxtum 1 − − − − − − − − − −

+2(m 4)φtum 3,t 2(m 4)φtum 3,xxt 2(m 4)φxum 3,xtt (m 4)φxxum 3,tt − − − − − − − − − − −

2(m 4)φxttum 3,x (m 4)φxxttum 3 2(m 4)φxxtum 3,t (m 4)φttum 3,xx − − − − − − − − − m − − − 2 4(m 4)φxtum 3,xt +(m 4)φttum 3 2a2 [(m j 2)(m j 3)φxujum j 1 − − − − − − − − − − − − j=0 +2(m j 3)φxujum j 2,x +(m j 3)φxxujum j 2 + ujum j 3,xx] − − − − − − − − − − m m j − 2 3a3 [(m k j 1)(m k j 2)φxujukum k j + ujukum k j 2,xx − − − − − − − − − − − − j=0 k=0 +(m k j 2)φxxujukum k j 1 + 2(m k j 2)φxujukum k j 1,x] − − − − − − − − − − − − m m j − 2 6a3 [(m k j 1)(k 1)φxujukum k j +(m k j 2)φxujuk,xum k j 1 − − − − − − − − − − − − − j=0 k=0 m 2 +ujuk,xum k j 2,x +(k 1)φxujukum j k 1,x] 2a2 [(j 1)(m j 2)φxujum j 1 − − − − − − − − − − − − − j=0 +(j 1)φxujum j 2,x +(m j 3)φxuj,xum j 2 + uj,xum j 3,x]=0, − − − − − − − − − where u = 0 if m < 0 and u ∂ (u ). As was the case in the Painleve´ section we must m m,x ≡ ∂x m transform (4.3) according to u(x,t) = v(x,t). The resulting equation will then follow from equation (4.40) with n = 2. As one can see from the recurrence relation for equation (4.2) the

146 recurrence relation for (4.40) would be unecessarily messy and therefore will be omitted. Iterating

through m values in the recurrence relation for equation (4.2) we obtain the coefﬁcients of the different powers of φ. Solving the equations for the coefﬁcient of the lowest order of φ in each

case for u0 and v0 we ﬁnd

√2φ √3φ u (x,t)= t and v (x,t)= t . (4.138) 0 −√ a 0 −2√ a − 3 − 5

Here again we do not solve for u2 or v2 at this time as we did previously when following the Painleve´ analysis method. Rather, following the procedure for the generalized Hirota expansion

method as outlined earlier in the section and expanding the u2 and v2 in x with the assumed forms

(i) (i) for the Γ (t) and Ω (t) we ﬁnd that u2 and v2 take the relatively simple forms

(1) 2 √2( 2k2 a3 + a2√ 2a3) √6(9a3 +2a2√6a3a5) u1(x,t)= − − and v1(x,t)= (4.139) − 6( a )3/2 36a √a a − 3 3 3 5

(1) where k2 is an arbitrary constant. Unlike the results for the KP equation and microstructure PDE

(i) (i) we cannot assume here that the kj (j = 1, 2, 3) in the φ are arbitrary. Rather we ﬁnd that only (1) (1) (2) (i) k1 , k2 , and k1 will be allowed to be arbitrary. Solving for the remaining kj in terms of the

(i) (i) coefﬁcients in their respective PDEs ((4.2) or (4.3)) and the previous km the φ are readily found to be

k √6 a ( 3k2a +6a a 2a2) φ(1)(x,t)=1+ exp k + k t 2 3 − 2 3 1 3 − 2 x (4.140) 1 2 − 3k2a +6a a 2a2 − 2 3 1 3 − 2

147 and

3a 6√2a a (32a a 6a2) φ(2)(x,t)=1+ exp k 3 t 3 5 1 5 − 3 x . (4.141) 1 − 2√ 3a − √ 3a (32a a 6a2) − 5 − 5 1 5 − 3

With this the remaining equations at each order of φ(i) are identically satisfied and thus we find the final solutions are given by

2k exp y(1)(x,t) √2( 2k a + a √ 2a ) u(1)(x,t)= 2 2 3 2 3 (4.142) { (1) } − 3/2 − −√ 2a (1 + exp y (x,t) ) − 6( a3) − 3 { } −

and, noting again that u(x,t)= v(x,t),

1/2 3k exp y(2)(x,t) √6(9a2 +2a √6a a ) u(2)(x,t)= 2 + 3 1 3 5 , (4.143) { (2) } 2√ 3a (1 + exp y (x,t) ) 36a3√a3a5 − 5 { } where

k √6 a ( 3k2a +6a a 2a2) y(1)(x,t)= k + k t 2 3 − 2 3 1 3 − 2 x (4.144) 1 2 − 3k2a +6a a 2a2 − 2 3 1 3 − 2

and

3a 6√2a a (32a a 6a2) y(2)(x,t)= k 3 t 3 5 1 5 − 3 x. (4.145) 1 − 2√ 3a − √ 3a (32a a 6a2) − 5 − 5 1 5 − 3

Again, we see that these solutions may be complex-valued but can be taken to be real with suitable

(1) choices of the arbitrary constants involved. For example, for u we may take a3 < 0 and a1 <

148 2 2 2 2 a2 k2 (2) 3√6a3 3a3 + . For u we may take a5 < 0, a3 < 0, and

We will now plot the solutions derived above for parameter sets which lead to real-valued solutions.

For the parameter set (k , k ,a ,a ,a )=(0, 1, 1, 1, 1) the solution (4.142) becomes 1 2 1 2 3 −

√2 3+ √2+(2√2 6)et+√6/5x u(1)(x,t)= − . (4.146) 6 2et+√6/5x +1

The plot of this solution for t = 5, 0, 5 is given in ﬁgure 4.12. −

Figure 4.12: Plot of the solution (4.142) for t = 5, 0, 5 on the interval [ 12, 12] with the choice of parameters (k , k ,a ,a ,a )=(0, 1, 1, 1, 1)−. − 1 2 1 2 3 −

149 As with the previous section, for some choices of parameter set (k1,a1,a3,a5) the term inside the square root in the solution (4.143) may oscillate between positive and negative values and thus will

only be real-valued in certain regions of the xt plane. For this reason we will only consider cases − where the solution is real-valued for all (x,t) R2. In particular, we consider the parameter set ∈ (k ,a ,a ,a )=(0, 1, 1, 1) for which the solution (4.143) becomes 1 1 3 5 −

√3 2√3 1/2 (2) √3 t+ x − u (x,t)= 1+ e− 2 √19 (4.147) 2

The plot of this solution for t = 5, 0, 5 is given in ﬁgure 4.13. −

Figure 4.13: Plot of the solution (4.143) for t = 5, 0, 5 on the interval [ 12, 20] with the choice − − of parameters (k ,a ,a ,a )=(0, 1, 1, 1). 1 1 3 5 −

150 CHAPTER 5: CONCLUSION

In the ﬁrst part of this dissertation we presented two methods for deriving the Lax pair for variable- coefﬁcient nlpdes, namely Khawaja’s Lax pair method and the extended Estabrook-Wahlquist

method. In doing so we determined the necessary conditions on the variable coefﬁcients for the variable-coefﬁcient nonlinear PDEs to be Lax-integrable. The latter technique is introduced

here in the context of space and time dependent coefﬁcients for the ﬁrst time. As the extended Estabrook-Wahlquist method requires many fewer assumptions and is algorithmic it proves to be a vast improvement over Khawaja’s Lax pair method.

As stated in the introduction, an accepted deﬁnition for integrability of a nonlinear PDE does not

currently exist in the literature. However, many nonlinear PDEs which have been classiﬁed as integrable share a remarkable number of properties. Perhaps the most important of these properties is the existence of a nontrivial Lax pair. We have derived variable-coefﬁcient extensions to several

well-known integrable nonlinear PDEs from the requirement that they possess nontrivial Lax representations as well as proving the nonexistence of a nontrivial Lax pair to an extension to a known

nonintegrable nonlinear PDE.

In the second part of this dissertation we gave a brief introduction to three distinct types of sin-

gular manifold methods: truncated Painleve´ analysis, truncated invariant Painleve´ analysis, and a generalized Hirota expansion method. These methods were then demonstrated on the well-known integrable KdV and KP-II equations. Plots of the derived solutions were given for various choices of the arbitrary constants and system parameters involved. Following these examples we employed each method of solution to derive nontrivial solutions to a microstructure PDE and two generalized Pochhammer-Chree equations. We found that the truncated Painleve´ analysis method failed to produce a solution for the second of the two Pochhammer-Chree equations (equation (4.3))

151 but was successful in all other cases considered. On the other hand, the invariant Painleve´ analysis and generalized Hirota expansion methods successfully produced solutions in all cases considered. Plots of the derived solutions were given for various choices of the arbitrary constants and system parameters involved.

Future work will be centered around the results presented in part 1 of this dissertation. Due to the algorithmic nature of the extended Estabrook-Wahlquist method the natural question of whether the whole procedure can be programmed arises. Although the results in part 1 were derived partially by hand, the entire process (or at least up to derivation of the determining equations for the Xi) can be programmed in a computer algebra system such as MAPLE or MATHEMATICA.

As a ﬁrst possible direction for future research we would like to look into developing a program (for MAPLE or MATHEMATICA) which, given a variable-coefﬁcient nlpde, would carry out the extended Estabrook-Wahlquist procedure.

Many PDEs which are integrable by this deﬁnition have been shown to possess a variety of other interesting properties, e.g. the existence of inﬁnitely many conserved quantities, a biHamiltonian representation, solvability by the Inverse Scattering Transform, etc..

It would be of interest to study these new extended systems to determine if they share the same properties common to integrable systems as their constant coefﬁcient predecessors.

As these generalized systems contain as limiting subcases the constant coefﬁcient equations from which they stem their study may lead to interesting results such as generalized biHamiltonian structures which in turn could determine hierarchies of variable-coefﬁcient nonlinear PDEs.

In [?]- [?] we considered variable-coefﬁcient extensions to the KdV, mKdV, NLS, DNLS, and PT- symmetric NLS equations. We have derived extensions to only a few of the many known PDEs in the ﬁeld of integrable equations.

152 As a third direction we may utilize the extended Estabrook-Wahlquist method to derive integrable extensions to many other known systems thus broadening the ﬁeld.

153 LIST OF REFERENCES

[1] M. J. Ablowitz and H. Segur, Solitons and the IST (SIAM, Philadelphia, 1981).

[2] P. G. Drazin and R. S. Johnson, Solitons: an introduction (Cambridge U. Press, Cambridge,

1989.

[3] M. J. Ablowitz and Z. Musslimani, Integrable Nonlocal Nonlinear Schrodinger Equation,

Physical Review Letters 110, (2013), 064105 (4 pages).

[4] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Stud. Appl. Math., 53 (1974) 249-315.

[5] V.E. Zakharov and A.B. Shabat, Functional Analysis and Its Applications, 8 (3): (1974) 226-

235

[6] R. Grimshaw and S. Pudjaprasetya, Stud. Appl. Math., 112 (2004) 271.

[7] G Ei, R. Grimshaw, and A. Kamchatov, J. Fluid Mech., 585 (2007) 213.

[8] G. Das Sharma and M. Sarma, Phys. Plasmas 7 (2000) 3964.

[9] E. G. Fan, Phys. Lett A294 (2002), 26.

[10] R. Grimshaw, Proc. Roy. Soc., A368 (1979) 359.

[11] N. Joshi, Phys. Lett., A125 (1987), 456.

[12] Y. C. Zhang, J. Phys. A, 39 (2006) 14353.

[13] E. G. Fan, Phys. Lett A375 (2011), 493.

[14] X. Hu and Y. Chen, J. Nonlin. Math. Phys., 19 (2012) 1250002

[15] S. F. Tian and H. Q. Zhang, Stud. Appl. Math., doi 10.1111/sapm.12026, 13 pages (2013).

154 [16] S. K. Suslov, arXiv: 1012.3661v3 [math-ph].

[17] Z. Y Sun, Y. T. Gao, Y. Liu and X. Yu, Phys.Rev E84 (2011) 026606.

[18] J. He and Y. Li, Stud. Appl. Math., 126 (2010), 1.

[19] U. Al Khawaja, J. Phys. A: Math. Gen. 39 (2006) 9679 9691.

[20] U. Al Khawaja, Journal of Mathematical Physics, 51(2010), 053506.

[21] M. Russo and S. Roy Choudhury, J. of Phys. Conf. Series 2014: 482(1): 012038. doi

10.1088/1742-6596/482/1/012038.

[22] M. Russo and S.R. Choudhury, Integrable Spatiotemporally Varying KdV and MKdV: Gen- eralized Lax Pairs and an Extended Estabrook Method, accepted for publication in Quarterly

of Applied Mathematics

[23] S.R. Choudhury and M. Russo, The Extended Estabrook-Wahlquist method, accepted for

publication in Physica D: Nonlinear Phenomena

[24] M. Russo and S.R. Choudhury, Integrable spatiotemporally varying NLS, PT-symmetric NLS,

and DNLS equations: Generalized Lax pairs and Lie algebras, Submitted

[25] M. Russo and S.R. Choudhury, Integrable generalized KdV, mKdV, and nonlocal PT- symmetric NLS equations with spatiotemporally varying coefﬁcients, Submitted

[26] F. B. Estabrook and H. D. Wahlquist, J. Math. Phys., 16 (1975) 1; ibid: 17 (1976) 1293.

[27] R. K. Dodd and A. P. Fordy, Proc. Roy. Soc. London A385 (1983) 389.

[28] R. K. Dodd and A. P. Fordy, J. Phys. A17 (1984) 3249.

[29] D. J. Kaup, Physica D1 (1980) 391.

155 [30] X. Yu, Y.-T. Gao, Z.-Y. Sun, and Y. Liu, N-soliton solutions, Baklund transformation and Lax

pair for a generalized variable-coefﬁcient ﬁfth-order Korteweg Vries equation, Phys. Scr. 81 (2010), 045402

[31] X. Yong, H. Wang, and Y. Zhang, Symmetry, Intregrability and Exact Solutions of a Variable-

Coefﬁcient Korteweg-de-Vries (vcKDV) Equation, International Journal of Nonlinear Sci- ence 14 (2012)

[32] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation. Physica D 95 (1996) 229-243.

[33] A. Fokas, On a Class of physically important integrable equations. Physica D 87 (1995) 145-

150.

[34] A. Salupere, K. Tamm, J. Engelbrecht and P. Peterson, Interaction of solitary deformation

waves in microstructured solids, Proc. 5th. Intl. Conf. on Nonlin. Wave Equations, Athens, Georgia, April 2007.

[35] J. Engelbrecht and F. Pastrone, Waves in microstructured solids with nonlinearities in mi-

croscale, Proc. Estonian Acad. Sci. Phy. Math., 52 (2003), 12-20.

[36] J. Engelbrecht, A. Berezovski, F. Pastroneand M. Braun, Waves in microstructured materials and dispersion, Phil. Mag., 85 (2005), 4127.

[37] J. Janno and J. Engelbrecht, Solitary waves in nonlinear microstructured materials, J. Phys. A, 38 (2005), 5159.

[38] J. Janno and J. Engelbrecht, And inverse solitary wave problem related to microstructured

materials, Inv. Problems, 21 (2005), 2019.

[39] I. L. Bogolubsky, Some examples of inelsatic soliton interaction. Comp. Phys., 31 (1977) 149-155.

156 [40] R. J. LeVeque, P. A. Clarkson and R. Saxton, Solitary wave interactions in elastic rods, Stud.

Appl. Math., 75 (1986) 95-122.

[41] R. Saxton, Existence of solutions for a ﬁnite nonlinearly hyperelastic rod, J. Math. Anal. Appl., 105 (1985) 59 - 75.

[42] Z. Weiguo and W. X. Ma, Explicit solitary wave solutions to generalized Pochhammer-Chree equations, Appl. Math. and Mech., 20 (1999) 666-674.

[43] L. Pochhammer, Biegung des KreiscylindersFortpﬂanzungs-Geschwindigkeit kleiner

Schwingungen in einem Kreiscylinder, Journal fur¨ die Reine und Angewandte Mathematik, vol. 81, p. 326, 1876.

[44] C. Chree, Longitudinal vibrations of a circular bar, Quarterly Journal of Mathematics, vol. 21, pp. 287298, 1886.

[45] J. Weiss, M. Tabor and G. Carnevale, The Painleve´ property for partial differential equations,

J. Math. Phys. 24 (1983), 522.

[46] R. Conte, Invariant painleve´ analysis of partial differential equations, Phyics Letters A, 140 (1989) 383-390.

[47] M. Russo, R. Van Gorder, S.R. Choudhury, Communications in Nonlinear Science and Nu- merical Simulation 18 (2013), no. 7, 1623-1634.

[48] M. Russo and S.R. Choudhury, Exact solutions of Pochammer-Chree and Boussinesq Equa-

tions by Invariant Painleve´ Analysis and Generalized Hirota Techniques, accepted for publication in Discontinuity, Nonlinearity, and Complexity

[49] M. Russo and S.R. Choudhury, Analytic Solutions of a Microstructure PDE and the KdV and

Kadomtsev-Petviashvili Equations by Invariant Painleve´ Analysis and Generalized Hirota Techniques, Submitted

157 [50] Y. T. Gao and B. Tian, Some two-dimensional and non-traveling-wave observable effects in

shallow water waves, Phys. Lett., A301 (2002), 74.

[51] B.B. Kadomtsev and V.I. Petviashvili, On the stability of solitary waves in weakly dispersive

media, Sov. Phys. Dokl. 15, 539-541 (1970).

[52] M. Hadac, S. Herr, and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, arXiv:0708.2011v3 (2009).

[53] J. Hammack, N. Scheffner, and H. Segur, Two-dimensional periodic waves in shallow water, J. Fluid Mech. 209, 567-589 (1989).

[54] G. Biondini and S. Chakravarty, Soliton solutions of the Kadomtsev-Petviashvili II equation, J. Math. Phys. 47, 033514:1-26 (2006).

[55] J. Boussinesq, Essai sur la theorie des eaux courantes, Memoires presentes par divers savants,

l’Acad. des Sci. Inst. Nat. France, XXIII, pp. 1680 (1877).

[56] D.J. Korteweg, G. de Vries, On the Change of Form of Long Waves Advancing in a Rect-

angular Canal, and on a New Type of Long Stationary Waves, Philosophical Magazine 39 (240): 422443 (1895).

[57] N.J. Zabusky, M.D. Kruskal, Interaction of ’Solitons’ in a Collisionless Plasma and the Re-

currence of Initial States, Phys. Rev. Lett. 15 (6): 240243 (1965)

158