Computation of Scaling Invariant Lax Pairs with Applications to Conservation Laws

COMPUTATION OF SCALING INVARIANT LAX PAIRS WITH APPLICATIONS TO CONSERVATION LAWS

by Jacob Rezac

A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulﬁllment of the requirements for the degree of Master of Science (Mathematical and Computer Sciences).

Golden, Colorado Date

Signed: Jacob Rezac

Signed: Dr. Willy Hereman Thesis Advisor

Golden, Colorado Date

Signed: Dr. Willy Hereman Professor and Interim Department Head Department of Applied Mathematics and Statistics

ABSTRACT

There has been a large amount of research on methods for solving nonlinear partial differential equations (PDEs) since the 1950s. A completely integrable nonlinear PDE is one which admits solutions, given specific constraints. These integrable nonlinear PDEs can be associated with a system of linear PDEs through a compatibility condition. Such a system is called a Lax pair. While Lax pairs are very important in the theory of solving nonlinear PDEs, few methods exist to compute them. This The- sis presents a method for generating Lax pairs for a specific class of nonlinear PDEs. Conservation laws, well-known from physics, are a related concept. Indeed, the existence of an infinite number of conservation laws for a nonlinear PDE also predicts it complete integrability. We discuss a method for the computation of conservation laws, given a PDE’s Lax pair. This method is based on work done by Drinfel’d and Sokolov in 1985, which does not seem to have been fully explored in literature. The goal of this Thesis is to demonstrate the efficacy of these two construction methods. As such, we also list a number of Lax pairs and conservation laws computed by the methods presented in the Thesis.

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TABLE OF CONTENTS

ABSTRACT ...... iii

LIST OF TABLES ...... vi

ACKNOWLEDGMENTS ...... vii

CHAPTER 1 INTRODUCTION ...... 1

CHAPTER 2 PRELIMINARIES ...... 5

2.1 Lax Pairs ...... 5

2.2 AKNS Scheme ...... 11

2.3 Gauge Transformations ...... 16

2.4 Conservation Laws ...... 21

CHAPTER 3 CONSTRUCTION OF LAX PAIRS ...... 25

3.1 Scaling Invariance ...... 25

3.2 Exhaustive Lax Pair Computation Methods ...... 28

3.3 Uniform Weight Construction ...... 34

3.4 Weak Lax Pairs and Triviality Concerns ...... 39

CHAPTER 4 THE CONSTRUCTION OF CONSERVATION LAWS FROM LAX PAIRS ...... 45

4.1 The Drinfel’d Sokolov Method for Computing Conservation Laws . . . 45

4.2 Triviality and Simpliﬁcation ...... 51

CHAPTER 5 FURTHER EXAMPLES ...... 55

5.1 Lax Pairs Computed with Exhaustive Methods ...... 55

5.1.1 Modiﬁed Korteweg-de Vries Equation ...... 55

5.1.2 Kaup-Kuperschmidt Equation ...... 56

5.1.3 Lax 5th-Order Equation ...... 57

5.1.4 Sawada-Kotera Equation ...... 57

5.1.5 Harry-Dym Equation ...... 58

5.1.6 Drinfel’d-Sokolov System ...... 59

5.2 Lax Pairs Computed by Uniform Weight Method ...... 59

5.2.1 Nonlinear Schr¨odingerEquation ...... 59

5.2.2 Gardner Equation ...... 60

5.3 Conservation Laws Produced by the Drinfel’d-Sokolov Method . . . . . 61

5.3.1 Korteweg-de Vries Equation ...... 61

5.3.2 Modiﬁed Korteweg-de Vries Equation ...... 61

5.3.3 Sine-Gordon Equation ...... 62

5.3.4 Sinh-Gordon Equation ...... 63

5.3.5 Nonlinear Schr¨odingerEquation ...... 63

CHAPTER 6 CONCLUSIONS AND FURTHER RESEARCH ...... 67

REFERENCES CITED ...... 69

APPENDIX - OUTLINE OF DRINFEL’D-SOKOLOV ALGORITHM AND USE OF MATHEMATICA ...... 73

LIST OF TABLES

Table 5.1 The ﬁrst four conservation laws for the KdV equation (2.4). . . . . 61

Table 5.2 The ﬁrst four conservation laws for the mKdV equation (2.31). . . 62

Table 5.3 The ﬁrst four conservation laws for the sine-Gordon equation (5.4). 63

Table 5.4 The ﬁrst four conservation laws for the sinh-Gordon equation (5.5). 64

Table 5.5 The ﬁrst four conservation laws for the NLS equation (5.2). . . . . 65

ACKNOWLEDGMENTS

Thanks to my Thesis advisor, Willy Hereman, for his countless hours of advice (mathematical and otherwise), for his friendship, and for constantly pushing me to be a better mathematician. Thanks to the Department of Applied Math and Statistics at the Colorado School of Mines, including all the faculty and staff who have helped me during my time here. In particular, thanks to my Thesis committee Drs. Paul Martin and Luis Tenorio, both of whom have taught me more math than I care to admit. Also thanks to Dr. Mark Hickman, whose brief sabbatical in Golden put the following work on a surer mathematical footing than could have occurred without him. Further thanks to the many students in my research group the last three years, Terry Bridgman, Jennifer Larue, Tony McCollom, Janeen Neri, Sara Clifton, Oscar Aguilar, Jon Tran, and Allen Voltz. Their work ethic, insightful ideas, and pointed questions all helped push this Thesis to completion. This research was partially funded by NSF research award no. CCF-0830783. Finally, thanks to my family and friends, who both distracted and encouraged me enough to complete this document, whether they realize it or not. Also to Golden’s numerous fine coffee shops (in particular Higher Grounds), which kept me caffeinated and gave me a place to work whenever I needed one.

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CHAPTER 1 INTRODUCTION

In the early 1950s, physicists Fermi, Pasta, and Ulam (FPU) investigated [1] an apparent paradox related to the field of statistical mechanics1. The research group was attempting to model wave motion in a nonlinear crystal lattice. Scientific under- standing at the time predicted that energy put into the system would become equally partitioned across the crystal lattice. In an unexpected turn of events, however, numerical experiments showed that nearly all of the energy put into the crystal would remain in its original position; e.g., if the energy were initially put into the lowest vibrational mode of a wave, all but 3% of the energy would remain in that same mode [1]. This problem plagued physicists for almost 10 years. In 1965, however, Zabusky and Kruskal [3] explored the problem in its continuous limit. Numerical experiments conducted by Zabusky and Kruskal revealed surprising interactions between solutions of the continuous FPU model: solitary wave solutions passed through each other, emerging with no changes in velocity, amplitude, or shape. Although the problem was nonlinear in nature, wave solutions interacted as though they were linear and obeyed the superposition principle. This discovery hinted at a solution to the FPU paradox. In the continuous limit, these non-interacting waves were analogous to energy not mixing in different wave modes of a nonlinear lattice. Zabusky and Kruskal called these waves solitons2 to emphasize both the solitary nature of the waves and the similarities these waves shared with particles (e.g., the electron or positron).

1In recent years [2] it has been argued that Tsingou should also be credited with these inves- tigations. Most of the numerical computations presented in the paper attributed to FPU were, in fact, done by Tsingou. 2Zabusky originally planned to call these waves solitrons. Solitron, however, was already the name of a manufacturing company (which still exists today) [4].

1 It was soon noticed that the equation studied by Zabusky and Kruskal was, in fact, a nonlinear partial differential equation (PDE) discovered much earlier in the context of water waves. This so-called Korteweg-de Vries (KdV) equation was derived in 1895, by Dutch mathematicians Korteweg and de Vries, to describe solitary wave behavior observed by naval engineer John Scott Russell in 1834 [5]. Soon, other equations with soliton solutions were found. The modified KdV (mKdV) for example, which is related to the KdV equation by a simple transformation, was investigated by the Zabusky-Kruskal team [3], as well as Miura [6], in the mid-’60s. The sine- Gordon equation, which had been known by plasma physicists for some time, was soon discovered [7] to have soliton solutions as well. As more and more equations of this type were uncovered, the study of exact solutions to nonlinear PDEs exhibiting soliton behavior quickly became a subject important to mathematicians. In a series of papers [6, 8] published in the 1960s and ‘70s, Gardner, Greene, Kruskal, and Miura (GGKM) found analytic solutions to describe the interaction of waves governed by the KdV equation. Importantly, GGKM developed a general method to find these solutions. The papers also describe other important analytic properties of the KdV equation and its solutions, such as conservation laws. In 1968, Lax proposed a formal technique [9] for finding soliton solutions to nonlinear evolution equations based on work done by GGKM. His technique involved relating the original nonlinear PDE to two linear operators via a compatibility condition. These linear operators, called a Lax pair, are the main topic of this Thesis. The work done by Lax was further generalized by Zakharov and Shabat [10] and Ablowitz, Kaup, Newell, and Segur (AKNS) [11]. Soliton solutions were found for more complicated nonlinear PDEs, such as the Nonlinear Schrödingerequation (NLS), and Lax pairs were again required for the solutions. The solution technique created by these researchers is called the Inverse Scattering Transform (IST). It has been very successful over the last 40 years, and has been called one of the most important techniques

2 developed in applied mathematics over that time. During the time-period Lax pairs were first being discovered, mathematicians be- gan to examine conditions under which soliton waves could occur. Many nonlinear waves which had been studied before the discovery of solitons had exhibited “shock wave” solutions, requiring jump boundary conditions and conservation law formu- lations. Applying this analogy to the new nonlinear waves, Miura [6], along with Kruskal, Zabusky, and Whitham [9], discovered nine conservation laws related to the KdV equation in the late 1960s. The first few conservation laws correspond to physical conserved quantities of the system - conservation of mass, momentum, and energy [12]. The rest, however, have no obvious physical meaning. More and more conservation laws were soon found, and it was proven that an infinite number existed. It was soon found that most PDEs having the property of infinite conservation laws also have a Lax pair and analytic soliton solutions [13]. Conservation laws, and their relationship to Lax pairs, are another subject discussed in this Thesis. The connections between exact solutions to a PDE and the existence of either Lax pairs or an infinite number of conservation laws are well-studied [13]. In the years since these properties were first discovered, there have been many attempts to algorithmically compute Lax pairs and conservation laws for a given PDE. In particular, the computation of conservation laws has been successful [12,14]. However, the algorithmic construction of Lax pairs has received much less attention [15]. This The- sis will outline algorithmic methods for the computation of both. The methods will require stringent properties for the PDEs being examined, but many of the equations important in nonlinear physics have these properties. The Thesis will proceed as follows: Chapter 2 will discuss the technical aspects of Lax pairs, conservation laws, and a related concept, gauge transformations. This Chapter will cover the theory and give concrete examples to further clarify the concepts. Chapter 3 will present two related methods for the computation of Lax pairs.

3 These methods are loosely based on past research, but have not appeared in literature. Particular attention is paid to a type of triviality often ignored by those researching Lax pairs [16]. Chapter 4 will discuss the computation of conservation laws from Lax pairs. Speciﬁcally, it will discuss a method based on work by Drinfel’d and Sokolov [17] from which conservation laws can be constructed algorithmically. Finally, Chapter 5 will list a number of Lax pairs and conservation laws constructed by the methods described in Chapters 3 and 4. Many of the Lax pairs are more general than those published in literature. The conservation laws listed in this chapter are compared to published results.

4 CHAPTER 2 PRELIMINARIES

The attempts in the 1960s and ’70s to create a general theory for solving nonlinear PDEs led to the discovery of a number of properties related to exactly solvable equations. Some of these properties will be discussed in this chapter and a number of examples will be given to further elucidate concepts.

2.1 Lax Pairs

Perhaps the most basic nonlinear PDE studied in applied mathematics is Burgers’ equation [18],

ut + uux − κuxx = 0. (2.1)

Here, and throughout this document, u(x, t) is a dependent variable, and x and t are spatial and temporal independent variables, respectively. Moreover, subscripts denote partial derivatives. Equation (2.1) developed as a simplification of the Navier- Stokes equations [19] has applications ranging from fluid flow to optical tomography and traffic flow [20,21]. A solution to (2.1) can be found by the so called Hopf-Cole3 transformation [23,24], which linearizes (2.1) into the (linear) heat equation,

H(v) = vt − κvxx = 0. (2.2)

In particular, the change of variables governed by ∂ ln v v u = −2κ = −2κ x (2.3) ∂x v reduces (2.1) to

2κ ∂ v − v H(v) = 0. v2 x ∂x

3While Hopf and Cole independently discovered this transformation in the early 1950s [18], it appears as though Florin found the same solution ﬁrst while studying properties of soil mechanics as early as 1948 [22].

5 ∂ ln v That is, if v solves (2.2), then u = −2κ ∂x solves (2.1). This Hopf-Cole solution to Burgers’ equation suggests a technique for solving nonlinear PDEs:

1. Find a transformation which changes a given nonlinear PDE into a linear one. 2. Solve the new linear equation. 3. Transform back to ﬁnd solutions to the original nonlinear problem.

This is exactly what the IST attempts to do, though in a much more complicated way. Of particular interest to the rest of this Thesis is step 1 - ﬁnding transformations. This process of ﬁnding transformations such as (2.3) will be generalized and made algorithmic in the sections that follow. To develop a linearization technique for nonlinear PDEs, we consider the KdV equation [5],

ut + 6uux + uxxx = 0, (2.4) for u(x, t). Note that the 6 in front of the uux-term can be scaled to any constant. The similarity between (2.4) and (2.1) suggests a linearizing transformation akin to the Hopf-Cole transform, (2.3). Indeed, consider the transformation ψ u = − xx , ψ for a new dependent variable ψ(x, t). Rewriting yields

ψxx + uψ = 0. (2.5)

Moreover, the KdV equation is invariant under a Galilean transformation [7], u → u + κ, for some constant κ. So, for some constant λ, we can write (2.5) as

ψxx − (λ − u)ψ = 0. (2.6)

Equation (2.6) is a well-known Sturm-Liouville equation, related to the Schr¨odinger

∂2 equation of quantum physics [7]. We then deﬁne the linear operator L = ∂x2 + uI by

6 Lψ = λψ. (2.7)

Note that I in the definition of L is the identity operator. While this sort of transformation specified the solution for Burgers’ equation, it is unlikely to do so for more complicated PDEs. To allow for additional flexibility, also consider an operator M to describe the time evolution of the eigenfunction ψ,

Mψ = ψt. (2.8)

4 Assuming λt = 0, consider the t-derivative of (2.7),

Ltψ + Lψt = λψt.

Substituting in from (2.7) and (2.8) gives that

Ltψ + L(Mψ) = λ (Mψ) = M(λψ) = MLψ,

(Lt + [L, M])ψ = 0,

where [L, M] = LM −ML is the commutator of L and M. Continuing, we can readily verify [15] that

3 Mψ = −4ψx − 6uψx − 3uxψ

ﬁts the role of (2.8). The preceding process suggests a general formulation for ﬁnding linear PDEs compatible with a nonlinear PDE. Let a pair of operators L and M be such that

Lψ = λψ and Mψ = ψt and, consequently,

(Lt + [L, M])ψ ˙=0, (2.9)

where ˙=means equal on solutions of the nonlinear PDE. Then L and M are known as a scalar operator Lax pair of the PDE. Moreover, (2.9) is called the operator Lax

4This assumption is related to solution theory regarding the linear Schr¨odingerequation and is required [5] for the IST to work.

7 equation. Example 1 Substituting the pair of operators

∂2 ∂3 ∂ L = + uI and M = −4 − 6u − 3u I. ∂x2 ∂x3 ∂x x into the Lax equation, (2.9), yields

(Lt + [L, M]) ψ = −(ut + 6uux + uxxx)ψ.

So, on solutions of the KdV equation (2.4), we see that the operators L and M are a Lax pair compatible with the KdV equation. The concept of the Lax pair was expanded further by both the Zakharov-Shabat team and the AKNS research group in the 1970s. Rather than searching for L and M operators consistent with a PDE, they considered n × n-matrices. In particular, they considered matrices X = X(λ, u, ux, uxx,...) and T = T (λ, u, ux, uxx,...) such that

XΦ = Φx and T Φ = Φt, (2.10) where Φ is a vector of eigenfunctions. Indeed, as X and T depended on the spectral parameter λ, AKNS and Zakharov and Shabat considered a more general scheme than that considered in the operator formalism discussed above. Note that, again, we required λt = 0. Following a similar technique as above, a compatibility condition relating X, T , and the original nonlinear PDE can be derived. Requiring Φxt = Φtx gives that

Φxt = XtΦ + XΦt = TxΦ + T Φx = Φtx.

Simplifying this and replacing Φt and Φx with (2.10) gives

(Xt − Tx + [X,T ])Φ = 0, where [X,T ] is the standard commutator,

[X,T ] = XT − TX.

8 Thus, we take

Xt − Tx + [X,T ] ˙=0 (2.11) to be the compatibility condition for a matrix Lax pair X and T . Here, 0 means the n × n zero-matrix (in the remainder of this Thesis, the distinction between the zero matrix and zero scalar will be clear from context). Equation (2.11) is called the matrix form Lax equation (or zero-curvature condition). Note that trivial pairs are avoided again by requiring compatibility with solutions of the PDE. The matrix formulation of the Lax equation will be used more commonly throughout the Thesis, so we say “Lax pair” to mean matrix form of the Lax pair. Example 2 Consider again the KdV equation, (2.4). Moreover, let Φ = [ψ φ]T be a vector-valued function such that ψ and φ are functions of u, ux, and higher- order x−derivatives of u. This type of dependence is said to be on the jet space of u. Then, we want to ﬁnd a matrix X which satisﬁes XΦ = Φx and a matrix T such that T Φ = Φt. If such X and T exist, there must be a functional relationship

5 between ψ, φ, ψx, and φx. So, we make a simple guess that φ = ψx. We assume φ and ψ are suﬃciently smooth so that φx = ψxx. Furthermore, we know from (2.6) that ψxx = λψ − uψ. Thus, we can replace ψxx with lower-order terms, giving

φx = λψ − uψ. Thus, there is enough information to construct a matrix X,

0 1 X = , λ − u 0 which satisfies Φx = XΦ. Finding the T matrix is not as immediate as finding X. The process, however, is the same. Assume again that φ = ψx. Then, again assuming that φ and ψ are sufficiently smooth, φt = (ψx)t = (ψt)x. However, we know that ψt = Mψ =

−4ψxxx − 6uψx − 3uxψ. So,

5This guess is motivated by the way second-order ordinary diﬀerential equations can be recast into matrix form.

9 φt = (−4ψxxx − 6uψx − 3uxψ)x.

This can be further reduced with the relations ψx = φ, ψxx = λψ − uψ, and ψxxx =

λψx − uxψ − uψx, yielding

2 2 φt = (−4λ + 2λu + 2u + uxx)ψ − uxφ.

Similarly,

ψt = −4ψxxx − 6uψx − 3uxψ

= −4(λφ − uφ − uxψ) − 6uφ − 3uxψ

= uxψ − (4λ + 2u)φ.

We have accomplished the goal of this example: φt and ψt are expressed in terms of φ and ψ after all x-derivatives of these functions have been removed. Thus, since

T Φt = [ψt φt] , we have computed the matrix T as ux −4λ − 2u T = 2 2 . −4λ + 2λu + 2u + uxx −ux A direct computation shows that these X and T satisfy the matrix Lax pair equation along with the KdV equation, (2.4). Indeed, after some algebra, substituting X and T into (2.11) gives

0 0 Xt − Tx + [X,T ] = − , ut + 6uux + uxxx 0 and substitution of (2.4) will give the zero-matrix. The above matrices can be generalized to generic α in the KdV equation,

ut + αuux + uxxx = 0. (2.12)

In this case,

0 1 X = 1 , (2.13) λ − 6 αu 0 and

10 1 1 6 αux −4λ − 3 αu T = 2 1 1 2 2 1 1 . (2.14) −4λ + 3 αλu + 18 α u + 6 αuxx − 6 αux Notice that substituting α = 6 into (2.13) and (2.14) generates the matrices found in Example 2, as expected.

2.2 AKNS Scheme

Finding a Lax pair for the KdV equation was somewhat intuitive. Other equations, however, do not produce compatible pairs so easily. Of particular interest, then, is work done by the AKNS group to construct integrable equations from classes of compatible Lax pairs. They took a reverse route: rather than constructing a pair for a given equation, these researchers started from general Lax pairs and tried to fit them to specific equations. In the simplified version of the AKNS method, only 2 × 2 matrices are considered (for a review of higher-dimensional matrices, see [25]). We will do the same. So, begin by assuming a general form of Lax matrices such that

−iκ q X = , (2.15) r iκ where κ is an eigenvalue and q(x, t) and r(x, t) are undetermined functions. The form of X comes from a generalized approach to the IST taken in [10]. For reasons which will become clear in section 2.3, we take κ to be the spectral parameter, rather than λ. Moreover, consider a very general T matrix,

AB T = , (2.16) CD where A, B, C and D are general functions of κ, and the jet space of u. Then, keeping in mind that the Lax equation is equivalent to XΦ = Φx and T Φ = Φt, where Φ = [ψ φ]T , we have

ψx = −iκψ + qφ, and ψt = Aψ + Bφ, (2.17)

φx = rψ + iκφ, φt = Cψ + Dφ. (2.18)

11 If we require symmetry of second-derivatives, ψxt = ψtx and φxt = φtx, and that

κt = 0, we ﬁnd

qtφ + qφt − iκψt = Axψ + Aψx + Bxφ + Bφx

rtψ + rψt + iκφt = Cxψ + Cψx + Dxφ + Dφx.

Replacing ψx, φx, ψt, and φt from their deﬁnition in (2.17) and simplifying yields

(Ax − iAκ + rB)ψ + (qA + Bx + iBκ)φ = (qC − iAκ)ψ + (qt − iBκ + qD)φ

(Cx − iCκ + rD)φ + (qC + Dx + iDκ)φ = (rt + rA + iCκ)ψ + (rB + iDκ)φ.

So, requiring that φ and ψ be non-zero, we ﬁnd the system of equations

Ax = qC − rB,

Bx + 2iκB = qt − (A − D)q,

Cx − 2iκC = rt + (A − D)r,

−Dx = qC − rB.

We can reduce this system of differential equations, without loss of generality [5], by taking D = −A. While there are a number of ways to find solutions to this system, we consider expanding A, B, and C into power series in κ (other authors e.g., [26] expand A, B, and C differently). In doing so, certain evolution equations will be found to hold on q or r, which will, in turn, determine the PDEs with which the Lax pairs are compatible. An example clarifies this procedure. Example 3 Expand A, B, and C up to the third power of κ,

2 3 A = A0 + A1κ + A2κ + A3κ , 2 3 B = B0 + B1κ + B2κ + B3κ , 2 3 C = C0 + C1κ + C2κ + C3κ ,

12 where Ai,Bi and Ci are functions on the jet space of q and r. While the reason behind expanding to the third order in κ is not clear a priori, we will ﬁnd that this expansion produces useful results6. Then, substituting this expansion into (2.11) and simplifying, gives

2 0 = (rB0 − qC0 + (A0)x) + (rB1 − qC1 + (A1)x) κ + (rB2 − qC2 + (A2)x) κ 3 + (rB3 − qC3 + (A3)x) κ , 2 qt = (2qA0 + (B0)x) + (2(iB0 + qA1) + (B1)x) κ + (2(iB1 + qA2) + (B2)x) κ 3 4 + (2(iB2 + qA3) + (B3)x) κ + 2iB3κ , 2 −rt = (2rA0 − (C0)x) + (2(iC0 + rA1) − (C1)x) κ + (2(iC1 + rA2) − (C2)x) κ 3 4 + (2(iC2 + rA3) − (C3)x) κ + 2iC3κ .

We hope to solve for A, B, C, q, and r which specify a Lax pair associated with a PDE. So, splitting these equations in powers of κ yields

0 κ : rB0 − qC0 + (A0)x = 0, qt = 2qA0 + (B0)x , −rt = 2rA0 − (C0)x , 1 κ : rB1 − qC1 + (A1)x = 0, 2(iB0 + qA1) + (B1)x = 0, 2(iC0 + rA1) − (C1)x = 0, 2 κ : rB2 − qC2 + (A2)x = 0, 2(iB1 + qA2) + (B2)x = 0, 2(iC1 + rA2) − (C2)x = 0, 3 κ : rB3 − qC3 + (A3)x = 0, 2(iB2 + qA3) + (B3)x = 0, 2(iC2 + rA3) − (C3)x = 0, 4 κ : 2iB3 = 0, 2iC3 = 0.

4 From the equations at order κ , we see that B3 = C3 = 0. Substituting B3 = C3 = 0

3 into the equations at order κ will give A3,x = 0, or A3 = a3, where a3 is some constant. Hence, the equations at order k3 reduce to

iB2 + a3q = 0 and iC2 + a3r = 0.

Consequently, B2 = ia3q and C2 = ia3r. Substituting these into the equations at

2 order κ gives A2,x = 0, or A2 = a2, for some constant a2. Then,

ia3qx + 2(iB1 + a2q) = 0 and ia3rx − 2(iC1 + a2r) = 0.

1 1 Thus, B1 = ia2q − 2 a3qx and C1 = ia2r + 2 a3rx.

6In general, there is a lack of motivation in the order of power series expansion. This is one drawback to the AKNS method.

13 1 We continue by solving the equations at order κ . These determine that (A1)x = 1 1 2 a3(qrx + qxr), so that A1 = 2 (a3qr + a1), for some constant a1. Then, from the remaining equations at order κ1,

1 1 2iB + a q + ia q + a q2r − q = 0, 0 2 1 2 x 3 2 xx and

1 1 2iC + a r − ia r + a qr2 − r = 0. 0 2 1 2 x 3 2 xx Solving gives

1 1 B = ia q − a q + ia (q2r − q ) , 0 2 1 2 x 3 2 xx and

1 1 C = ia r + a r + ia (qr2 − r ) . 0 2 1 2 x 3 2 xx Finally, from the equations at order κ0, we see

1 1 (A ) = a (q r + qr ) + ia (q r − qr ) , 0 x 2 2 x x 4 3 xx xx so 1 1 A = a + a qr + ia (q r − qr ) , 0 0 2 2 4 3 x x

0 where a0 is some constant. Moreover, the remaining equations at order κ specify q and r as evolution equations. Speciﬁcally,

1 1 1 1 q = 2a q + ia q + a q2r + q + ia 3qq r − q , (2.19) t 0 2 1 x 2 2 xx 2 3 x 2 xxx and

1 1 1 r = −2a r + ia r + a qr2 + r + ia (3qrr − r ) . (2.20) t 0 2 1 x 2 2 xx 2 3 x xxx

Substituting a0 = a1 = a2 = 0, a3 = −4i, and r = −1 into (2.19) and (2.20) yields

14 qt + 6qqx + qxxx = 0,

the KdV equation. The goal of this exercise was to represent T as a cubic polynomial on the jet space of u. From the above method, we have found q, r, A, B, and C which form a valid Lax pair when substituted into (2.15) and (2.16). Thus, we have constructed a Lax pair for the KdV equation,

−iκ q X = , (2.21) AKNS −1 iκ and

3 2 2 −4iκ + 2iqκ − qx 4qκ + 2iqxκ − 2q − qxx TAKNS = 2 3 . (2.22) −4κ + 2q 4iκ − 2iqκ + qx Substituting this back into the Lax equation (2.11) yields

0 q + 6uu + u (X ) − (T ) + [X ,T ] = t x xxx , AKNS t AKNS x AKNS AKNS 0 0 so the pair is indeed valid. A number of other Lax pairs can be found this way. Important examples include the NLS equation, which comes from expanding A, B and C as second-order polyno- mials and the sine-Gordon equation, which comes from expanding in inverse powers of κ. For a larger list of these Lax pairs, as well as a more detailed version of the above discussion, see [5]. The AKNS scheme was successful in helping to classify integrable systems. It also broadened the form of Lax pairs mathematicians were willing to use. However, the whole approach had one major flaw: if a researcher had an equation which needed to be solved, the scheme would not necessarily produce a Lax pair for it. Attempts to fix the problem require much creativity on the part of a researcher, and were not guaranteed to work [27]. Moreover, some require techniques and language difficult for non-experts to interpret [28]. The methods described in Chapter 3 are an attempt to fix this. Before considering new computation techniques, however, we discuss two

15 more important properties of integrable PDEs.

2.3 Gauge Transformations

At first glance, the above sections seem to be surprising, if not contradictory. In particular, two separate Lax pairs have been given for the same equation. Given that both of these pairs are valid, some questions arise: is there a relationship between these two pairs? Is one pair better than the other? If so, can one pair easily be trans- formed into the more useful pair? The first of these questions will be addressed by the following section on gauge transformations. The other questions will be addressed in the following Chapters. Since we know that more than one matrix Lax pair exists for the KdV equation, we hypothesize that this is true in general. In particular, we look at the vector Φ used in the definition of X and T , (2.10); if a different Φ, say Φˆ is used, it seems likely that a different X and T , say Xˆ and Tˆ will be required. Thus, consider a nonsingular matrix G such that Φˆ = GΦ, where Φ = [ψ φ]T , as before. Then, we can relate X and Xˆ by noting that ˆ −1 ˆ ˆ ˆ Φ = G (Φ)x + GxΦ = (GX + Gx)Φ = (GX + Gx)G Φ = XΦ. x Rewriting this, we see that

ˆ ˆ −1 −1 XΦ = (GXG + GxG )Φ. (2.23)

Similarly, we ﬁnd that

ˆ ˆ −1 −1 T Φ = (GT G + GtG )Φ. (2.24)

Thus, for any pair of matrices which satisfy (2.11), an inﬁnite number of equivalent pairs may be found through a gauge transformation of the form (2.23)-(2.24). In general, constructing a gauge between two given Lax pairs (or, more generally, from one known Lax pair to a pair of a given form) is nontrivial. In the simplest cases, however these gauge transformations are fairly straightforward to create.

16 Example 4 Consider again the two Lax pairs found for the KdV equation. Let

0 1 ux −4λ − 2u X = ,T = 2 2 , λ − u 0 −4λ + 2uλ + uxx + 2u −ux

and

3 2 2 ˆ −iκ u ˆ −4iκ + 2iuκ − ux 4uκ + 2iuxκ − 2u − uxx X = , T = 2 3 . −1 iκ −4κ + 2u 4iκ − 2iuκ + ux

Rewriting equations (2.23) and (2.24), we need to ﬁnd a nonsingular matrix G such that

ˆ ˆ XG = GX + Gx and TG = GT + Gt.

Because both X and Xˆ are linear in u, let us assume that G is a matrix with only constant entries g11, g12, g21, g22,

g g G = 11 12 . (2.25) g21 g22

Since Gx = Gt = 0, this signiﬁcantly reduces the gauge relationship. So, we have

−ig κ − g u −ig κ + g u XGˆ = 11 21 12 22 , ig21κ − g11 ig22κ − g12 and

g λ − g u g GX = 12 12 11 . g22λ − g22u g21 These must match element-wise, which yields

g11 = −ig12κ + g22u, (2.26)

g21 = ig22κ − g12, (2.27)

ig11κ − g21u = −g12λ + g12u, (2.28)

ig21κ − g11 = g22λ − g22u. (2.29)

Substituting g11 and g21 from (2.26)-(2.27) into (2.28)-(2.29) yields

17 2 g12(κ + λ) = 0, 2 g22(κ + λ) = 0.

If we set λ = −κ2, this is satisﬁed. Thus, if G is nonsingular, we have that

−ig κ + g u g G = 12 22 12 , (2.30) ig22κ − g12 g22 and λ = −κ2, will relate X and Xˆ. Moreover,

2 2 det(G) = g12 − 2ig12g22κ + g22u, so as long as g12 or g22 is nonzero, G will be a legitimate gauge transformation.

However, g22 multiplies a u term in the (1, 1)-element of (2.30). Since we assumed that G was a constant matrix, we must set g22 = 0. This yields

−ig κ g G = 12 12 . −g12 0

To match the gauge transformation found in literature [15], take g12 = 1. This gives

−iκ 1 G = . −1 0 In fact, using this G gives

−iκ −κ2 − λ + u GXG−1 = . −1 iκ Substituting λ = −κ2 gives Xˆ, as expected. More complicated gauge transformations can also be found. In the next example, we consider a more complicated case, for which a constant gauge matrix cannot be found. Rather than turning (2.23) and (2.24) into a simple algebraic system, we will be forced to work with a system of diﬀerential equations. Example 5 Consider the mKdV equation [6],

2 ut + αu ux + uxxx = 0.

18 Common values for α are ±6 and ±1. In this example, we will consider the case α = −6,

2 ut − 6u ux + uxxx = 0. (2.31)

Consider the following two Lax pairs from the mKdV equation,

0 1 X = 2 (2.32) λ + u + ux 0 with

2 −2uux − uxx −4λ + 2u + 2ux T = 2 2 4 2 . −4λ − 2λu − 2λux + 2u + 4u ux − 2uuxx − uxxx 2uux + uxx (2.33)

This pair can be computed in the same way as were X1 and T1 from Example 4. A second pair [5] is

−ik u Xˆ = (2.34) u ik with

3 2 2 3 ˆ −4ik − 2iku 4k u + 2ikux + 2u − uxx T = 2 3 3 2 , (2.35) 4k u − 2ikux + 2u − uxx 4ik + 2iku

obtained through the AKNS-scheme. Now, we want to ﬁnd a nonsingular matrix G

ˆ −1 −1 such that X = GXG + GxG . We ﬁrst naively consider a constant gauge matrix, as in Example 4,

g g G = 11 12 . g21 g22 Equating terms element-wise as before, we get

g11 = −ig12κ + g22u,

g21 = ig22κ + g12u, 2 g12(λ + u + ux) = −ig11κ + g21u, 2 g22(λ + u + ux) = ig21κ + g11u.

19 Substituting g11 and g21 from the ﬁrst two equations in the above system into the last two equations gives

2 g12(λ + ux) = −g12κ , 2 g22(λ + ux) = −g22κ .

Assuming ux is non-constant, the only solution is g12 = g22 = 0. However, this would also mean that g11 = g21 = 0, leading to G = 0. Thus, the choice of a constant matrix is too limited, and must be modiﬁed. Because the sets of Lax pairs have u2 as their highest order, we assume G is linear in u and its ﬁrst derivative,

g (u, u ) g (u, u ) G = 11 x 12 x , (2.36) g21(u, ux) g22(u, ux)

for arbitrary functions g11, g12, g21, and g22. Then, substituting G into (2.23) gives the system of equations

ˆ XG − GX − Gx = 0.

ˆ Substituting X and X gives 4 equations in terms of g11, g12, g21, g22, u, and spectral parameters. Speciﬁcally, the following must be satisﬁed

ig12κ − g22u + g12,x + g11 = 0, (2.37)

ig22κ + g12u − g22,x − g21 = 0, (2.38) 2 ig11κ − g21u + g11,x + g12 λ + u + ux = 0, (2.39) 2 ig21κ + g11u − g21,x − g22 λ + u + ux = 0. (2.40)

Assume for the moment that g12 and g22 are non-constant. Say, e.g., they are

functions of u. Then, (2.37) and (2.38) require that g11 and g21 be quadratic functions of u. However, this violates the assumption that G is linear in u. Thus, we assume

that g12 and g22 are constants, say g12 = α and g22 = β. Substituting these values into (2.39) and (2.40) yields

20 2 ig11κ − g21u + g11,x + α(λ + u + ux) = 0, (2.41) 2 ig21κ + g11u − g21,x − β(λ + u + ux) = 0, (2.42)

while (2.37) and (2.38) give that g11 = βu − iαk and g21 = αu + iβk. Now replacing these values of g11 and g21 into the remaining equations and simplifying gives

2 α(κ + λ) + (β + α)ux = 0, 2 β(κ + λ) + (β + α)ux = 0.

One solution to this system is λ = −k2 and (α + β) = 0. That is, α = −β. Thus, the system (2.37)-(2.40) is satisﬁed if

2 g12 = −g22 = α, g11 = −α(iκ + u), g21 = α(−iκ + u), and λ = −κ .

Substituting the above into G gives

−α(iκ + u) α G = . α(−iκ + u) −α For α = −1, this simpliﬁes into

ik + u −1 . ik − u 1

2.4 Conservation Laws

The concept of a conservation law is well-known from physics. Conservation of mass, energy, and momentum are common phrases even among non-specialists. While this Thesis focuses primarily on Lax pairs, conservations law are also discussed. We introduce the concept now, in anticipation of what will follow in Chapters 4 and 5. In particular, a conservation law for a given PDE is an equation of the form

ρt + Jx ˙=0. (2.43)

21 The function ρ(x, t) is called a conserved density and the function J(x, t) is the associated flux. Both ρ and J are functions on the jet space of u. As before, ˙= means equality on solutions of the PDE. Earlier we saw that a PDE with a Lax pair could be solved exactly with the IST. Similarly, the existence of an infinite number of conservation laws predicts integrabil- ity7 [13]. Moreover, conservation laws can be used to find other properties of interest for a PDE. For example, the Miura transformation, which helped lead to the discovery of a Lax pair for the KdV equation, was discovered [7] in the context of conservation laws. Furthermore, the knowledge of a conservation law is often helpful for finding a numerical solution to a PDE. Thus, finding conserved quantities is an important topic. In section 4, we will discuss a construction technique based on Lax pairs. In this section, however, we will demonstrate the concept of conservation laws and their relationship to Lax pairs with an example. Example 6 Consider again the KdV equation, (2.4). It is well-known that there are an infinite amount of conservation laws related to the KdV equation [6]. The first two conservation laws are simple:

2 (u)t + 3u + uxx x ˙=0, 2 3 2 (u )t + 4u − ux + 2uuxx x ˙=0.

These are related to conservation of mass and momentum, respectively. In view of both, we might expect a pattern to form, with the next density being ρ = u3. So, given this density, we try to ﬁnd a ﬂux, J, by solving

3 2 u t + (J)x ˙= − 3u (6uux + uxxx) + (J)x = 0.

That is,

2 (J)x = 3u (6uux + uxxx) .

7The converse is not always the case. The Burgers’ equation, for example, can be solved exactly through linearization, but only has a ﬁnite number of conservation laws [29].

22 However, there is no way to express a J (on the jet space of u) which satisﬁes the above equation. So, the guess of ρ = u3 was wrong. In fact, a nontrivial computation [9] shows that 3 1 2 9 4 2 2 1 2 u − ux + u − 6uux + 3u uxx + uxx − uxuxxx ˙=0. 2 t 2 2 t So, we see that even in simple cases, conservation laws are non-trivial to construct. Many methods, however, have been developed with this construction in mind (for a survey of these methods, see, e.g., [12] or [14]) One of these methods, developed by Zakharov and Faddeev [10] in 1971, is related to a PDE’s Lax pair. We will demonstrate their method by algorithmically developing the above conservation laws for the KdV equation. Example 7 Recall that for the KdV equation, (2.4), the Lax operator L (2.6) was given by,

Lψ − λψ = ψxx − (λ − u)ψ = 0.

Making the substitution ψ = eiκx+σ(x,t) turns this into

iκx+σ 2 e 2iκσx + u + (σx) + σxx = 0,

where we have used the substitution λ = −κ2. Assuming eiκx+σ 6= 0, this reduces to

a Riccati equation in terms of σx,

2 2iκσx + u + (σx) + σxx = 0.

Similarly, from

ψt = Aψ + Bψx,

the same substitution and simpliﬁcation gives

σt = A − (iκ + σx) B. (2.44)

Note that, in Example 2, we set up our Lax pair this way, setting A = ux and B = 4κ2 − 2u. So, using standard techniques for solving Riccati equations [5], we

23 expand µ = σx as

∞ X µn µ = , (2iκ)n n=1 where µn are functions on the jet space of u. Substituting this into (7) will give [5]

µ1 = −u,

µ2 = ux, 2 µ3 = −(u + uxx), ... n−1 ! n+1 X µn+1 = (−1) µpµn−p + (µn)x , n ≥ 2. p=1

Note that (µk)x = µn−1, for odd n. So, even numbered µis are total derivatives and we consider them to be trivial. A more rigorous deﬁnition of triviality will be

2 given in Chapter 4. Using A = ux and B = 4κ − 2u and substituting the above expansion for µ into (2.44), we get

" ∞ # " ∞ !# X µn X µn = u − 4κ2 − 2u + iκ . (2.45) (2iκ)n x (2iκ)n n=1 t n=1 x Equating terms in negative odd powers of κ will give a listing of the conservation laws for the KdV equation. Note that this example requires the existence of a Lax pair. However, there are equations with a ﬁnite number of conservation laws but without a Lax pair. For a review of equations of that type, see, e.g., [12].

24 CHAPTER 3 CONSTRUCTION OF LAX PAIRS

It is common, in differential equation theory, to use symmetry arguments to find characteristics of a given differential equation. Lie-point symmetries, for example, can be used to generate solutions to a differential equation from an already-known solution [30]. Other types of symmetry, specifically scaling or (dilation) symmetry, have been used to construct conservation laws [12] and Lax pairs in operator form [15]. We will use this type of symmetry to generate Lax pairs in matrix form as well. Recall that a Lax pair is only valid if it satisfies the Lax equation, (2.11), when evaluated on the PDE. This suggests that properties of a Lax pair are associated with properties of the PDE. One property held by many integrable PDEs is scaling symmetry invariance. As such, we will construct candidate Lax pairs which have the same symmetry properties as the PDE we are examining. This restricts both the class of PDEs with which we can work and the type of Lax pairs we can construct for those PDEs. However, we will find Lax pair construction methods based on scaling invariance to be adept at constructing Lax pairs in this restricted class. In the next sections, we discuss the concept of scaling invariance, as well as two specific methods for the construction of Lax pairs.

3.1 Scaling Invariance

Before considering the construction of Lax pairs, we must elaborate on the meaning of scale invariance. Simply, a PDE is scale invariant if applying a scaling transformation returns the original PDE, multiplied by a constant. Example 8 Consider, again, the KdV equation, (2.4). Let us assume that the KdV equation is scaling invariant under a symmetry

(x, t, u) → (κ−ax, κ−bt, κcu) = (ξ, τ, υ),

25 where the exponents a, b, and c must be determined. Substituting this transformation into (2.4), we have

∂u ∂u ∂3u ∂ h υ i υ ∂ h υ i ∂ h υ i + 6u + = + 6 + ∂t ∂x ∂x3 ∂t κc κc ∂x κc ∂x3 κc " # 1 ∂ ∂τ 1 ∂ ∂ξ 1 ∂3 ∂ξ 3 = υ + 6 υ υ + υ κc ∂τ ∂t κ2c ∂ξ ∂x κc ∂ξ3 ∂x 1 ∂ ∂ t 1 ∂ ∂ h x i = υ + 6 υ υ κc ∂τ ∂t κb κ2c ∂ξ ∂x κa " # 1 ∂3 ∂ h x i3 + υ κc ∂ξ3 ∂x κa 1 ∂υ 1 ∂υ 1 ∂3υ = + 6 υ + . κc+b ∂τ κ2c+a ∂ξ κc+3a ∂ξ3

Thus, for the KdV equation to be scaling invariant, we must have

c + b = 2c + a = c + 3a. (3.1)

This system of equations is under determined, so we set a = 1 and ﬁnd b = 3 and c = 2. Thus, the KdV equation scales like

(x, t, u) → (κ−1x, κ−3t, κ2u) = (ξ, τ, υ).

Furthermore, every monomial of the equation scales with a κ−5. Because of this, the KdV equation is scaling invariant. The calculations required to find scaling symmetries were by no means complicated. Doing the same computations with a computer algebra system, however, is not so straightforward. To simplify the process of finding scaling symmetries, we introduce the concept of a variable’s weight. Using this tool, we can reduce the process of finding scaling symmetries to one of solving an algebraic equation such as (3.1). In doing so, we turn the problem into linear algebra which can be easily done by a computer program.

Deﬁnition 3.1.1 (Weight). The weight of a variable in a PDE, denoted W (·), is the exponent of κ in the variable’s scaling symmetry. By convention, W (x) = −1,

26 W (constant) = 0.

This deﬁnition can be extended to ﬁnd the weights of a derivative as well. In

∂ particular, for some dependent variable ν, W ∂ν = −W (ν). Thus, by convention, ∂ W ∂x = 1. This tool can also be extended to find the weight of more complicated ∂ terms. For example, W (ut) = W (u) + W ∂t . With this definition in hand, we compute the same symmetry as above, but in a much simpler manner. Example 9 Assume the KdV equation (2.4) is weight invariant. To find the weights of each variable, we set up a system of algebraic equations by equating the weights of each monomial in the PDE. Moreover, we require that each term in the PDE has an equal weight. This gives

∂ ∂ ∂ W (u) + W = 2W (u) + W = W (u) + 3W . ∂t ∂x ∂x

∂ Note that this is the same system of equations as (3.1), with W ∂x = −a, ∂ ∂ W ∂t = −b, and W (u) = c. Since, by convention, W ∂x = 1, we get that ∂ −1 −3 2 W (u) = 2 and W ∂t = 3. We found above that (x, t, u) → (κ x, κ t, κ u), so this result is unsurprising. Weights and scaling symmetries will be used extensively in the construction techniques in the next section. Another important tool in the construction of Lax pairs is the computation of monomials of a certain weight. If we assume weights are always nonnegative and integer8, this process is simple. Consider,

∂ e.g., a PDE such that W (u) = W ∂x = 1. Then, we can list every combination on the jet space of u of a certain weight. For example, every weight 1 term is of the

form c1u, where c1 is some constant. Similarly, every weight 2 term is either of the

2 form c2ux or c3u , for arbitrary constants c2 and c3. No other combination involving

u, ux, uxx,... can be of weight 2. The weight approach will prove to be very important in Lax pair construction techniques.

8This is not always appropriate. See, e.g., [12], for an example of non-integer weights

27 3.2 Exhaustive Lax Pair Computation Methods

We discuss the computation of Lax pairs with an exhaustive search technique. Assume that if a PDE has a certain scaling invariance, then that same scale must be present in Lax pairs and the Lax equation. That is, assume that the Lax pair inherits the symmetry properties of the PDE. We further assume that each element of X and T is weight invariant (though we do not initially specify what those weights will be). To justify this technique, consider the Lax pairs already presented in Chapter 2.

∂ 9 Assuming W (λ) = W ∂x will produce weight-invariant elements in each example given. To begin, assume that the X and T matrices are such that

W (X ) W (X ) W (T ) W (T ) W (X) = 11 12 and W (T ) = 11 12 , (3.2) W (X21) W (X22) W (T21) W (T22) where, e.g., W (X11) is the weight of each monomial in the (1, 1) element of X. Then, we require that each element of the matrix Lax equation is scale invariant. So, substituting X and T from (3.2) into the Lax equation (2.11) gives the overdetermined system of equations

∂ ∂ W (X ) + W = W (X ) + W (T ),W (T ) + W = W (T ) + W (X ), 11 ∂t 12 21 11 ∂x 11 11 ∂ ∂ W (X ) + W = W (X ) + W (T ),W (T ) + W = W (T ) + W (X ), 12 ∂t 12 22 12 ∂x 11 12 ∂ ∂ W (X ) + W = W (X ) + W (T ),W (T ) + W = W (T ) + W (X ), 21 ∂t 22 21 21 ∂x 21 11 ∂ ∂ W (X ) + W = W (X ) + W (T ),W (T ) + W = W (T ) + W (X ), 22 ∂t 22 22 22 ∂x 21 12

9The reader may notice that W (λ) = 2 in Example 1. However, λ does not actually play the role of the spectral parameter in this example. Indeed, the spectral parameter is κ such that −iκ2 = λ. Hence, W (κ) = 1, as expected.

28 W (T12) + W (X21) = W (X11) + W (T11),W (T12) + W (X21) = W (X12) + W (T21),

W (T12) + W (X22) = W (X11) + W (T12),W (T12) + W (X22) = W (X12) + W (T22),

W (T22) + W (X21) = W (X21) + W (T11),W (T22) + W (X21) = W (X22) + W (T21),

W (T22) + W (X22) = W (X21) + W (T12).

Solving this will give the required weight of each element of the X and T matrices. Speciﬁcally, we have

∂ W ∂x W (X12) W (X) = ∂ ∂ (3.3) 2W ( ∂x ) − W (X12) W ∂x and

∂ ∂ ∂ W ∂t W ∂t + W (X12) − W ∂x W (T ) = ∂ ∂ ∂ , (3.4) W ∂t − W (X12) + W ∂x W ∂t

where W (X12) is undetermined. The reader should compare this with the Lax pairs listed in, e.g., Example 2 and Example 3. With the required form of X and T speciﬁed, the construction method proceeds quite quickly. We generate candidate X and T matrices, whose elements have weights as described above, with undetermined coeﬃcients. Then, substituting X and T into the Lax equation, we form a system of equations to solve for the undetermined equations. Example 10 We make this method concrete by again looking at the KdV equation

∂ (2.4). For this example, set W (X12) = 0. Assuming W (λ) = W ∂x = 1, we construct candidate Lax pairs of the form

1 0 W (X) = , 2 1 such that c1λ c2 X = 2 , (3.5) c3λ + c4u −c1λ

29 and

3 4 W (T ) = , 2 3 such that

T T T = 11 12 , T21 −T11 where

3 T11 = c5λ + c6λu + c7ux, 2 T12 = c8λ + c9u, 4 2 2 T21 = c10λ + c11λ u + c12λux + c13u + c14uxx.

Here, ci, i = 1,..., 14 are undetermined constants. Note that X and T are assumed to be traceless, which can be assumed without loss of generality [31]. With this candidate Lax pair in hand, we substitute (3.5) and (10) back into the Lax equation (2.11), evaluate on the PDE, and find the coefficients which force the Lax equation to be zero on the PDE. So, we have four equations to solve, one from each matrix element. The first equation, for example, is

4 2 2 (c2c10 − c3c8) λ + (c2c11 − c4c8 − c3c9) λ u + (c2c13 − c4c9) u

+ (c2c12 − c6) λux + (c2c14 − c7) uxx = 0.

This equation, along with three others of that type, is diﬃcult to solve. However, since we have further requirements that λ 6= 0 and u 6= 0, we may split these equations by powers of λ and on the jet space of u. This reduces the above equation, for example, to a set of algebraic equations: c10c2 = c3c8; c2c11 = c4c8 + c3c9; c2c13 = c4c9; c6 = c2c12; and c7 = c2c14. Solving these, along with the similar equations from the other three matrix elements of the Lax equation, gives a number of solutions. Most of these solutions, however, are trivial (e.g., one of the matrices is entirely constant or zero). One of

2 c10 c11 c11 the more useful solutions is c1 = 0; c2 = − 2 ; c3 = ; c4 = ; c5 = 0; c6 = 0; c7 = c11 2 c10

30 2 2 2 c10 c10 c11 c11 1; c8 = −2 3 ; c9 = 2 2 ; c12 = 0; c13 = −2 ; c14 = − ; c10 6= 0; and c11 6= 0. c11 c11 c10 c10 Substituting this back into the candidate Lax pair, (3.5) and (10), gives

" c10 # 0 − c2 X = 11 1 c11 2 (c10λ + 2c11u) 0 2 c10 and

T T T = 11 12 , T21 −T11 where

T11 = ux, c10 2 T12 = −2 3 c10λ − c11u , c11 2 4 2 c11 2 T21 = c10λ + c11λ u − 2u + uxx . c10

Speciﬁcally, if we choose c10 = 4, c11 = 2i, and rescale the spectral parameter as λ2 = −iλ, we get the Lax pair in Example 2. Thus, based on this example, a method for Lax pair construction could proceed as follows:

1. Find the scaling symmetry of the PDE. 2. Build a candidate Lax pair based on this symmetry, speciﬁed by weight matrices (3.3) and (3.4). 3. Substitute the candidate Lax pair into the Lax equation and solve for undetermined coeﬃcients. 4. Filter out trivial solutions. 5. Substitute the non-trivial solutions into the candidate Lax pair.

This process has only one non-algorithmic aspect: the choice of weights for the candidate Lax pairs is determined by the (unknown) weight of the (1, 2)-element of the X matrix. One possible criterion for the selection of this weight will be addressed in the Section 3.3. However, the method proposed above is nearly algorithmic, and the selection of weights has a reasonable amount of choice. Moreover, as shown in Section 5.1, this method works for a large number of PDEs.

31 To further describe the eﬃcacy of the method, we treat the transcendental sine- Gordon equation. This example will illustrate two important aspects of the computation method. First, it deals with the case of a zero-weighted u, and, as such, a

∂ negatively-weighted ∂t . Moreover, the equation is non-polynomial. This, too, will cause some complications. Example 11 The sine-Gordon equation [32],

uxt = sin u,

is quite diﬀerent than the examples above, and those in Section 5.1. Trying to ﬁnd the weights of the PDE, for example, is trickier than usual. We have

∂ ∂ W (u) + W + W = W (sin u). ∂x ∂t While there are many ways to deal with this weighting [33], one simple way is to treat the weight of sin u as zero. Similarly, this choice gives that W (u) = 0 Then,

∂ ∂ assuming, as always, that W ∂x = 1 will give that W ∂t = −1. Each of these - the zero weight of u and the negative weight of t - are things we have not yet encountered. As an example of why these new weights cause problems, consider the construction of a matrix element of weight 1. Certainly, we should have a λ term. However, what

2 −2 about a λu term? Or a λu term? Or even a λ uxxx term? It is all-but impossible to choose a cutting-oﬀ point, at least algorithmically. For this Thesis, we will deal with this problem somewhat arbitrarily, and further research is warranted. The other main issue in constructing a candidate Lax pair involves the form of the PDE. Before, when the PDEs were polynomial on the jet space of u, it made sense for each matrix element to also be polynomial on the jet space of u. Here, however, the sin u term makes that assumption invalid. Thus, the candidate pairs will contain undetermined functions of u, as well as undetermined combinations of polynomial terms. As we will see, this will require the solving of diﬀerential equations as well as algebraic equations.

32 With these considerations in mind, we assume that each element of the X matrix has weight 1 and that each element of the T matrix has weight −1. Then, assume there are functions a and b such that W (a(u)) = W (b(u)) = 0. Based on the known Lax pair given in [5], we consider the candidate Lax pair10

c λ c u X = 1 2 x c3ux −c1λ and

" a(u) b(u) # λ λ T = b(u) a(u) . λ − λ Putting these into the Lax equation and separating by powers of λ and terms on the jet space of u, we get the system of equations

0 a (u) = (c2 − c3)b(u),

2c1b(u) = −c2 sin u,

= c3 sin u, 0 b (u) = −2c2a(u)

= 2c3a(u).

One solution to this mixed system of algebraic and diﬀerential equations is

1 1 1 c = −i, c = −c = − , a(u) = i cos u, and b(u) = i sin u. 1 2 3 2 4 4

Hence,

1 −iλ − 2 ux X = 1 2 ux −iλ and

1 cos u sin u T = i . 4λ sin u − cos u This is exactly the Lax pair given in [5]. Since the candidate Lax pair was based on that Lax pair, this is expected.

10This candidate pair can quite easily be extended to hold, say, more polynomial terms in the X matrix. For the purposes of this example, however, we ignore these terms.

33 The above is a toy example. Other problems with transcendental PDEs, such as Liouville’s equation [33], do not admit Lax pairs so easily.

3.3 Uniform Weight Construction

The method for Lax pair construction presented in the previous section allows for Lax pair matrices to have elements of different weights. This is a very general solution technique and, given a perfect nonlinear algebraic equation solver, would be sufficient for the construction of scaling invariant Lax pairs. However, due to computational complexity, even the most advanced solvers cannot handle some basic examples. In this section, we discuss a more structured method, which requires a certain weight-invariance in each element of the X matrices and a, possibly different, weight-invariance in each element of the T matrix. Consider again the defining weight equations, (3.3)-(3.4). For each element of

∂ the matrix X to have equal weight, it must be true that W ∂x = W (X12) = ∂ ∂ 2W ( ∂x ) − W (X12). Hence, W (X12) = W ∂x . So the overall weight of the X matrix must be equal to this quantity as well. By a similar argument, it follows that the

∂ weight of the T matrix must be equal to the weight of ∂t . Note that this method won’t

work for every PDE. The KdV equation, for example, requires that either W (X12)

∂ or W (X21) be diﬀerent from W ∂x . We will see, however, that there are a number of important PDEs for which this method works, and so we continue describing this method. Once the weights of X and T are established, expand both Lax matrices in a ﬁnite power series in λ. That is, set

∂ ∂ W ( ∂x ) W ( ∂t ) X X X = X(i)λi and T = T (j)λj i=0 j=0

34 11 ∂ ∂ We will assume , for the rest of the discussion, that W ∂x < W ∂t . Since we have ∂ 1 ∂ 1 assumed that W (X) = W ∂x , W (T ) = W ∂t , and W (λ) = 1, the weights of each X(i) and T (j) must be changed accordingly. It is not hard to see that W (X(i)) =

∂ 1 (j) ∂ 1 W ∂x − i and W (T ) = W ∂t − j will suffice. In this way, we have expanded X and T in terms of a sum of matrices whose elements all have the same weight. Thus, we can find the equations which must be satisfied by X(i) and T (j) to ensure that X and T will satisfy the Lax equation. To find these relationships, we substitute X and T into the Lax equation and separate in powers of λ,

 ∂   ∂   ∂   ∂  W ( ∂x ) W ( ∂t ) W ( ∂x ) W ( ∂t )  X (i) i  X (j) j  X (i) i  X (j) j  X λ  −  T λ  +  X λ   T λ  i=0 j=0 i=0 j=0 t x (3.6)  ∂   ∂  W ( ∂t ) W ( ∂x )  X (j) j  X (i) i −  T λ   X λ  ˙=0. j=0 i=0

As diﬀerential operators are linear, the only diﬃculty in the expansion comes from

∂ products in the commutator term. To simplify this expansion, assume W ∂x = 1. We have taken this assumption to be standard, so we may simplify as such without loss of generality. As a further simplifying step, recall that a matrix element with

W ∂ weight zero is simply a constant. Thus, each element of X(1) and T ( ( ∂t )) is a constant and their derivatives are zero. With these two considerations in mind, we separate (3.6) by powers of λ to get

1 h (1) W ∂ i X h (k) W ∂ −k i X ,T ( ( ∂t )) ˙=0 and X ,T ( ( ∂t ) ) ˙=0, (3.7) k=0 which we call the kinematic equations, along with

(0) (0) (0) (0) X t − T x + X ,T ˙=0

and

11This assumption is based on Lax pairs found in literature.

35 1 (i) X h (k) W ∂ −i−k i ∂ − T + X ,T ( ( ∂t ) ) ˙=0 i = 1,...,W − 1, x ∂t k=0 which we call the dynamic equations. At this point, the power of this method is clear: though we have restricted the form of the Lax pair, we have written down a relationship which must hold true for any Lax pair of this type. In particular, the kinematic equations (3.7) can be solved quite easily. The solution is then used to reduce the complexity of the dynamic equations (3.3). Once the kinematic equation is solved, we find a Lax pair as before. We construct candidates of each X(i) and T (j) and use the simplified dynamic equations to find the undetermined coefficients. To demonstrate this method, we will find the Lax pair for the mKdV equation. Example 12 Consider the mKdV equation, (2.31). For the purposes of this example, take α = −6 so that the equation is

2 ut − 6u ux + uxxx = 0.

As described by the method above, we ﬁrst ﬁnd the weights of the mKdV equation. Therefore we solve the system of equations

∂ ∂ W (u) + W = 3W (u) + W , ∂t ∂x ∂ ∂ W (u) + W = W (u) + 3W , ∂t ∂x ∂ ∂ 3W (u) + W = W (u) + 3W ∂x ∂x

∂ ∂ for the weights of u, ∂x , and ∂t . The third of these equations is a direct consequence ∂ of the ﬁrst two, so the system is under-determined. Thus, set W ∂x = 1. Then, ∂ the second equation gives W ∂t = 3, so the ﬁrst equation gives W (u) = 1. Thus, ∂ 1 1 ∂ 1 1 we require W (X) = W ∂x = and W (T ) = W ∂t = (3) and assume that X and T take the form X = X(0) + X(1)λ and T = T (0) + T (1)λ + T (2)λ2 + T (3)λ3. Using

36 (3.7) the kinematic equations are

[X(1),T (3)] = 0, (3.8) [X(0),T (3)] + [X(1),T (2)] = 0.

To facilitate the solution of this system of equation, we ﬁrst assume a form for X(i), for i = 0, 1 and T (j), for j = 0,..., 3, such that X and T are both traceless. Thus, set

" (i) (i) # " (i) (i) # (i) X11 X12 (i) T11 T12 X = (i) (i) and T = (i) (i) . X21 −X11 T21 −T11 Solving (3.8) gives a number of possible solutions. After sorting through the trivial ones, we have

(1) (3) (1) (3) (1) X11 T12 (1) X11 T21 X12 = (3) ,X21 = (3) , T11 T11 (0) (3) (3) (1) (3) (2) (1) (2) (3) (0) X11 T11 T12 + X11 T11 T12 − X11 T11 T12 X12 = 2 , (3) T11 (0) (3) (3) (1) (3) (2) (1) (2) (3) (0) X11 T11 T21 + X11 T11 T21 − X11 T11 T21 X21 = 2 . (3) T11

Substituting these relationships back into the candidate for X gives

X X X = 11 12 , X21 −X11 where

(0) (1) X11 = X11 + X11 λ, (0) (3) (3) (1) (3) (2) (1) (2) (3) ! (1) (3) ! X11 T11 T12 + X11 T11 T12 − X11 T11 T12 X11 T12 X12, = + λ, (3) 3 (3) (T11 ) T11 and

37 (0) (3) (3) (1) (3) (2) (1) (2) (3) ! (1) (3) ! X11 T11 T21 + X11 T11 T21 − X11 T11 T21 X11 T21 X21 = + λ. (3) 3 (3) (T11 ) T11 The T matrix remains unchanged. Now that the commutator relationships have been solved, we further specify the candidate matrices with scale invariance. Speciﬁ- cally, since W (X) = 1, we have

" 1 # c1λ + c2u 2 (c1c3c4λ + (c2c3c4 − c1c3c5 + c1c4c6) u) c4 X = 1 . 2 (c1c4c7λ + (c2c4c7 − c1c5c7 + c1c4c8) u) −(c1λ + c2u) c4

Furthermore, since W (T ) = (3)1, we have

T T T = 11 12 . T21 −T11 where

3 2 2 3 T11 = c4λ + c5λ u + c11u + c12ux λ + c9u + c10uux + c13uxx, 3 2 2 3 T12 = c3λ + c6λ u + c16u + c17ux λ + c14u + c15uux + c18uxx, 3 2 2 3 T21 = c7λ + c8λ u + c21u + c22ux λ + c19u + c20uux + c23uxx.

Finally, these candidate X and T matrices are substituted back into the matrix Lax equation and we solve for the undetermined coeﬃcients. Again, there are many

3 2 solutions to this, one of which is c2 = 0; c3 = 0; c4 = −4c1; c5 = 0; c6 = −2c1c14; c7 = 2 c1 0; c8 = −8 ; c9 = 0; c10 = 0; c11 = 2c1; c12 = 0; c13 = 0; c15 = 0; c16 = 0; c17 = c14

1 1 c1 1 −c1c14; c18 = − c14; c19 = 4 ; c20 = 0; c21 = 0; c22 = 4 ; and c23 = −2 . Substi- 2 c14 c14 c14 tuting these coeﬃcients into the candidate pairs, we have

1 c1λ 2 c14u X = 2 u −c1λ c14 and

3 3 2 2 2 3 1 −4c1λ + 2c1λu −2c1c14λ u − c1c14λux + c14u − c14 2 uxx T = 2 2 2 3 3 3 2 , (−4c λ u + 2c1λux + 2u − uxx) 4c λ − 2c1λu c14 1 1

38 for undetermined c1 and c14. Particular choices of these coeﬃcients give nicer answers.

For example, setting c1 = −i and c14 = 2 gives

−iλ u X = u iλ and

3 2 2 3 −4iλ − 2iλu 4λ u + 2iλux + 2u − uxx T = 2 3 3 2 . 4λ u − 2iλux + 2u − uxx 4iλ + 2iλu Notice that this is the same Lax pair given in Example 5 with λ replaced by κ. Comparatively, this method is computationally simple. Unfortunately, only a handful of equations seem to admit Lax pairs of this type. See Section 5.2 for two speciﬁc examples.

3.4 Weak Lax Pairs and Triviality Concerns

The methods above appear to be promising. One important topic - that of triviality in the computed Lax pairs - still needs to be addressed. In a series of papers published in the 1990s and early 2000s, Sakovich [34,35], Marvan [36], and Calogero and Nucci [16] deﬁne the concept of strong and weak Lax (or ‘true and fake’) Lax pairs. This represents a diﬀerent type of triviality for Lax pairs than discussed in Chapter 2. Essentially, a Lax pair is weak if it is gauge equivalent to a matrix without a spectral parameter. Otherwise, it is strong. To demonstrate this concept, we consider an example of a weak Lax pair given by Marvan for Burgers’ equation, [36]. Example 13 Consider again Burgers’ equation, (2.1),

ut − uxx − κuux = 0.

For this example, take κ = 1. A Lax pair for Burgers’ equation which can be produced by the methods presented above (or found in literature, e.g., [35,36]) is

1 1 λ λ + 2 u X = 1 2 −λ + 2 u −λ and

39 1 1 2 λu λu + 2 u + ux T = 1 2 . 4 −λu + 2 u + ux −λu It is straightforward to verify that substituting these X and T into the Lax equation yields 1 0 ut − uxx − uux Xt − Tx + [X,T ] = , 4 ut − uxx − uux 0 which is the zero matrix upon evaluation on the original PDE. In other words, by triviality methods discussed thus far, this is a valid and non-trivial Lax pair. Now, consider the gauge matrix " √ √ # 1 λ + √1 λ − √1 G = √ λ √ λ . 2 λ − √1 λ + √1 λ λ Gauging X and T with equations (2.23) and (2.24) gives

1 1 ˆ −1 1 1 + 2 u X = GXG = 1 2 −1 + 2 u −1 and

1 1 2 ˆ −1 u 2 u + u + ux T = GT G = 1 2 . 4 2 u − u + ux −u These are exactly the original X and T , but with the spectral parameter removed. For applications, these weak Lax pairs are useless. For example, when carrying out the IST to ﬁnd solutions to a nonlinear PDE, the existence of a spectral parameter is crucial [5]. Moreover, the method for constructing conservation laws presented in Chapter 4 will only work if a spectral parameter is present in a Lax pair. In many ways, the lack of a spectral parameter in a Lax pair takes away any implication of integrability for a nonlinear PDE12. The number of weak Lax pairs that are found by the computation methods described in this chapter is of concern, and a good solution has yet to be found. On the positive side, there are algorithmic methods for detecting

12There are equations admitting weak Lax pairs which are, indeed, completely integrable. How- ever, these weak Lax pairs do not lead to solutions via the IST. See [37] for further discussion of this.

40 if a Lax pair is weak. We will outline these now, following [31] and [34].

Consider a generic evolution equation, ut = h(u, ux, uxx,...) with Lax pair X and def T . Moreover, deﬁne the covariant derivative as ∇M = Mx −[X,M] for any matrix M and the matrix X from the Lax pair. Note that this is not related to the ∇ operator from Calculus. We use, however, it to be consistent with [35]. Furthermore, deﬁne the matrix C as

def 2 C = Xu − ∇ (Xux ) + ∇ (Xuxx ) − ....

Here, e.g., ∇2M = ∇(∇(M)), and so on for higher powers of ∇. Then, it can be shown [34] that the zero-curvature representation of the evolution equation can be written in terms of C. Because this new representation is dependent on an invariant PDE, it is no surprise that C has invariant properties of its own. To ﬁnd this invariance, deﬁne the cyclic basis of C as the sequence of linearly independent matrices C, ∇C, ∇2C,..., ∇n−1C. Then, the closure equation of the cyclic basis is

n n−1 ∇ C = a0Ca1∇C + ... + an−1∇ C.

The invariance here comes from the constants ai. Every valid Lax pair of a PDE will have the same ai. Moreover, if one or more of these ai have a spectral parameter in it, that parameter cannot be removed from the Lax pair - e.g., it is a strong Lax pair. This is the key idea to filtering out weak Lax pairs. All that is required is the computation of the closure equation and examination of the constants in it; due to the invariance of cyclic bases, the existence of an explicit gauge transform which removes a spectral parameter from a Lax pair it is not required when testing for triviality. We clarify these concepts with an example. Example 14 Consider again the weak Lax pair for the Burgers’ equation given in (2.1), with κ = 1. We first find the first few covariant derivatives for the X matrix,

41 1 0 1 C = , 4 1 0 1 −1 −1 ∇C = λ , 4 1 1 u −1 −1 ∇2C = λ . 8 1 1

The first two of these matrices are linearly independent. The third, however, can be represented in terms of the first two. Thus, the closure equation of this cyclic basis is 1 ∇2C = u∇C, 2 with the coefficient on C being 0. Note, importantly, that this closure equation is free of any spectral parameters. It follows, then, that X and T are weak Lax pairs, as expected. We now apply this method to a strong Lax pair for Burgers’ equation. This, again, is given in [35] (though earlier references can be found). Consider the pair13

1 λ + 4 u 0 X = 1 1 − λ + 4 u and

2 1 2 1 −2λ + 8 u + 4 ux 0 T = 1 2 1 2 1 . −2λ + 2 u 2λ − 8 u − 4 ux We compute the covariant derivatives,

1 1 0 C = , 4 0 −1 0 0 ∇C = 1 , − 2 0 2 0 0 ∇ C = 1 , −(λ + 4 u) 0

13Note that the exhaustive search method of Lax pair computation will indeed construct this pair, though with considerable diﬃculty and in a large amount of time.

42 and derive the closure equation,

1 ∇2C = 2λ + u ∇C, 2 where the coeﬃcient on C is 0 again. Because the closure equation necessarily has a spectral parameter in it, X and T are strong Lax pairs. The above is an important warning about the eﬃcacy of the computational methods described in this chapter. It is easy to construct Lax pairs which appear to be nontrivial which are, in reality, weak pairs. However, with diligence, weak Lax pairs can be avoided, or at least detected.

CHAPTER 4 THE CONSTRUCTION OF CONSERVATION LAWS FROM LAX PAIRS

In this chapter, we discuss a method to construct conservation laws based on Lax pairs. As discussed in Chapter 2, several method exist for algorithmic computation of conservation laws for nonlinear PDEs. In the following section, we outline one of these methods, based on work done in the early 1980s [17] by Drinfel’d and Sokolov (DS). The method takes advantage of the similarity between the equation which defines a Lax pair, (2.11), and the equation which defines a conservation law, (2.43), to convert from the former to the latter. This approach, which we name after its discoverers, is algorithmic and is based on a simple recursive formula. However, it requires a specific form of Lax pair and will only admit conservation laws for a narrow class of PDEs. Perhaps for this reason, the method has been, to the author’s knowledge, largely ignored in literature.

4.1 The Drinfel’d Sokolov Method for Computing Conservation Laws

Consider again the deﬁning equations for conservation laws and Lax pairs,

ρt + Jx ˙=0 and Xt − Tx + [X,T ] ˙=0.

Examining these equations, it is clear that if [X,T ] = 0, the Lax equation represents a conservation law with density ρ = X and ﬂux J = −T . Thus, given a Lax pair, X and T , the goal of the DS method will be to force the commutator term to vanish. Before describing how to do this, we put some restrictions on the Lax pair. Consider a Lax pair of the form14

X = X0 − X1λ (4.1)

14It seems obvious to try to expand this in higher (or lower) powers of λ. This approach, however, has yet to prove fruitful.

45 and

m X i T = Tiλ i=0 where X0, an oﬀ-diagonal matrix, and Ti are matrices containing functions on the jet space of u, X1 is a constant diagonal matrix, m is an undetermined integer, and λ is the spectral parameter. Example 15 Consider the KdV equation with a Lax pair given by (2.21)-(2.22). We can decompose the X matrix into the form of (4.1) where

0 u X = (4.2) 0 −1 0 and

i 0 X = . (4.3) 1 0 −i

With an appropriate Lax pair at hand, we return to the task of annihilating the commutator term. Recall that the commutator of two diagonal matrices is always zero. Thus, to make the commutator in (2.11) vanish, we use a diagonalization technique.

Theorem 4.1.1 (Drinfel’d-Sokolov [17]). Let S be a matrix-valued series of the form

∞ X −i S = I + Γiλ , (4.4) i=1 where I is the appropriately-sized identity matrix and Γi are matrices whose elements are on the jet space of u. Then, we deﬁne X˜ = SXS−1 to be of the form

∞ ˜ X −i X = Dx − X1λ + Piλ , (4.5) i=0 ∂ where the Pi are diagonal matrices and Dx is a matrix of ∂x operators. Equating XS˜ = SX in powers of λ gives the following recurrence relation

46 n−1 0 X Pn + [Γn+1,X1] = ΓnX0 − Γn − PiΓn−i, (4.6) i=0

which allows each Γi and Pi to be solved recursively.

Proof. First, recall Cauchy’s product formula.

Theorem 4.1.2 (Cauchy’s Product Formula [38]). Consider two series with terms

an and bn, which are assumed to converge as n → ∞. Then,

∞ ! ∞ ! ∞ n X X X X an bn = akbn−k. n=0 n=0 n=0 k=0 Using this, expand both sides of XS˜ = SX. Speciﬁcally, to use Cauchy’s product, we must re-index X˜ to

∞ ˜ X −(j+1) X = Dx − X1λ + P0 + Pj+1λ j=0 and S to

∞ X −(j+1) S = I + Γj+1λ . j=0

For what follows, we take Γ0 = I. Thus, expanding these power series, we have

∞ ! ∞ ! ˜ X −(j+1) X −(j+1) XS = Dx − X1λ + P0 + Pj+1λ I + Γj+1λ j=0 j=0 ∞ ∞ X 0 −(j+1) X −j = DxI + Γj+1λ − X1λ − X1Γj+1λ + P0 j=0 j=0 ∞ ∞ X −(j+1) X −(j+1) + P0Γj+1λ + Pj+1λ j=0 j=0 ∞ ! ∞ ! X −(j+1) X −(j+1) + Pj+1λ Γj+1λ . j=0 j=0

Expanding the product of sums in the last term with Cauchy’s product formula gives

47 ∞ ∞ ˜ X 0 −(j+1) X −j XS = Γj+1λ − X1λ − X1Γj+1λ + P0 j=0 j=0 ∞ ∞ X −(j+1) X −(j+1) + P0Γj+1λ + Pj+1λ j=0 j=0 ∞ j X X −(j+2) + Pk+1Γj−k+1λ . j=0 k=0

Similarly, we expand SX as

∞ ! X −(j+1) SX = I + Γj+1λ X0 − X1λ j=0 ∞ ∞ X −(j+1) X −j = X0 − X1λ + Γj+1X0λ − Γj+1X1λ . j=0 j=0

Equating these and simplifying, we get

∞ ∞ ∞ X 0 −(j+1) X −j X −j Γj+1λ − X1Γj+1λ + Γj+1X1λ j=0 j=0 j=0 ∞ ∞ ∞ X −(j+1) X −(j+1) X −(j+1) + P0Γj+1λ + Pj+1λ − Γj+1X0λ j=0 j=0 j=0 ∞ j X X −(j+2) + Pk+1Γj−k+1λ + P0 − X0 j=0 k=0 = 0.

Setting n = j + 1 and splitting these in powers of λ yields

P0 = X0 at order λ0 and

n−1 0 X Pn + [Γn+1,X1] = ΓnX0 − Γn − PkΓn−k k=0 at order λn. This is exactly the recursion relation given in (4.6).

48 With this theorem in hand, we can ﬁnd the ﬁrst few conservation laws for the KdV equation.

Example 16 It was shown by Drinfel’d and Sokolov [17] that if P0 = X0 then the elements of each matrix Pi are a conserved density of the given PDE. Thus, continuing with the KdV equation (2.4), we solve the recurrence relation (4.6) for n = 0,..., 3.

First, we let P0 be

0 0 P = . 0 0 0

Then, the ﬁrst few Pi matrices are

1 −u 0 P = i , 1 2 0 u

1 u 0 P = − x , 2 4 0 0

i (u2 + u ) 0 P = xx . 3 8 0 −u2

1 1 i 2 i 2 Thus,, e.g., ρ1 = 2 iu, ρ2 = − 4 ux, ρ3,1 = − 8 u , and ρ3,2 = 8 (u + uxx) are valid densities. Using only the recurrence relationship (4.6), we can create an infinite number of conserved densities. The corresponding fluxes could be constructed by the same process, using the matrix T from the Lax pair with the same S matrix. However, this calculation quickly grows unwieldy due to complicated matrices being processed. Thus, we take a simpler approach. Specifically, given a ρ, we can take the integral of (2.43) with respect to x, giving Z Z ρt dx + Jx dx ˙=0.

So, rearranging,

49 Z J = − (ρi)t dx, (4.7)

where ut and the x-derivatives of u are replaced from the PDE. Note that this integral can be difficult to evaluate. For a straightforward way of evaluating integrals of this type, see [39] Example 17 Finally, we finish the derivation of conservation laws for the KdV equation. As an example, we take a non-zero element from the first computed matrix density, 1 ρ = iu 1 2 and put it into (4.7). This gives

Z 1 J = D iu dx 1 t 2 Z 1 = iu dx 2 t 1 Z = − i 6uu + u dx 2 x xxx 1 = − i 3u2 + u . 2 xx

Thus, we have now constructed a ρ1 and J1 such that

(ρ1)t + (−J1)x ˙=0.

The ﬂuxes for the next two conservation laws follow similarly. We have 3 1 J = iuu + iu 2 2 x 4 xxx

1 associated with ρ2 = − 4 iux. Also, 1 1 1 J = − iu3 + iu2 − iuu . 3,1 2 8 x 4 xx

i 2 associated with ρ3,1 = 8 u and 1 7 1 1 J = − iu3 − iu2 − iuu − iu 3,2 2 8 x 2 xx 8 xxxx

50 i 2 associated with ρ3,2 = 8 (u + uxx). These conservation laws are valid, but fail to match the conservation laws given in literature for the KdV equation. In the next section, we discuss the reasons for this.

4.2 Triviality and Simpliﬁcation

While the above densities and fluxes are valid, there are two important points to consider: trivial conservation laws (to be defined precisely below) are constructed and the conservation laws given in Example 17 do not match exactly with published results, e.g., [12,14]. Two types of trivial conservation law are considered in [12]. The first type occurs when the conservation law equation (2.43) is satisfied even when not evaluated on solutions to the PDE. A simple example of this is

(uxx)t + (−uxt)x = 0, which holds for any sufficiently smooth u. For the purposes of this Thesis, we will ignore this type of trivial conservation law. The second type occurs when all the terms from the density (or flux) can be moved into the flux (or density). That is, a conservation law is trivial if either ρ = 0 or J = 0. An example of this is

1 3 1 iux + i uux + uxxx ˙=0. 4 t 2 4 x This is trivial because

1 1 iux = iut , 4 t 4 x so we can rewrite the conservation law as

1 3 1 (0)t + − iut − iuux − iuxxx ˙=0. 4 2 4 x

The second of these examples may look familiar. In fact, it was ρ2 computed in Example 17. Moreover, in Chapter 2, we came across this listing of conservation

51 laws, ignoring every other density for this reason. Note that we only have triviality when all the terms can be moved from, e.g., the density into the ﬂux. For example,

i 2 consider ρ3,2 = 8 (u + uxx). The (uxx)t term is equivalent to (uxt)x, which can be moved into the ﬂux. However, since the u2 term cannot be moved by similar means, the conservation law is non-trivial. It is, however, equivalent to ρ3,1 for this reason. To make the process of detecting these trivial conservation laws easier, we must ﬁrst introduce the so-called Euler operator.

Deﬁnition 4.2.1. The Euler operator (sometimes called the Euler-Lagrange operator or variational derivative) for f = f(x, u(M)(x)), a function of x, u(x), and up to M- derivatives of u(x) is [39]

M X ∂k ∂f L u(x)f = − k , ∂x ∂ukx k=0 k k where ukx = ∂ ux .

Using this, we state without proof15 a theorem which will aid us in detecting trivial conservation laws.

Theorem 4.2.1 (Poole [12]). A function f = f(x, u(M)(x)) is exactly integrable if and only if Lu(x)f = 0. That is, f = Fx, for some function F if and only if the Euler-operator applied to f is identically zero.

Using the Euler operator, we can now algorithmically detect a trivial conservation law of the second type. Speciﬁcally, assume some density, ρ, is such that Lu(x)ρ = 0.

(ρ) (ρ) Then, ρ = ∂Jx , for some function J . But, this means

(ρ) (ρ) ρt + Jx = Jx + (J)x = (0)t + J + Jt , t x and so the conservation law is trivial. Thus, we know that if the density of a conservation law is exact, the conservation law is trivial. We will use this result in picking out trivial conservation laws and removing them from consideration.

15See [12] for a thorough discussion and proof of this theorem.

52 Finally, we must remove terms from conserved densities which are linear combinations of terms equivalent to other terms up to divergence. That is, elements whose Euler-operators are equivalent. To illustrate this, we consider an example. Example 18 Consider the list of terms

3 2 {u , ux, uuxx, uxxxx}. (4.8)

To ﬁnd divergence-equivalent terms, apply the Euler-operator to each of these terms separately,

3 2 2 Lu(x){u , ux, uuxx, uxxxx} = {3u , −2uxx, 2uxx, 0}.

2 Thus, the term uxxxx is exact and can be removed. Similarly, ux + uuxx is exact

2 because Lu(x)ux + Lu(x)uuxx = −2uxx + 2uxx = 0. Thus, we remove the terms with highest-order derivatives and (4.8) simpliﬁes into

3 2 {u , ux}.

The final simplification step involves the normalization of the term with highest- degree exponents. That is, forcing the coefficient in front of the highest-degree term to be 1. Example 19 Consider each of the three conservation laws found in Example 17. The first conservation law is 1 i (u) + 3u2 + u = 0. 2 t xx x After normalization, this becomes

2 (u)t + 3u + uxx x .

The second conservation law is 1 i [(u ) + (6uu + u ) ] = 0. 4 x t x xxx x As discussed, this conservation law is trivial, so we remove it. The third conservation law is

53 1 − i u2 + 4u3 − u2 + 2uu , 8 t x xx x which simpliﬁes to

2 3 2 (u )t + 4u − ux + 2uuxx x = 0.

See Appendix A for an outline of the symbolic algorithm to simplify conservation laws. More examples of conservation laws produced by this method are in section 5.3.

54 CHAPTER 5 FURTHER EXAMPLES

In this chapter, we provide Lax pairs and conservation laws computed with the methods presented in this Thesis. In particular, the ﬁrst section lists Lax pairs computed by the exhaustive search method given in Section 3.2. Section 5.2 lists Lax pairs computed by the uniform weight method given in Section 3.3. The ﬁnal section lists conservation laws computed by the Drinfel’d-Sokolov method presented in Chapter 4.

5.1 Lax Pairs Computed with Exhaustive Methods

We now give Lax pairs constructed by the methods discussed in Section 3.2. For each equation, weights and working Lax pairs will be given, as well as simpliﬁcations and comparisons to literature.

5.1.1 Modiﬁed Korteweg-de Vries Equation

The modiﬁed Korteweg-de Vries (mKdV) equation,

2 ut − 6u ux + uxxx = 0,

has the following weights:

∂ ∂ W = 1,W (u) = 1, and W = 3. ∂x ∂t We generate a Lax pair of c1λ c2u X = 1 u −c1λ c2 and

T T T = 11 12 , T21 −T11 where

55 3 3 2 T11 = −4c1λ + 2c1λu , 2 2 3 T12 = −4c1c2λ u − 2c1c2λux + 2c2u − c2uxx, 1 2 2 3 T21 = − 4c1λ u − 2c1λux − 2u + uxx . c2

This can be simpliﬁed to the Lax pair found in Example 12 with c1 = −i and c2 = 1.

5.1.2 Kaup-Kuperschmidt Equation

The Kaup-Kuperschmidt (KK) equation, 1 5 u + γ2u2u + γu u + γuu + u = 0, t 5 x 2 x xx xxx xxxxx has weights

∂ ∂ W = 1,W (u) = 2, and W = 5. ∂x ∂t We generate a Lax pair with

" 2 1 c2 5 # 1 −25c1λ −25c2λ − 2 γ u 500 c1 X = 2 2 c1 γ 25 25c1λ c2 and

T T T = 11 12 , T21 −T11 where

5 1 1 1 T = c λu2 + c λu − γ2uu − γu , 11 1 2 γ 1 xx 25 x 20 xxx

2 2 5 1 2 2 c2 2 1 c2 1 c2 5 3 T12 = c2λ u + c2λ uxx − γ λuux − γλuxxx + 2 γ u 2 γ 25 c1 10 c1 12500 c1 1 c2 4 2 9 c2 4 1 c2 3 2 γ ux + 2 γ uuxx + 2 γ uxxxx, 625 c1 5000 c1 500 c1 2 c1 2 5 1 T21 = − u + uxx . c2 2 γ

It is unknown if this Lax pair has been reported in literature. The pair given in [40] is related to the above pair, but the two are certainly distinct. The pair constructed

56 in this section, however, is quite diﬀerent from pair given in [41].

5.1.3 Lax 5th-Order Equation

For the Lax 5th-order equation, 3 u + γ2u2u + 2γu u + γuu + u = 0, t 10 x x xx xxx xxxxx with weights of

∂ ∂ W = 1,W (u) = 2, and W = 5, ∂x ∂t we generate a Lax pair with

iλ − 1 c u X = 16 1 8γ −iλ 5c1 and

T T T = 11 12 , T21 −T11 where

4 2 1 1 T = −16iλ5 + iγλ3u + γλ2u − iγλ 3γu2 + 10u − γ (3γuu + 5u ) , 11 5 5 x 50 xx 50 x xxx 1 1 1 T = c λ4u − iλ3u − λ2 γu2 + 5u + iλ (3γuu + 5u ) 12 1 2 x 20 xx 40 x xxx 1 + 3γ2u3 + 30γu2 + 40γuu + 50u , 800 x xx xxxx 4 1 4 2 2 2 T21 = − γ 800λ − 40γλ u + 3γ u + 10γuxx . 125 c1

When c1 = −16, γ = 10, and λ = −iλ, this is equivalent to the Lax pair presented in [42].

5.1.4 Sawada-Kotera Equation

The Sawada-Kotera (SK) equation, 1 u + γ2u2u + γu u + γuu + u = 0, t 5 x x xx xxx xxxxx has

57 ∂ ∂ W = 1,W (u) = 2, and W = 5. ∂x ∂t In this case, we obtain

" 1 c2 2 2 5 # 1 −25c1λ − 2 (3125λ c1 + γ u) 125 c1 X = 2 2 c1 γ 25 25c1λ c2 and

T T T = 11 12 , T21 −T11 where

1 T = 25c λ γu2 + 10uu − γ2 (γuu + 5u ) , 11 25γ 1 xx x xxx 1 c2 1 2 2 2 2 T12 = 2 3125c1λ γu + 10uxx − 250γ c1λ (γuux + 5uxxx) 3125 c1 γ 4 2 3 2 + 5γ γ u + 5γux + 15γuuxx + 25uxxxx , 2 c1 2 T21 = − γu + 10uxx . c2γ

5.1.5 Harry-Dym Equation

The Harry-Dym equation,

3 ut + u uxxx = 0,

with weights

∂ ∂ W = 1,W (u) = 1, and W = 6, ∂x ∂t admits a Lax pair of

" c1c2 1 2 # c1λ + 2 u (c1c4λ + c1c3λu) c3 c5 X = c1c2c5 c1c2 −4 2 − c1λ + 2 u c3 c3 and

" 6 c2c5 5 7 6 5 2 1 c3 3 # c5λ + 2 λ u c4λ + c3λ u + c2λ u − u uxxx c3 2 c5 T = 2 . c2c5 5 6 c2c5 5 −4 λ −(c5λ + 2 λ u) c3 c3 This diﬀers from the Lax pair given in [5].

58 5.1.6 Drinfel’d-Sokolov System

Consider the Drinfel’d-Sokolov system of equations [29],

u + 3vv = 0, t x (5.1) vt + vux + 2uvx + 2vxxx = 0.

In equation (5.1), both u(x, t) and v(x, t) are both dependent variables. While systems of equations have not yet been considered, they are an important and common examples of complete integrable PDEs. The Drinfel’d-Sokolov system has weights of

∂ ∂ W = 1,W (u) = 2,W (v) = 2, and W = 3. ∂x ∂t

The Drinfel’d-Sokolov system admits a Lax pair of

" 2 c2 # c1λ c2λ + 2 (u − v) 4c1 X = 2 c1 − −c1λ c2 and

" 1 2 1 c2 2 # c1λv + vx c2λ v + 2 (uv − v + c1vx + 2vxx) 2 4 c1 T = 2 . c1 1 − v c1λv − vx c2 2 To the author’s knowledge, no matrix Lax pair for this equation is reported in the literature.

5.2 Lax Pairs Computed by Uniform Weight Method

This section will list Lax pairs computed by the methods presented in Section 3.2, Section 3.3.

5.2.1 Nonlinear Schr¨odingerEquation

The NLS equation,

2 ∗ iut − uxx − 2u u = 0, (5.2) where u(x, t) is a complex function and ∗ denotes complex conjugate has weights of

59 ∂ ∂ W = 1,W (u) = W (u∗) = 1, and W = 2. ∂x ∂t

We generate a Lax pair with c1λ c2u X = 1 ∗ , u −c1λ c2 and

2 2 ∗ −2ic1λ + iuu −2ic1c2λu − ic2ux T = 1 ∗ ∗ 2 2 ∗ . (−2ic1λu + iu ) 2ic λ − iuu c2 x 1

This is exactly the Lax pair reported in [5] with c1 = −i and c2 = 1.

5.2.2 Gardner Equation

Consider the Gardner equation [43],

2 2 ut + αuux + α ux + u ux + uxxx = 0. (5.3)

A simple transformation [43] relates the Gardner equation (5.3) to the mKdV equation (2.31). It’s no surprise, then, that similar methods can be used to construct Lax pairs for each equation. The Gardner equation has has weights of

∂ ∂ W = 1,W (u) = 1,W (α) = 1, and W = 3. ∂x ∂t

Assuming an X with uniform weight of 1 and a T with uniform weight of 3 yields

−iλ − 6 c (α + 2u) X = 5 1 5 (α + 2u) iλ 144c1 and

T T T = 11 12 T21 −T11, where

60 1 T = −4iλ3 + iλ(5α2 + 2αu + 2u2), 11 6 1 T = c −24λ2(2u + α) − 24iλu + 5α3 + 12α2u + 6αu2 + 4u3 + 12u , 12 5 1 x xx 5 2 3 2 2 3 T21 = − −24λ (α + 2u) + 24iλux + 5α + 12α u + 6αu + 4u + 12uxx , 864c1 where c1 is a free constants. This Lax pair has the same form (with diﬀerent constant coeﬃcients) as (2.34)-(2.35) under the transformation given in [43].

5.3 Conservation Laws Produced by the Drinfel’d-Sokolov Method

We will show conservation laws from four PDEs. These conservation laws were computed with the DS method discussed in Chapter 4. In this section, we adopt the

∂nu notation unx = ∂xn for n > 3.

5.3.1 Korteweg-de Vries Equation

We return to the KdV equation, (2.4), and list additional conservation laws. Using

X0 and X1, as given in Chapter 4, the ﬁrst 4 conservation laws are listed in table Table 5.1. These are comparable to the conservation laws found in Section 2.4. They can also be compared to those found in [5].

Table 5.1: The ﬁrst four conservation laws for the KdV equation (2.4).

Order Density(ρ) Flux(J) 2 1 u 3u + uxx 2 3 2 2 u 4u − ux + 2uuxx 3 1 2 9 4 2 2 1 2 3 u − 2 ux 2 u − 6uux + 3u uxx + 2 uxx − uxuxxx 4 2 1 2 24 5 2 2 3 2 4 u − 2uux + 5 uxx 5 u − 18u ux + 4u uxx + 2uxuxx + 16 2 1 2 2 5 uuxx − 4uuxuxxx − 5 uxxx + 5 uxxuxxxx

5.3.2 Modiﬁed Korteweg-de Vries Equation

Consider again the mKdV equation, (2.31). The X part of the Lax pair constructed in Chapter 3 can be decomposed as

61 0 u i 0 X − X λ = − λ. 0 1 u 0 0 −i The DS method gives the conservation laws in Table Table 5.2

Table 5.2: The ﬁrst four conservation laws for the mKdV equation (2.31).

Order Density(ρ) Flux(J) 2 4 2 1 u −3u − ux + 2uuxx

6 2 2 3 2 4 − 4u + 6u ux + 10u uxx − uxx 2 u − uuxx + uxuxxx − uu4x

9 11 − u8 − 45u4u2 + u4 + 6u5u 2 x 2 x xx 6 2 2 1 2 2 2 2 3 u + 5u ux + 2 uu4x − 22uuxuxx − 23u uxx − 14u uxuxxx 1 1 1 − 3u3u + u u − u u + uu 4x 2 xx 4x 2 x 5x 2 6x

24 126 − u10 − 112u6u2 − u2u4 5 x 5 x 406 154 + 8u7u − u3u2u − u4u2 xx 5 x xx 5 xx 2 2 3 4 − 22uxuxx + 44uuxx + 28u uxuxxx 8 4 2 21 2 u + 14u u + uu uxx 129 628 x 5 x − u3u + uu u u 4 5 x xxx 5 x xx xxx 14 2 2 1 + u u − uu6x 106 129 5 xx 5 + u2u2 + uu2u 5 xxx 5 x 4x 208 72 6 + u2u u + u2u u + u3u 5 xx 4x 5 x 5x 5 6x 1 1 1 − u u + u u − uu 5 xx 6x 5 x 7x 5 8x

5.3.3 Sine-Gordon Equation

The sine-Gordon equation [5],

uxt − sin u = 0. (5.4)

62 with matrix X decomposed as

1 0 − 2 ux i 0 X0 − X1λ = 1 − λ 2 ux 0 0 −i admits conservation laws with the DS method. The ﬁrst four of these are listed in table Table 5.3. These are equal to the conservation laws given in [33].

Table 5.3: The ﬁrst four conservation laws for the sine-Gordon equation (5.4).

Order Density(ρ) Flux(J) 2 1 ux 2 cos(u) 4 2 ux + 4uxuxxx −4 sin(u)uxx 6 2 2 2 4 3 ux − 20uxuxx + 8uxu5x − (8u4x − 32uxuxx) sin(u) + 14ux cos(u)

16 4 3 − 16uxuxx + 11uxx − 12uxuxxuxxx 8 4 2 336 2 5 ux − 56uxuxx + uxuxxuxxx 5 2 4 − 24uxu4x + 4u6x sin(u) 224 2 2 64 + uxuxxx + uxu7x 8 5 5 − 3u6 − 50u2u2 − 72u3u cos(u) 5 x x xx x xxx

5.3.4 Sinh-Gordon Equation

The sinh-Gordon equation [5], a close relative of the sine-Gordon equation, is

uxt − sinh(u) = 0. (5.5)

The X Lax pair for the sinh-Gordon can be decomposed as [5]

1 0 2 ux i 0 X0 − X1λ = 1 − λ. 2 ux 0 0 −i The conservation laws given in Table Table 5.4 are equivalent to those given in [33].

5.3.5 Nonlinear Schr¨odingerEquation

Consider again the NLS equation, (5.2). We can write the X from the Lax pair of the NLS as [5]

63 Table 5.4: The ﬁrst four conservation laws for the sinh-Gordon equation (5.5).

Order Density(ρ) Flux(J) 2 1 ux −2 cosh(u) 4 2 ux − 4uxuxxx 4 sinh(u)uxx 6 2 2 2 4 3 ux + 20uxuxx + 8uxu5x −8 (4uxuxx + u4x) sinh(u) + 14 cosh(u)ux

16 3 4 − 11uxx − 16uxuxx − 12uxuxxuxxx 8 4 2 336 2 5 ux + 56uxuxx + uxuxxuxxx 5 2 4 − 24uxu4x − 4u6x sinh(u) 224 2 2 64 + uxuxxx − uxu7x 8 5 5 + 3u6 + 50u2u2 + 72u3u cosh(u) 5 x x xx x xxx

0 u i 0 X − X λ = − . 0 1 u∗ 0 0 −i The software we developed to compute conservation laws does not accept complex PDEs. As such, to compute the conservation laws given in Table 5.5, we rewrote (5.2) as the system

2 ∗ ut − 2αu u + αuxx = 0, ∗ ∗ 2 ∗ ut + 2αu(u ) − αuxx = 0, where α was replaced by i after calculations. These three conservation laws are the same as those given in [44].

64 Table 5.5: The ﬁrst four conservation laws for the NLS equation (5.2).

Order Density(ρ) Flux(J) 2 ∗ ∗ 1 |u| −i(uux − u ux) ∗ 4 ∗ ∗ 2 ∗ 2 uuxx − |u| −i(uuxxx − uxuxx − 4|u| uux) 3 − i − u2(u∗)3u − 4u3(u∗)2u∗ − u∗u2u∗ x x 2 x x 3 ∗ 2 1 ∗ 2 ∗ ∗ + uux(ux) − (u ) uxuxx + 2uu uxuxx 2u3(u∗)3 − 6uu∗u u∗ 2 4 x x 3 9 1 3 − 5u2(u∗)2 − u(u∗)2u + uu∗u u∗ + u2u∗u∗ + u(u∗)2u x xx 2 x xx 2 x xx 4 xxx − 6u2u∗u∗ + uu∗ 1 xx xxxx + 2u2u∗u∗ + u u∗ xxx 4 x xxxx 1 − uu∗ 4 5x

3 ∗ 4 ∗ 3 3 4 ∗ 3 ∗ i 24u (u ) ux + 6(u ) ux + 64u (u ) ux − 5u4(u∗)4 + 10u3× ∗ 2 2 ∗ 2 ∗ ∗ 2 − 18u(u ) u u − 212u u ux(u ) ∗ ∗ 2 ∗ 2 ∗ x x x × 5u (ux) + 3(u ) uxx 3 ∗ 3 ∗ 3 − 60u (ux) − 12u(u ) uxuxx 2 ∗ 2 − 106u2(u∗)2u∗u − 19u (u∗)2u + u − 19(uxx) x xx x x xx 2 ∗ 2 ∗ 3 ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ − 120u (u ) uxuxx − 216u u uxuxx − 28u u + (u ) uxu x xxx x 2 ∗ ∗ ∗ ∗ − 52u u u − 20u uxuxxu ∗ ∗ ∗ 3 x x xx xx − 10u uxxxx + 8(u ) uxx ∗ ∗ ∗ 2 + 110uuxuxxuxx + 52uux(uxx) 4 2 ∗ − 10u2(u∗)3u − 10u∗u u∗u − 10|u| 2uxxuxx xxx x x xxx + 29u(u∗)2u + 30uu∗u∗ u ∗ x xxx xx xxx + uxuxxx 3 ∗ 2 ∗ ∗ 2 ∗ − 48u (u ) uxxx − 20u uxuxxx ∗ 2 ∗ ∗ ∗ ∗ ∗ − u 19(ux) uxx − u6x + 72uuxuxuxxx + 40uu uxxuxxx 2 ∗ ∗ ∗ 2 ∗ ∗ ∗ 3 2 + 68u uxxuxxx − (u ) uxuxxxx + 20u uxuxxx − 6(u ) ux ∗ ∗ ∗ ∗ ∗ 2 ∗ ∗ + 12uu uxuxxxx + 20uu uxuxxxx + (u ) uxxxx + 52uxuxuxx 2 ∗ ∗ ∗ 2 2 ∗ ∗ + 40u uxuxxxx + u(u ) u5x + 12u u u5x ∗ ∗ + uxu6x − uu7x

CHAPTER 6 CONCLUSIONS AND FURTHER RESEARCH

The preceding was an overview of construction methods for various properties of integrable nonlinear PDEs. A novel method, based on scaling symmetry, was discussed for the computation of Lax pairs, with good results. An old method for constructing conservation laws was also discussed. Both methods used properties inherent to the equations being discussed to turn a complicated problem into an algebraic problem. Both computational methods have been shown to work on a variety of integrable nonlinear PDEs. Considering that most completely integrable PDEs are closely-related to the ones discussed above, this is indicative of the usefulness of the computational methods discussed. This Thesis by no means closes the topics of research related to the construction of Lax pairs and conservation laws as described above. Huge swaths of material were ignored, or not discussed in this document. For example, this Thesis only focused on 2×2 matrix Lax pairs. However, n × n matrices are required for Lax pairs in higher spatial dimensions. Moreover, both systems of equations and PDEs of dimension higher than (1 + 1) were not adequately discussed. Both of these extensions would provide ample research opportunity. There are obvious relationships between the Drinfel’d-Sokolov method for conservation law computation and the method of Lax pair computation presented in this Thesis which were not discussed. Either method could be generalized by replacing current sums with Laurent-type expansions in λ rather than Taylor-type expansions. Either method could be generalized by considering, more in-depth, non-polynomial PDEs. Furthermore, the problems associated with weak and strong Lax pairs and their relationships with gauge transforms were not adequately discussed. Finally, inverting the processes described above, to cre-

67 ate classes of PDEs related to a Lax pair or set of conservation laws, would be a worthwhile undertaking.

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72 APPENDIX - OUTLINE OF DRINFEL’D-SOKOLOV ALGORITHM AND USE OF MATHEMATICA

The goal of this program is to construct, simplify, and verify the validity of conservation laws for a given Lax Pair. To do this, we use the method proposed by Drinfel’d and Sokolov as well as tools from calculus of variations described in Chapter 4. The algorithm used to achieve this goal is as follows:

1. Construct conserved densities with Drinfel’d’s and Sokolov’s recurrence relationship, (4.6).

2. Verify that constructed densities are valid.

(a) Test if a density is trivial;

(b) Test that a density is exact.

3. Simplify densities

(a) Test for terms in density which are total divergences;

(b) Test for divergence-equivalence of linear combinations of terms.

4. Construct ﬂuxes of conservation law based on densities.

Analytic methods of each of these steps are described above. Thus, all that remains is to discuss the implementation of these methods in Mathematica.

1. Simpliﬁcation Process

We use the following algorithm to remove divergence-equivalent terms from

the densities. In the description, let ρL be a list containing each term of the conserved density ρ.

73 Step 1 Apply the Euler-operator to ρL. If any elements are identically zero in the

Euler space, remove the corresponding element from ρL.

Step 2 If there is only one element in ρL, do nothing to it and stop. The simpliﬁed ρ is simply that element.

Step 3 Let S = {} and B = {} be empty sets. Let B contain the ﬁrst non-zero

element of the list LuρL and let S contain the corresponding element of

ρL. This is the ﬁrst element of the density.

Step 4 For an element ρi ∈ ρL, check if Luρi is in the span of B. If it is, let

B = B ∪ {Luρi} and S = S ∪ {ρi}. Otherwise, throw out element ρi.

Continue this process for all elements in ρL. P Step 5 Set the simpliﬁed ρ to ρsimp = Si, where Si are the elements of S.

2. Coeﬃcient Fixing

The goal of this process is to ﬁnd coeﬃcients for the elements of the density which will make a density exact. We display some of the Mathematica code used to do this before explaining it.

validDensitiesCoeff=Inner[Times,Table[c[i],{i,1, Length[validDensities[i1]]}], (validDensities[i1])/.Plus->List,Plus];

coeffList=stripper[EulerD[D[Expand[validDensitiesCoeff],t] /.rule,u[x,t],{x,t}]];

solveRule=Flatten[Solve[Map[#==0&,Table[Coefficient [EulerD[D[Expand[validDensitiesCoeff],t]/.rule, u[x,t],{x,t}],coeffList[[i]]],{i,1,Length[coeffList]}]],

74 Table[c[i],{i,1,Length[validDensitiesCoeff]}]]];

validDensities[i1]=(validDensitiesCoeff/.solveRule)/.{c[i_]->1};

Step 1 First, put undetermined coeﬃcients on each of the elements of the den-

sity, validDensities[i1]. This is done in two steps. First, the density is converted to a list of elements. Then, the Inner function is used to simultaneous multiply each element by an unknown coeﬃcient and add these elements together.

Step 2 Next, a stripper function is used to ﬁnd which elements on the jet space of u are present in the density.

Step 3 A list of equations to solve for each of the unknown coeﬃcients is created.

This is done with the command Coefficient. This command ﬁnds the coeﬃcients of the density which are multiplying the same type of element on the jet space of u.

Step 4 Finally, the list of equations created in Step 3 is solved and applied to the density. Any still-unknown coeﬃcients are normalized to one.

3. Normalization

The ﬁnal piece of code involved the normalization of the highest-order terms of a density. Again, we will ﬁrst display the interesting bit of code, and then explain it.

multiplier[i1,1]=(First[validDensities[i1,1]/.{Plus->List}])/ Cases[First[validDensities[i1,1]/.{Plus->List}], Except[_Complex]][[1]];

Step 1 The density being considered is ﬁrst converted to a list. The ﬁrst element of this list will always be the highest-order term due to Mathematica’s

75 ordering procedures, and so it is selected.

Step 2 The pattern matching function Cases is then used to ﬁnd the elements in this element which are not numbers. This is done with the Except command.

Step 3 The highest order element is then divided by its non-numerical element to ﬁnd the coeﬃcient which is in front of it.

The program used to construct these conservation laws is available from the author upon request.