Numerical Study of the KP Solitons and Higher Order Miles Theory of the Mach Reﬂection in Shallow Water

Numerical study of the KP solitons and higher order Miles theory of the Mach reﬂection in shallow water

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Yuhan Jia, M.S, B.S.

Graduate Program in Mathematics

The Ohio State University

2014

Dissertation Committee:

Yuji Kodama, Advisor Barbara Keyﬁtz Fei-Ran Tian

Abstract

In 1970, two Russian physicists Kadomtsev and Petviashvili proposed a two- dimensional nonlinear dispersive wave equation to study the stability of the solitary wave solution under the inﬂuence of weak perturbations transverse to the propagation direction. The equation is now referred to as the KP equation. The KP equation arises as the leading order approximation of certain physical systems under weak nonlinearity, weak dispersion and quasi-two dimensionality assumptions, and admits several exact solutions, called KP solitons, that are regular, non-decaying and localized along distinct lines in the two-dimensional plane, the xy-plane.

The main part of this thesis concerns numerical study of the KP solitons for their application to the Mach reflection problem in shallow water. This problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall. In 1977, Miles proposed a theory to explain the phenomenon, and predicted a fourfold amplification of the incident wave. His theory is based on an asymptotic analysis, and in this thesis we show that his results can be interpreted in terms of the KP solitons.

Since Miles presented his theory, there have been several numerical studies as well as water tank experiments to conﬁrm his theory. However, they all found considerable discrepancies between their results and Miles’ predictions, in particular, they could not obtain fourfold ampliﬁcation of the incident wave. In this thesis, we consider the

ii problem starting from the three-dimensional Euler equation for the irrotational and incompressible ﬂuid. We still use the assumptions, weak nonlinearity, weak dispersion and quasi-two dimensionality, but derive the higher order corrections to the leading order KP equation, which we refer to as the higher order KP equation in this thesis.

Employing an asymptotic perturbation theory called the normal form theory, we study the higher order KP equation, and ﬁnd the higher order corrections to the KP solitons. We then perform numerical simulation of the full Euler equation for the

Mach reﬂection phenomena, and conﬁrm that the solutions to the higher order KP equation well describe the phenomena. Thus, we extend the Miles theory including the higher order corrections, which we call the higher order Miles theory, and show that the results obtained from the extended theory are in good agreement with the numerical and experimental results in the literature.

In this thesis, we construct a higher order KP soliton solutions from the result of the numerical simulation to demonstrate the validity of the higher order KP equation.

We also consider the stability of the KP solitons which concerns the robustness of the KP solitons and certain convergence issues of the initial value problem of the KP equation. We ﬁnd an orbital instability of a solitary wave solution due to the existence of the phase shifts propagating along the transversal direction of the soliton.

iii Acknowledgments

I would like to express my very great appreciation to my advisor, Professor Yuji

Kodama, for his patient guidance and valuable advice during my research work. His

ﬁnancial support and enthusiastic encouragement have been very much appreciated.

I would also like to thank Professor Chui-Yen Kao for her numerous assistance in the numerical simulation. I would like to oﬀer my special thanks to Professor Barbara

Keyﬁtz and Professor Fei-Ran Tian for the useful critiques and suggestions. Pertinent advice given by Dr. Shelley Quinn has also been much appreciated. My grateful thanks are also extended to Dr. Thomas Kerler and Mr. Roman Nitze for their assistance in keeping my progress on schedule. I would also like to thank the computer support group in the department of mathematics and the Ohio Supercomputer Center for their help for oﬀering me with the resources in running the numerical program. I am particularly grateful for the corrections and timely help given by Larry Tano-Long.

Finally, I wish to thank my parents for their support and encouragement throughout my study. Thanks again to all who helped me.

iv Vita

2004 ...... B.S. Mathematics, University of Science and Technology of China 2006 ...... M.S. Mathematics, Kansas State University 2009 ...... M.A. Statistics, The Ohio State University 2006-present ...... Graduate Teaching Associate, The Ohio State University.

Fields of Study

Major Field: Mathematics

Specializations: Nonlinear dispersive wave equations

v Table of Contents

Page

Abstract ...... ii

Acknowledgments ...... iv

Vita...... v

List of Tables ...... viii

List of Figures ...... ix

1. Introduction and overview ...... 1

2. Basic equations of shallow water waves ...... 10

2.1 The Euler equation ...... 10 2.2 Boussinesq approximation for the Euler equation ...... 11 2.3 KP equation and higher order terms ...... 15

3. The Miles theory and the KP theory for the Mach reﬂection ...... 19

3.1 The Mach reﬂection in shallow water ...... 19 3.2 Miles’ models of the interaction of the oblique solitary waves . . . . 22 2 3.2.1 Weak interaction: κ = sin Ψ0 ∼ O(1) ...... 23 2 3.2.2 Strong interaction: κ = sin Ψ0 ∼ O() ...... 26 3.2.3 The resonant interaction ...... 27 3.3 KP soliton solutions ...... 30 3.3.1 One-line soliton ...... 31 3.3.2 Y-shape soliton ...... 35 3.3.3 X-shape soliton: (2143)-soliton ...... 40 3.3.4 (3142)-soliton ...... 42

vi 3.3.5 X-shape soliton versus (3142)-soliton ...... 43 3.4 Miles solitary waves in terms of the KP solitons ...... 45 3.5 Miles vs. KP in the numerics ...... 49 3.5.1 The amplitude amplification factor aM ...... 49 ai 3.5.2 The angle Ψ∗ corresponding to the stem length ...... 51 3.5.3 The angle Ψr of the reflective wave ...... 52 3.5.4 The amplitude ar of the reflective wave ...... 53

4. Numerical study of the Euler equation ...... 55

4.1 Higher order scheme for the Euler equation ...... 55 4.2 Pseudospectral method and scheme accuracy ...... 58 4.3 Initial value problem and boundary condition of oblique incident waves 61 4.4 Numerical simulation of the Mach reﬂection vs. the regular reﬂection 65

5. Higher order Miles theory ...... 68

5.1 The higher order KP equation ...... 69 5.1.1 Normal form of the higher order KdV equation ...... 69 5.1.2 Normal form of the higher order KP equation ...... 75 5.2 Higher order KP solution from numerical results by minimization . 80 5.3 Numerical results reconsidered with higher order correction . . . . 92 5.3.1 The amplitude amplification factor aM ...... 93 ai 5.3.2 The angle Ψ∗ corresponding to the stem length ...... 95 5.3.3 The angle Ψr of the reflective wave ...... 97 5.3.4 The amplitude ar of the reflective wave ...... 98 5.3.5 Asymptotic simulations ...... 100

6. Stability of some KP solitons ...... 105

6.1 Stability of one-line soliton with transverse perturbations ...... 106 6.1.1 Phase shift perturbation ...... 107 6.1.2 Amplitude perturbations ...... 112 6.2 Stability of Y-shape soliton ...... 116

Bibliography ...... 122

vii List of Tables

Table Page

4.1 The L2 error between two successive time steps, dt and dt/2...... 60

5.1 The minimum L2 error between the higher order (3142)-soliton solution of the KP equation and the numerical results of the Mach reﬂection. . 85

5.2 The minimum L2 error between the higher order X-shape soliton solution of the KP equation and the numerical results of the regular reﬂection...... 89

5.3 Critical angle for the diﬀerent amplitude of the incident wave. . . . . 93

viii List of Figures

Figure Page

2.1 The three-dimensional view of a solitary wave(left) and its contour plot(right)...... 14

o o 2.2 Contour plot of line soliton with Ψ0 = 0 (left), Ψ0 = 15 (middle) and o Ψ0 = 30 (right). The widths of the soliton in the latter two cases are 96.6% and 86.6% of that of the original soliton...... 18

3.1 The schematic regular reﬂection...... 20

3.2 The schematic Mach reﬂection...... 21

3.3 The contour plot of the surface elevation η in (3.7). The solitary waves

N1 and N2 have the phase shifts due to the interaction...... 25

3.4 The contour plot(left) and the chord diagram(right) of [i, j]-soliton. . 34

3.5 The three-dimensional view(left), the contour plot(middle) and the chord diagram(right) of two types of Y-shape soliton solution to the

KP equation. The k-parameters are (k1, k2, k3) = (−1, −0.25, 0.75). . 35

3.6 The plot of hi(p) versus p for diﬀerent k values k1, k2 and k3...... 37

3.7 The X-shape soliton graph at t = −20(left), t = 0(middle) and t = 20(right). The k-parameters that used to generate this soliton solution

are (k1, k2, k3, k4) = (−0.5, −0.25, 0.25, 1) and the intersection is placed at the origin at t =0...... 40

3.8 Example of the chord diagram of X-shape soliton solution with π = (2143)...... 41

ix 3.9 The (3142)-soliton graph at t = −80, 0, 150 from left to right. The

k-parameters are (k1, k2, k3, k4) = (−0.5, −0.3, 0.2, 1)...... 42

3.10 Example of the chord diagram of (3142)-soliton solution...... 42

3.11 Contour plot of the X-shape soliton symmetric about the x-axis. . . . 44

3.12 Contour plot of the (3142)-soliton symmetric about the x-axis. . . . . 44

3.13 The amplitude ampliﬁcation factor aM . The left ﬁgure show the result ai obtained by Miles theory and the right one by the KP theory. The 3

are the numerical results with the incident amplitude ai = 0.3. . . . . 50

3.14 The angle Ψ∗ corresponding to the stem. The left ﬁgure show the result obtained by Miles theory and the right one by the KP theory. The 3

are the numerical results with the incident amplitude ai = 0.3. . . . . 51

3.15 The angle Ψr of the reﬂective wave. The left ﬁgure show the result obtained by Miles theory and the right one by the KP theory. The 3

are the numerical results with the incident amplitude ai = 0.3. . . . . 52

3.16 The amplitude ar of the reﬂective wave. The left ﬁgure show the result obtained by Miles theory and the right one by the KP theory. The 3

are the numerical results with the incident amplitude ai = 0.3. . . . . 54

4.1 a = 0.05...... 61

4.2 a = 0.1...... 61

4.3 The contour plot of the initial V-shape wave surface in the whole domain. 62

4.4 The partial wave surface without boundary modiﬁcation around y =

±LY ...... 62

o 4.5 Surface of the Mach stem with a = 0.3, Ψi = 20 and t = 350. . . . . 66

o 4.6 Surface of the regular reﬂection with a = 0.3, Ψi = 50 and t = 350. . 66

4.7 The crest line of the Mach reﬂection case with t = 0, 10, 20,..., 150. . 66

4.8 The crest line of the regular reﬂection case with t = 0, 10, 20,..., 100. 66

x 5.1 Contour plot of line soliton with higher order correction for Ψ0 = o o o 0 (left), Ψ0 = 15 (middle) and Ψ0 = 30 (right). The widths of the soliton in the latter two cases are 98.2% and 93.2% of the original soliton...... 78

5.2 A contour plot of the water wave pattern of the numerical simulation. The incident soliton has amplitude 0.05 and angle 15o...... 83

5.3 Plots (1)-(7) show the diﬀerences between the higher order (3142)- soliton solution of the KP equation and the numerical results of the

Mach reflection. Plot (1) is obtained by minimization on x14 with initial fixed value of x13. Plot (2) is obtained by minimization on x13 with fixed value of x14 from last step. (3),(4),(5),(6) and (7) repeat the same previous procedures where plot (6) is omitted due to the similarity with plot (7)...... 86

5.4 Contour plot of the leading order and the higher order KP (3142)- soliton solution(left column) and their diﬀerences to the numerical results(right column) of the Mach reﬂection...... 87

5.5 A contour plot of the simulation of the water waves. The incident soliton has amplitude 0.05 and angle 30o...... 88

5.6 Plots of the diﬀerence between the higher order X-shape soliton solution of the KP equation and the numerical results of the regular

reflection. Plot (1) is obtained by minimization on x12 with initial fixed value of x34. Plot (2) is obtained by minimization on x34 with fixed value of x12 from last step. (3) and later iterations repeat the same previous procedures...... 90

5.7 Contour plot of the leading order and the higher order X-shape soliton solution(left column) and their diﬀerences to the numerical results(right column) of regular reﬂection...... 91

5.8 aM vs. K with a = 0.3...... 94 ai

5.9 aM vs. K with a = 0.1...... 94 ai

5.10 Ψ∗ vs. Ψi with a = 0.3...... 96

xi 5.11 Ψ∗ vs. Ψi with a = 0.1...... 96

5.12 Ψr vs. Ψi with a = 0.3...... 98

5.13 Ψr vs. Ψi with a = 0.1...... 98

5.14 ar vs. K with a = 0.3...... 99 ai

5.15 ar vs. K with a = 0.1...... 99 ai

5.16 aM vs. K with a = 0.1...... 101 ai

5.17 Ψ∗ vs. Ψi with a = 0.1...... 101

5.18 ar vs. K with a = 0.1...... 102 ai

5.19 Ψr vs. Ψi with a = 0.1...... 102

5.20 aM vs. K with a = 0.05...... 103 ai

5.21 Ψ∗ vs. Ψi with a = 0.05...... 103

5.22 ar vs. K with a = 0.05...... 103 ai

5.23 Ψr vs. Ψi with a = 0.05...... 103

6.1 Initial solitary wave with a local phase shift...... 107

6.2 d = 0.1...... 108

6.3 d = 0.2...... 108

6.4 d = 0.3...... 109

6.5 d = 0.4...... 109

6.6 d = 0.5...... 109

6.7 d = 0.1 phase shift to right...... 111

6.8 d = 0.2 phase shift to right...... 111

xii 6.9 An amplitude dent in the middle of the soliton crest and an artiﬁcial

amplitude hump is placed atx ¯w = −35...... 112

6.10 An amplitude dent in the middle of the soliton crest and an artiﬁcial

amplitude hump is placed atx ¯w = −40...... 113

6.11 An amplitude dent in the middle of the soliton crest and an artiﬁcial

amplitude hump is placed atx ¯w = −45...... 113

6.12 Initial soliton has a smaller amplitude in the middle...... 114

6.13 Initial soliton has a larger amplitude in the middle...... 114

6.14 The initial(partial) chord diagram corresponding to (6.8)...... 117

6.15 Numerical simulation of KP equation with initial value u(x, y, 0) given by (6.8) and the exact Y-shape soliton solution by minimization. . . . 119

2 6.16 The average L2 error between the exact solution and the numerical results versus time...... 120

xiii Chapter 1: Introduction and overview

The motion of a non-viscous, incompressible and irrotational water flow under the force of gravity located above an impermeable bottom with free surface boundary is described by the Euler equation. It can be reduced to a pair of evolution equations with respect to the water surface elevation from the still water and velocity potential on the water surface. The complexity of solving the Euler equation directly leads to various model equations derived from asymptotic expansions, some of which are physically meaningful and mathematically interesting. A special type of water wave, called solitary wave, is described as a permanent wave in terms of keeping its surface elevation of a finite amplitude and travelling at a constant speed in a long time. It was first observed and documented by John Scott Russell in 1834 but was considered as a contradiction to Airy wave theory known as linear wave theory[1] that a water wave with finite amplitude can not propagate without changing its shape and speed.

This issue was resolved by Boussinesq[8] and Rayleigh[49] who showed independently that neglecting the dispersive terms in Airy’s shallow water wave equation led to the conﬂict between Russell’s discovery and Airy’s wave theory prediction. The equation for the surface elevation η(x, t) with the space variable x and the time t is deduced based on a long wave and small amplitude regime, and it contains both the nonlinear

1 and dispersive terms,

2 3 η2 1 2 ηtt = C [ηxx + ( )xx + h ηxxxx] (1.1) 0 2 h0 3 0 √ where C0 = gh0 is the speed of the solitary wave with g the constant acceleration from gravity and h0 the asymptotic water depth. Note that the Boussinesq equation includes both right- and left-running waves. If we consider just the right-running wave in a far ﬁeld, Boussinesq equation (1.1) can be reduced to the Korteweg-de

Vries(KdV) equation,

3 η2 1 2 ηt + C0[ηx + ( )x + h ηxxx] = 0 (1.2) 4 h0 6 0

which has a commonly used non-dimensional form in a moving frame

ut + 6uux + uxxx = 0. (1.3)

It was ﬁrst derived by Korteweg & de Vries[32] in 1895 and also found to have a family

of periodic solutions called cnoidal waves. Solitary waves can be obtained when the

periodicity of the cnoidal waves approaches inﬁnity. It is only until 1954 that the

existence theory of solitary waves is established by Friedrichs and Hydes[16].

Although Russell’s observation and numerous experimental evidence consolidate

the impression that a solitary wave is stable in the sense that it maintains its original

form and travels with a constant speed over a long time, it is not until the 1970’s that

a rigorous proof of the stability of a solitary wave solution to the KdV equation is ﬁrst

established by Benjamin[4] and then improved by Bona[5] later. They showed that a

solitary wave solution to the KdV equation with a small perturbation maintained the

identity of the solitary wave solution at a later time up to a phase shift. The study

of stability problems had been extended to the generalized KdV equation

p ut + (uxx + u )x = 0 for p = 2, 3, 4. (1.4)

2 by Pego and Weinstein[47], followed by the asymptotic study of the amplitude and the velocity of the solitary wave by Martel and Merle[37], and the asymptotic stability of the sum of N solitary waves for subcritical generalized KdV equation (1.4) was proved under a similar architecture of the proof to a single solitary wave solution by Martel, Merle and Tsai[38]. Another crucial discovery in the KdV theory is the interaction property between solitary wave solutions. That is, those solitary waves with diﬀerent amplitudes interact without changing their characteristics such as the amplitude and the speed excluding the phase shift. This particle-like phenomenon was discovered by Zabusky and Kruskal[61] in their numerical simulation of the interaction of solitary waves and they coined the name “solitons” for the solitary waves. Subsequently, Gardner et cl(GGKM)[18], Hirota[22] and Whitham[56] have constructed the analytical solutions of the KdV equation and proved the N-soliton interaction property.

The approximate models for the water wave equations have been studied both theoretically and numerically for over a century, however it is only less than four decades ago when a rigorous justification of these models as approximate models be- gan. One-dimensional shallow water equations with a flat bottom were first analyzed in the 1970’s in Lagrangian coordinates[45][46][24]. The initial data were taken from a Sobolev class with the assumption of small perturbation of the still water and the solution of the aforementioned shallow water equations was proved to converge to that of the water wave equations as the small perturbation approaches zero, locally in time. The cases with infinite and finite water depth, and variable bottom topography were considered by Yosihara[60]. Craig[12] in 1985 showed that the solution of water wave equations exists over a large time interval with the same assumption

3 of small and analytic data in Lagrangian coordinate. This was a necessary and sig- niﬁcant eﬀort in order to compare the solutions of the water wave equations in the

KdV and one-dimensional Boussinesq limits. Kano and Nishida[25] also derived the

KdV and Boussinesq equations from the expansion of the Euler equations by drop- ping the higher order terms of a small dimensionless parameter and proved that their solutions approximate the water waves up to the same order of error as the dropped terms. The restriction of small and analytic data assumed by the earlier literatures has been removed, since Wu[57] in 1997 showed the existence and uniqueness of the solutions to the one-dimensional water wave problem with infinite water depth and locally in time. Wu[58] also derived a similar result that the two dimensional water wave problem was uniquely solvable in Sobolev space without the assumption of small data. The case of finite water depth was done by Lannes[34] for both one dimensional and two dimensional water wave equations under the Eulerian coordinate, and the large time existence for the solutions of the water wave equations in two dimensional space with flat bottom was completed by Bona, Colin and Lannes[6]. Their recent works[35][2] in 2008 showed that the justification of the approximate models to the water wave equations could be done in a systematic way by expressing the existence time and the energy bounds in terms of five parameters, representing still water depth, wave surface elevation, wave length in the longitudinal direction, wave length in the transverse direction and bottom topography. Note that the aforementioned literature about rigorous justification on validating the existence and uniqueness of the solution to the water wave equations and the convergence of the solution of the approximate models to that of the water wave equations relies upon a compact support region and

4 ﬁnite energy constraint. Therefore, their results may not be applicable to a solitary wave in two dimensional space.

In the two-dimensional case with the space variables (x, y), a solitary wave is a particular wave which is non-decaying and localized along a distinct line in the xy- plane. A solitary wave solution to the KdV equation can be considered as a special two-dimensional solitary wave with its crest localized along the y-axis. The stability of such solitary wave solution to the KdV equation under a transverse disturbance is initiated by Kadomtsev and Petviashvili in 1970 [23]. There are two versions of the

KP equation

(ut + 6uux + uxxx)x + 3σuyy = 0.

The case with σ = −1(called KP-I equation) corresponds to the presence of a strong surface tension and the case with σ = 1(called KP-II equation) corresponds to the weak or absence of a surface tension in the long wave model of the water wave problem.

The instability of the solitary wave solution of the KP-I equation has been proven under either periodic or localized disturbance in the direction of propagation[20][50][51].

Well-posedness for the KP-I equation has also been studied by Molinet, Saut and

Tzvetkov[44]. In this thesis, we are interested in the asymptotic models to the water wave equation with zero surface tension, therefore our studies are based on the KP-II equation (we call it simply the KP equation here and after). By integrating the KP equation with respect to x, we can get another form of KP equation which has a nonlocal term

−1 ut + 6uux + uxxx + 3Dx uyy = 0 (1.5)

5 −1 where we assume u(x, y, t) → 0 as x → +∞, and deﬁne the nonlocal operator Dx by Z +∞ −1 Dx v(x, ··· ) := − v(s, ··· )ds. x Then, if u and all its x derivatives vanishes as x → −∞, the equation (1.5) becomes

Z +∞ uyy(x, y, t) dx = 0. (1.6) −∞

The Cauchy problem for the KP equation has been discussed in many literatures

[14][15][7][53] under the assumption that the non-local term exists and the initial data belongs to L2(R2). However, a two-dimensional solitary wave with a local transverse disturbance usually does not satisfy the above assumption. For instance, if the disturbance changes the amplitude of the solitary wave, then the condition (1.6) does not hold. With this type of disturbance as the initial condition, the KP equation is not well-posed. Mizumachi and Tzvetkov[43] first considered this type of stability in an infinite domain in x and a periodic domain in y, i.e. (x, y) ∈ R×S1. They claimed that if the initial value with the small disturbance is sufficiently close to an exact KdV soliton solution, then in the sense of L2(R × S1) norm the solution stays arbitrary close to the exact KdV solution with a phase shift. Then, Mizumachi extended the stability study over an infinite domain for both x and y, i.e. (x, y) ∈ R2. His recent results[42] have revealed that the solitary wave solution to the KdV equation under small transverse perturbations in L2(R2) is stable in the sense that the amplitude of the line soliton converges to that of the line soliton at the initial time and the phase shift of the crest approaches to y = ±∞ at a finite speed. This gives a rigorous mathematical proof of the problem originally started by Kadomtsev and Petviashvili.

6 The interactions of solitary waves form complex patterns in the xy-plane. Miles[40]

[41] is the ﬁrst to give a theoretical discussion about the interaction of two oblique solitary waves. Two diﬀerent situations have been considered[40] according to the angle between the two interacting solitary waves. If the interacting angle is large, the overlapping region is small, hence the interaction is expected to be weak. This situation is then called weak interaction and the wave is modeled as a linear superposition of the two solitons plus a higher order interaction term. If the interacting angle is small, it is called strong interaction and the wave is handled according to the transformation technique from Whitham ([56] $ 17.2). In [41], Miles studied resonantly interacting solitary waves among three oblique solitary waves. He derived the formula of the ratio between the maximum amplitude of the intersection and that of the incident wave for all these cases. According to his theory, the maximum amplitude ratio can be four fold. There are many numerical simulation (for examples,

[3][17][27][36][39][52][54][59]) of water waves with various amplitude and angle of the incident waves. However, there are considerable discrepancies between these numerical results and the predictions drew from Miles theory. This is the main motivation of this thesis.

The rest of this thesis is organized as follows.

In chapter 2, we consider the Euler equation for inviscid, irrotational and incompressible ﬂow where the water surface tension is neglected. With the assumption of small wave amplitude and weak dispersion, a Boussinesq-type system of equations with respect to the surface elevation can be obtained from the Euler equation. One can also derive the Benney-Luke equation with respect to the velocity potential. Then under the quasi-two dimensionality and a far ﬁeld unidirectional approximation, we

7 derive the KP equation as the leading order approximation. We also include the higher order terms of the KP equation. In chapter 3, we brieﬂy review the Miles theory of the interaction of the two oblique solitary waves. We also give a brief review of the recent result on the exact solutions of the KP equation, called the KP solitons, developed in [9][10][11][26][29][30]. Then, we discuss some two-dimensional wave patterns generated by certain exact KP solitons, including one-line soliton, Y-shape soliton with resonant condition, X-shape soliton and another KP soliton related to the Mach reﬂection phenomena. The Miles theory is reconsidered based on the KP equation. Chapter 4 concerns the numerical simulation of the Euler equation. The numerical solution is based on the higher order amplitude approximation of the Euler equation, and using the pseudospectral method originally developed by Tanaka[54].

The scheme has been known as one of the best numerical scheme to compute the

Euler equation, and it assumes that the amplitude of the wave is small(week nonlinearity). Then, we perform a numerical simulation of the Mach reflection problem. In chapter 5, we use normal form theory to study the higher order KP equation derived in chapter 2. Then, we show that certain soliton solutions to the higher order KP equation can give a good approximation to the numerical simulation of the Mach reflection phenomenon obtained in chapter 4. Here, we solve an inverse problem in the sense that we construct a soliton solution of the higher order KP equation from the numerical data. Chapter 6 is a numerical study of the two dimensional stability problem. We first test the stability of line solitons with a transverse disturbance and explain the results using the recent study by Mizumachi[42]. Then we perform a numerical simulation with the initial value containing only a fraction of the exact

8 Y-shape soliton solution. Our numerical discoveries show that the solution develops into the Y-shape exact soliton solution up to a phase shift of the initial one.

9 Chapter 2: Basic equations of shallow water waves

We expand the Euler equation under the assumptions of small amplitudes and long waves and derive the approximated equation including next order terms. The

Boussinesq equation and the Benny-Luke equation are derived as the leading order approximation, both of which admit solitary wave solutions. We further assume a quasi-two dimensionality which leads to another approximated model, the KP equation. Although it admits solitary wave solution, it is no longer rotationally symmetric.

2.1 The Euler equation

The water body under consideration is in the domain

3 Ωt = {(x, y, z) ∈ R : 0 ≤ z ≤ h0 + η(x, y, t)},

where z = 0 indicates a ﬂat bottom, h0 the quiescent water depth and η(x, y, t) the surface elevation from the undisturbed water surface. We consider the ﬂow is

˜ ˜ ∂ ∂ ∂ irrotational, i.e., ∇ × u = 0 where u is the velocity of the ﬂow and ∇ = ( ∂x , ∂y , ∂z ). Thus, there exists a potential function φ such that u = ∇˜ φ. We also assume that the

ﬂow is incompressible, that is, ∇˜ · u = 0. Then the surface wave is described by the

Euler equation which consists of four equations, one for the interior water body, one at the impermeable bottom boundary(assume a ﬂat bottom) and two at the water

10 surface ˜ 4φ = 0 for 0 < z < h0 + η

φz = 0 at z = 0  (2.1) 1 ˜ 2 φt + |∇φ| + gη = 0  2 at z = h0 + η ηt + ∇φ · ∇η = φz  ∂ ∂ ˜ ˜ 2 where ∇ = ( ∂x , ∂y ), and ∆ = ∇ . Let λ0 be a typical wave length, a0 be the maximum water surface elevation from the quiescent water, and λ0 be the time scale where C0

√ 0 0 0 0 0 0 C0 = gh0 is the phase velocity. The non-dimensional variables {x , y , z , t , η , φ } are deﬁned as λ x = λ x0, y = λ y0, z = h z0, t = 0 t0 0 0 0 C 0 (2.2) 0 a0 0 η = a0η , φ = λ0C0φ . h0 Then the Euler equation in the non-dimensional form is given by (drop the prime 0)

φzz + β∆φ = 0 for 0 < z < 1 + αη

φz = 0 at z = 0 1 1 α  φ + α|∇φ|2 + φ2 + η = 0  (2.3) t 2 2 β z  at z = 1 + αη 1 ηt + α∇φ · ∇η = φz  β  where the parameters α and β are given by

a h 2 α = 0 and β = 0 . h0 λ0

Hence, α stands for the ratio between the maximum water elevation from the still water surface and the water depth, and β stands for the squared ratio between the water depth and the typical wave length.

2.2 Boussinesq approximation for the Euler equation

It is Ursell[55] who points out the diﬀerent wave equations according to α

β, α β and α ∼ β. α β implies the weak nonlinearity which leads to a

11 linear dispersive wave equation. β α implies the weak dispersive wave which

indicates a nonlinear hyperbolic equation. The balance α ∼ β is called the Boussinesq approximation which is of our interests.

Let be a small positive parameter described in the Boussinesq approximation, i.e.,

α ∼ β = O(). (2.4)

We employ the asymptotic perturbation theory to derive the approximate equations.

Notice that the solution to the first two equations in (2.3) can be formally written as φ(x, y, z, t) = cos(zpβ∆)f(x, y, t) z2 z4 z6 (2.5) = f − β ∆f + β2 ∆2f − β3 ∆3f + O(4) 2 24 720 where f(x, y, t) := φ(x, y, 0, t). Then, we insert this solution into the last two equations in (2.3). The first equation at the surface gives α β αβ β2 η + f + |∇f|2 − ∆f + (∆f)2 − ∇f · ∇(∆f) − 2η∆f + ∆2f t 2 2 t 2 t 24 t β3 1 − ∆3f + α2β η(∆f)2 − η∇f · ∇(∆f) − η2∆f (2.6) 720 t 2 t αβ2 + 3|∇(∆f)|2 + ∇f · ∇(∆2f) + 4η∆2f − 4∆f∆2f = O(4). 24 t and the second gives β αβ β2 β3 η + ∆f + α∇ · (η∇f) − ∆2f − ∇ · (η∇(∆f)) + ∆3f − ∆4f t 6 2 120 5040 (2.7) 1 αβ2 − α2β η∇(∆f) · ∇η + η2∆2f + ∇ · η∇(∆2f) = O(4). 2 24 From equation (2.6), we can obtain a formula of η up to the first order of in terms

of f. Then we insert this formula into equation (2.6) to eliminate η in the second

order terms. We repeat this procedure to get the desired order for the formula of η

in terms of function f. We have, up to the second order,

α 2 β η = −ft − |∇f| + ∆ft 2 2 (2.8) αβ β2 − (∆f)2 + 2f ∆f − ∇f · ∇(∆f) − ∆2f + O(3) 2 t t 24 t 12 Now, we can use formula (2.8) to eliminate η in the higher order terms in formula

(2.6) and (2.7). Then, from the above two equations, we derive an equation for f

β 1 f − 4f = − α (∇ · (f ∇f) + ∇f · ∇f ) + 4f − 42f tt t t 2 tt 3 αβ 1 2 2 − (4f) − 2∇f · ∇(4f) + 2ft4ft − ft4 f (2.9) 2 2 t α2 β2 1 − ∇ · (|∇f|2∇f) − (42f − 43f) + O(3) 2 24 tt 5 Remark: The terms up to the order of O() in equation (2.6) and (2.7) give a

Boussinesq-type system

α 2 β 2 η + ft + |∇f| − ∆ft = O( ) 2 2 (2.10) β η + ∆f + α∇ · (η∇f) − ∆2f = O(2) t 6 This is exactly equation (1.1) if one takes the derivative with respect to t in the ﬁrst equation, eliminates ηt from both equations, and transforms equation (1.1) into non- dimensional coordinate using (2.2). Eliminating η from (2.10) or equivalently replace

4ftt in (2.9) with ftt = 4f, the terms up to the order of O() in (2.9) gives the

Benney-Luke equation

β 2 (1 − 3 4)ftt − 4f + α (∇ · (ft∇f) + ∇f · ∇ft) = O( ). (2.11)

We further consider a far ﬁeld unidirectional solitary waves and introduce a moving reference frame coordinate

ξ = x cos Ψ0 + y sin Ψ0 − t, τ = t, (2.12)

where Ψ0 is the angle between the crest of the solitary wave and positive y-axis oriented in the counterclockwise direction from positive y-axis(Figure 2.1).

13 Figure 2.1: The three-dimensional view of a solitary wave(left) and its contour plot(right).

Then, equation (2.9) has the form

3 1 2F + αF 2 + βF = O(2), (2.13) τ 2 ξ 3 ξξξ which is a (potential) KdV equation with F (ξ, τ) = f(x, y, t). The solution to the leading order of (2.13) is

4β 4k2β F (ξ, τ) = k tanh k ξ − τ 3α 6 where k is a free parameter. In the coordinate of (x, y, t), we have

4β 4βk2 f(x, y, t) = k tanh k x cos Ψ + y sin Ψ − (1 + )t (2.14) 3α 0 0 6

Water elevation η up to the leading order of (2.8) has the form

η(x, y, t) = −φt + O() = −ft + O() 4β 4βk2 (2.15) = k2 sech2 k x cos Ψ + y sin Ψ − (1 + )t + O() 3α 0 0 6

14 2.3 KP equation and higher order terms

We have made two assumptions in the water wave expansion which are small am-

plitude and long wave in formula (2.4). If we further assume quasi-two dimensionality, √ ζ = γy, γ = O()

which means a weak dependence on y variable and consider a far ﬁeld unidirectional

moving frame with respect to x-axis,

ξ = x − t, τ = t (2.16)

then equation (2.9) has the form β α2 2f + 3αf f + f + γf − f 3 τξ ξ ξξ 3 ξξξξ ζζ 2 ξ ξ αβ β2 + (f f − f f ) − f + αγ (f f + 2f f ) 2 ξξ ξξξ ξ ξξξξ 30 ξξξξξξ ξ ζζ ζ ξζ βγ + f − α (2f f + f f ) − βf − 2f = O(3) 6 ξξζζ ξ ξτ τ ξξ ξξξτ ττ We also calculate (2.8) with the change of variables (2.16) and obtain α β η = f − f − (f )2 − f + O(2) ξ τ 2 ξ 2 ξξξ

−1 −1 Deﬁne v := fξ and introduce D := ∂ξ as a formal integral operator. Then 1 β v = − 3αvv + v + γD−1v + O() τ 2 ξ 3 ξξξ ζζ 1 β2 2βγ v = ∂ 9α2v2v + αβ(2vv + 3v v ) + v + v ττ 42 ξ ξ ξξξ ξ ξξ 9 ξξξξξ 3 ξζζ −1 −1 2 −3 +3αγ(vD vζζ + ∂ζ D (vvζ )) + γ D vζζζζ + O().

Then we obtain β 2v + 3αvv + v + γD−1v τ ξ 3 ξξξ ζζ 19 53 5 βγ γ2 + β2v + αβ v v + vv + v − D−3v 180 ξξξξξ 12 ξ ξξ 6 ξξξ 2 ξζζ 4 ζζζζ (2.17) 5 3 1 + αγ vD−1v + 2v D−1v − ∂ D−1(vv ) + v D−2v = O(3). 4 ζζ ζ ζ 4 ζ ζ 2 ξ ζζ

15 and the water surface elevation η is given by

−1 α 2 β 2 η = v − D vτ − v − vξξ + O( ) 2 2 (2.18) α β γ = v + v2 − v + D−2v + O(2) 4 3 ξξ 2 ζζ Notice that this formula is corrected up to order . Indeed, the α, β and γ terms

will appear in the normal form transformation (chapter 5) in deriving the solution of

higher order KP equation.

The ﬁrst line of (2.17) of order O() is the KP equation and the terms in the

second and third lines are of order O(2) which is a higher order correction to the KP

equation. With a variable change

2 ξ = pβ X, ζ = pβγ Y, τ = −6pβ T, v = u, (2.19) 3α

we ﬁnd the standard KP equation with higher order terms

−1 uT = 6uuX + uXXX + 3D uYY 19 3 3 53 5 + u + u − D−3u + u u + uu 60 XXXXX 2 XYY 4 YYYY 6 X XX 3 XXX (2.20) −2 5 −1 −1 3 −1 9 +u D u + uD u + 4u D u − ∂ D (uu ) + O( 2 ) X YY 2 YY Y Y 2 Y Y

and 2 1 2 1 αη = u + u2 − u + D−2u + O(3). (2.21) 3 9 9 XX 3 YY

In terms of the non-dimensional physical variables (x, y, t) and u, we have

1 1 1 2 X = √ (x − t),Y = √ y, T = − √ t, v = u (2.22) β β 6 β 3α and in terms of the physical variables (˜x, y,˜ t˜) andη ˜, we have

1 1 C0 3 X = (˜x − C0t˜),Y = y,˜ T = − t,˜ u = η˜ (2.23) h0 h0 6h0 2h0

16 With formula (2.23), the KP equation in the physical coordinate is given by

2 3C0 C0h0 C0 η˜t˜ + C0η˜x˜ + η˜η˜x˜ + η˜x˜x˜x˜ + η˜y˜y˜ = 0. 2h0 6 x˜ 2

As a particular solution, we have one-line soliton solution in the form, s 3a a 1 2 0 0 2 ˜ η˜ = a0sech 3 x˜ +y ˜tan Ψ0 − C0 1 + + tan Ψ0 t − x˜0 , (2.24) 4h0 2h0 2 where a0 > 0, Ψ0 andx ˜0 are arbitrary constants. To compare with formula (2.15), a rotation free solitary wave, we rewrite (2.24) in the following form s 3a a η˜ = a sech2 0 x˜ cos Ψ +y ˜sin Ψ − C cos Ψ 1 + 0 0 4h3 cos2 Ψ 0 0 0 0 2h 0 0 0 (2.25) 1 + tan2 Ψ t˜− x cos Ψ . 2 0 0 0

Notice that the width of the line-soliton solution to the KP equation, proportional

q 3 4h0 to cos Ψ0, depends on the angle Ψ0. If we considerη ˜ with the same amplitude 3a0

o o o a0 = 0.1 and h0 = 1 but diﬀerent angle Ψ0 = 0 , 15 , 30 , the width of the soliton is proportional to 3.65, 3.53 and 3.16, respectively. Figure 2.2 shows the contour plots of a line-soliton in the three cases. It is noticeable that the width of the soliton becomes shorter as the angle becomes larger which is not physical and should be corrected when we apply the KP equation to a physical problem.

17 o o Figure 2.2: Contour plot of line soliton with Ψ0 = 0 (left), Ψ0 = 15 (middle) and o Ψ0 = 30 (right). The widths of the soliton in the latter two cases are 96.6% and 86.6% of that of the original soliton.

Note that using the following identity

2 − 1 1 2 2 cos Ψ = (1 + tan Ψ ) 2 = 1 − tan Ψ + O( ), (2.26) 0 0 2 0 we obtain the velocity of the solitary wave up to O(2), i.e.

a0 1 2 a0 2 cos Ψ0 1 + + tan Ψ0 = 1 + + O( ), 2h0 2 2h0 which does not depend on the angle up to O(2).

We have derived several approximated model equations from the leading order terms of the Euler equation under the assumptions of small amplitude and long wave.

With the quasi-two dimensionality assumption, we obtain the KP equation and its solitary wave solution. However, the width of the solitary wave is dependent on the angle which is not physical. It is necessary to include the higher order terms in the

KP equation to obtain an angle correction which will be discussed in chapter 5.

18 Chapter 3: The Miles theory and the KP theory for the Mach reﬂection

In this section, we first give a historical background of the Mach reflection phenomena observed in shallow water. Then, we briefly review the Miles theory about the interaction of oblique solitary waves and the reflection of an obliquely incident wave onto a vertical wall which is divided into two cases, the regular reflection and the Mach reflection. We also describe several wave patterns generated by the soliton solutions to the KP equation using the explicit formulas of those solutions developed in the recent papers (see for examples, [9][10][11][26][29][30]). We then discuss the

Miles theory from the KP point of view, and show that KP soliton solutions provide systematic description of the interaction of solitary waves. Finally, the numerical results are compared with the theoretical predictions from the Miles theory and the

KP theory.

3.1 The Mach reﬂection in shallow water

The Mach reﬂection phenomenon was ﬁrst recognized as a characteristic of shock waves by Ernst Mach and was studied extensively in gases (see for example [33]).

The Mach reﬂection in water waves was ﬁrst observed in Perroud’s experiment[48]

o in 1957 where three cases of reﬂection were given: (1) Ψ0 < 20 : the wave crest was

19 bent so that it was perpendicular to the wall and no reﬂective wave was observed;

o o (2) 20 < Ψ0 < 45 : the incoming wave, the reﬂective wave and the third wave

o perpendicular to wall were all present; (3) Ψ0 > 45 : only the incoming wave and

the reﬂective were present. We show schematic ﬁgures 3.1 and 3.2 to illustrate those

observations.

Figure 3.1: The schematic regular reﬂection.

The picture on the left in ﬁgure 3.1 is a semi-inﬁnite solitary wave propagating

parallel to the horizontal boundary to the right. Once it meets the point O at which

the second boundary forms an angle Ψ0 from the horizon, it produces a reﬂective soli-

tary wave and their intersection is on the boundary. This is the case (3) in Perroud’s

experiment and is called the regular reﬂection.

20 Figure 3.2: The schematic Mach reﬂection.

If the angle Ψ0 gets smaller, then the regular reflection will transit into the Mach reflection phenomenon. The characteristic is that the intersection point of the incident solitary wave and the reflective wave are no longer on the boundary. Instead it forms a third wave, called the Mach stem, extended to the boundary. In fact, this is the case for both (1) and (2) in Perroud’s experiment although the reflective wave was not observed when the angle is less than 20o.

In ﬁgures 3.1 and 3.2, if one places an analogous semi-inﬁnite solitary wave in the

opposite side of the boundary, we would obtain the blue ﬁgures as a mirror image of

the black ones. This can be considered as two oblique solitary waves initially inter-

secting at point O and forming a V-shape wave proﬁle. Additionally, the interaction

at a later time, shown in the pictures on the right, can be considered as combining the

two oblique boundaries horizontally. Miles[40][41] in 1977 discussed the interaction

of two oblique solitary waves and derived the formula of the amplitude ratio between

the waves run up at the wall and the incident wave, where the maximum ratio is four

21 fold. Melville[39] tested the amplitude ratio by experiment in 1979 and concluded

that both experiment results from his and Perroud’s casted considerable doubt on

Miles’ model to describe the Mach reﬂection. There are also numerical simulations

(see for examples, literatures[3][17][27][52][54] [?][?]) for the Mach reﬂection problem, and all those studies obtained the similar results as Melvilles observation. That is, they could not observe Miles’ prediction of four fold amplitude ampliﬁcation at the

Mach stem. To discuss further the Mach reﬂection problem, we ﬁrst give a brief review of the Miles theory about the interaction of two oblique solitary waves.

3.2 Miles’ models of the interaction of the oblique solitary waves

In the study of the interaction of two oblique solitary waves, Miles[40][41] was the ﬁrst to give an analysis based on the leading order terms of water wave equations

(2.9). He derived the formula for the amplitude ampliﬁcation factor and for the angles of interest under three diﬀerent situations.

In Miles’ ﬁrst article[40], the water wave is considered in a far ﬁeld moving reference coordinate   ξ1 = x cos Ψ1 + y sin Ψ1 − t   ξ2 = x cos Ψ2 + y sin Ψ2 − t (3.1)    τ = t Notice that the reference frame is similar to that introduced in section 2.2 formula

(2.12) under the one line soliton regime. Therefore, each solitary wave in its own

territory has its velocity potential and surface elevation in the form of formula (2.14)

and (2.15). With the special relation of small parameters α and β considered by

22 3 Miles, which is β = 4 α, we have

Fn(ξn, τ) = lntanh (lnθn) (3.2)

2 2 ηn(ξn, τ) = lnsech (lnθn) (3.3)

α 2 where θn = ξn − 2 lnτ with some constant ln for n = 1, 2. With the coordinate change from (x, y, t) to (ξ1, ξ2, τ), the leading order terms from (2.9) becomes

3 3 −4κ∂ ∂ f + (∂ + ∂ ) 2∂ f + α[ (∂ f)2 + (∂ f)2 1 2 1 2 τ 2 1 2 2 (3.4) 1 +(3 − 4κ)(∂ f)(∂ f)] + β(∂ + ∂ )3f = O(2) 1 2 3 1 2

2 1 where κ = sin Ψ0 with Ψ0 = 2 |Ψ1 −Ψ2|, and ∂n = ∂/∂ξn for n = 1, 2. Miles separately discussed the solutions to the above equation based on the value of κ.

2 3.2.1 Weak interaction: κ = sin Ψ0 ∼ O(1)

2 When κ = sin Ψ0 ∼ O(1), i.e., the angle formed by the two solitary waves is relatively large, the dominant term in (3.4) is −4κ∂1∂2f. Truncating the higher order terms in (3.4) leads to

∂1∂2f = O() ⇒ f(ξ1, ξ2, τ) = F1(ξ1, τ) + F2(ξ2, τ) + O()

⇒ η(ξ1, ξ2, τ) = (∂1 + ∂2)f + O() = η1(ξ1, τ) + η2(ξ2, τ) + O()

The above formula indicates that the leading order solution can be expressed as a

superposition of two solitary wave solutions and this implies that their interactions are

weak up to a higher order eﬀect. Hence, it is called weak interaction when κ ∼ O(1).

To study the higher order interaction eﬀect, a next order interaction term is included

in the formula of f,

2 f(ξ1, ξ2, τ) = F1(ξ1, τ) + F2(ξ2, τ) + αF12(ξ1, ξ2, τ) + O( ).

23 Inserting this function to (3.4) and considering that F1 and F2 satisfy (2.13), one can

derive the equation for F12

(3 − 4κ)(∂1 + ∂2)∂1F1∂2F2 − 4κ∂1∂2F12 + O() = 0 (3.5)

Integrating (3.5) yields the solution

3 F = − 1 (∂ + ∂ )F F + O(). 12 4κ 1 2 1 2

The leading order term in F12 can be absorbed in the variable of F1 and F2

2 f = F1(ξ1, τ) + F2(ξ2, τ) + αF12(ξ1, ξ2, τ) + O( ) 3 = F (ξ , τ) + F (ξ , τ) + α − 1 (F ∂ F + F ∂ F ) + O(2) (3.6) 1 1 2 2 4κ 2 1 1 1 2 2 2 = F1(ξ1 + χ2, τ) + F2(ξ2 + χ1, τ) + O( ),

3 where χn = α( 4κ −1)Fn(ξn, τ). χ1, χ2 indeed gives the phase shift of F2 and F1. Using formula (2.8) up to O(), the water elevation can be expressed by

2 η = N1(ξ1 + χ2, τ) + N2(ξ2 + χ1, τ) + αIN1N2 + O( ), (3.7)

1 3 1 2 2 3 where Nn = (∂n − 3 β∂n)Fn + 4 α(∂nFn) + O( )(n = 1, 2) and I = 2κ − 3 + 2κ. The eﬀect of higher order interaction in the far ﬁeld can be interpreted as follows.

1. Near N1 in the far ﬁeld, N2 ∼ 0 implies η ∼ N1(ξ1 + χ2, τ).

(a) For y > 0, then F2 > 0 implies χ2 > 0. That is, the crest of solitary wave

1 is shifted to the left.

(b) For y < 0, then F2 < 0 implies χ2 < 0. That is, the crest of soliton wave

1 is shifted to the right.

2. Near N2 in the far ﬁeld, N1 ∼ 0 implies η ∼ N2(ξ2 + χ1, τ).

24 (a) For y > 0, then F1 < 0 implies χ1 < 0. That is, the crest of solitary wave

2 is shifted to the right.

(b) For y < 0, then F1 > 0 implies χ1 > 0. That is, the crest of solitary wave

2 is shifted to the left.

Figure 3.3 shows how the solitary waves shift their positions from a superposition under the consideration of a higher order interaction term.

Figure 3.3: The contour plot of the surface elevation η in (3.7). The solitary waves N1 and N2 have the phase shifts due to the interaction.

A reﬂection of an incident wave at a rigid wall can be modelled by setting Ψ2 =

2 −Ψ1 = Ψ0, κ = sin Ψ0, F1 = F2 and N1 = N2 ≡ N in (3.6) and (3.7). Then from

(3.7), the maximum run-up at a wall is given by

aM 3 2 = 2 + I ai = 2 + 2 − 3 + 2 sin Ψ0 ai. (3.8) ai 2 sin Ψ0 where aM is the non-dimensional maximum amplitude of the wave run-up at the wall

a0 and ai is the non-dimensional amplitude of the incident wave , i.e., ai = α. h0

25 2 3.2.2 Strong interaction: κ = sin Ψ0 ∼ O()

2 When κ = sin Ψ0 ∼ O(), all the terms in equation (3.4) are equally important.

Miles called this case strong interaction of two solitary waves and considered two unequal-amplitude solitary waves in the form of (3.3). Applying the transformation given by Whitham[56]

2 f = D ln G(ξ1, ξ2), η = D ln G(ξ1, ξ2) + O(), (3.9)

where D = ∂1 + ∂2 and ∂n = ∂/∂ξn , equation (3.4) can be expressed in a bilinear form

of function G(ξ1, ξ2)

3 3 3 2 2 3 α[{GD − (DG)}(D − 4l1∂1 − 4l2∂2)G + 3{(D G) − (DG)(D G)}] (3.10) 2 − 16l1l2κ{G∂1∂2G − (∂1G)(∂2G)} = O( )

A solution to the leading order of (3.10) can be found in the form G = 1 + G1 + G2 +

2δ e G1G2 where

2 −2lnξn 1 4κ − 3α(l1 − l2) Gn = e for n = 1, 2 and δ = ln 2 . (3.11) 2 4κ − 3α(l1 + l2)

Then the formula (3.9) gives

2 2 2 2δ 2 2 2 1 l1G1 + l2G2 + (l1 − l2) G1G2 + e {(l1 + l2) + l2G1 + l1G2}G1G2 η = 2δ 2 . (3.12) 4 (1 + G1 + G2 + e G1G2)

2 Consider the special case of two symmetric waves, which have equal amplitudes l1 =

2 l2 = 1, opposite angles Ψ2 = −Ψ1 = Ψ0 and same travelling speed c1 = c2 = c. Then, (3.12) becomes 2δ 1 G1 + G2 + e {4 + G1 + G2}G1G2 η = 2δ 2 . 4 (1 + G1 + G2 + e G1G2)

1 We treat G1 and G2 as the variables of function Z := 4 η with the conditions that

−δ G1 > 0 and G2 > 0. First, ﬁnd the critical values of Z which gives G1 = e and

26 −δ G2 = e . Then, use the second derivative test for the two-variable cases, we have −δ 2 2 − e Z Z − Z | −δ −δ = > 0 G1G1 G2G2 G1G2 G1=e ,G2=e 4e−3δ(1 + e−δ)4 1 Z | −δ −δ = − < 0 G1G1 G1=e ,G2=e 2e−2δ(1 + e−δ)2

−δ We ﬁnd that the maximum amplitude aM can be obtained when G1 = e and

−δ G2 = e , which gives

aM 4 = q (3.13) ai 1 1 + 1 − 2 kM

sin Ψ0 where kM = √ . This is the k parameter in the theoretical result about the am- 3ai

plitude ampliﬁcation factor and Miles used √Ψ0 instead. One should note that the 3ai regular and real solution f in (3.9) for l1 = l2 exists only when κ > 3α, and this

solution corresponds to the regular reﬂection, that is, the solution represents two

obliquely interacting solitary waves with a constant phase shift.

3α 2 Remark: Deﬁne κ± = 4 (l1 ± l2) . If κ = κ−, then for the case l1 6= l2, the expression of δ in formula (3.11) leads to e2δ = 0. Hence, function G consists of three

terms. For the case l1 = l2, κ = κ− = 0 leads to Ψ0 = 0 which degenerates the two

2δ solitary waves into one. If κ = κ+ = 3α, then e becomes inﬁnity and the solution

breaks. The oblique interaction of two solitary waves at the value κ = 3α leads to a

resonant phenomenon, and we discuss the resonant interaction in the next section.

3.2.3 The resonant interaction

The resonant interaction between two oblique solitary waves is studied in [41],

where Miles identiﬁed it as the case corresponding to the Mach reﬂection. Shown in

ﬁgure 3.2 is an incident wave with angle −Ψ0 moving in the arrow direction L1. After

encountering a wall, it generates a reﬂective wave moving in the direction L2 and a

third wave perpendicular to the wall called Mach stem moving in the direction of L3.

27 We can express the argument in the surface elevation formula (3.3) in the form of

Ln · (x, y) − wnt where

α L = (l cos Ψ , l sin Ψ ), and w = l c = l (1 + l2 ). (3.14) n n n n n n n n n 2 n

Then, the resonant condition requires

L3 = L1 + L2, and w3 = w1 + w2. (3.15)

With small angle assumption, cos Ψn and sin Ψn are approximated by 1 and Ψn, respectively. Then, the resonant conditions gives

l = l + l 3 1 2 (3.16) l3Ψ3 = l1Ψ1 + l2Ψ2

Note that Ψ3 = 0, then the second formula becomes

l1Ψ1 + l2Ψ2 = 0. (3.17)

When κ = κ+, the small angle assumption leads to

1 3α κ = sin2 Ψ ∼ Ψ2 = (Ψ − Ψ )2 = (l + l )2 0 0 4 2 1 4 1 2

Then we have, √ Ψ2 − Ψ1 = 3α(l1 + l2).

Multiplying Ψ2 and applying (3.17), one gets Ψ2 and Ψ1, i.e.,

√ √ √ Ψ2 = l1 3α = l1 3ai, and Ψ1 = −l2 3ai. (3.18)

Let the amplitude of the incident wave be 1, i.e., l1 = 1, and Let Ψ1 = −Ψ0, formula

(3.18) and (3.16) give the value of the rest parameters as follows,

√ {l1, l2, l3} = {1, kM , 1 + kM }, {Ψ1, Ψ2, Ψ3} = 3ai{−kM , 1, 0} (3.19)

28 Ψ0 where kM = √ . To ﬁnd the formula of Ψ∗, the angle formed by the wall and 3ai the trace of the resonant point, consider a moving reference frame with a velocity

c∗ = c∗(cos Ψ∗, sin Ψ∗). Then, all the three solitary waves are stationary in this

moving frame and we can ﬁnd,

cn = c∗ cos(Ψn − Ψ∗) for n = 1, 2, 3.

Using formula (3.14) for the velocity, the ratio between c1 and c2 can be written as

1 2 1 + 2 αl1 cos(Ψ1 − Ψ∗) 1 2 = . 1 + 2 αl2 cos(Ψ2 − Ψ∗) 1 By invoking Ψn − Ψ∗ = O(α 2 ), we have up to O(α) that 1 1 − 1 (Ψ − Ψ )2 1 1 1 + α(l2 − l2) ∼ 2 1 ∗ ∼ 1 − (Ψ − Ψ )2 + (Ψ − Ψ )2. 1 2 1 2 1 ∗ 2 ∗ 2 1 − 2 (Ψ2 − Ψ∗) 2 2 Then, we obtain 2 2 1 α l2 − l1 Ψ∗ = (Ψ1 + Ψ2) + 2 2 Ψ2 − Ψ1

Substituting l1 and l2 from (3.18) into the above formula, Ψ∗ can be expressed sym- metrically as follows, 1 ra Ψ = (Ψ + Ψ + Ψ ) = i (1 − k ). (3.20) ∗ 3 1 2 3 3 M

The Mach reﬂection case corresponds to Ψ∗ > 0 which leads to kM < 1 from formula

(3.19). According to (3.3), the amplitude ratio between the Mach stem and the incident wave is given by 2 aM l3 2 = 2 = (1 + kM ) . (3.21) ai l1 The amplitude ratio given by Miles from formula (3.8), (3.13) and (3.21) are  3 2 + − 3 + 2 sin2 Ψ a Weak Interaction  2 sin2 Ψ 0 i  0 aM  4 = Regular reﬂection kM > 1 (3.22) ai q 1 1 + 1 − k2  M  2 (1 + kM ) Mach Reﬂexion kM < 1

29 Ψ0 where kM = √ . 3ai

3.3 KP soliton solutions

In this section, we review several special patterns of the KP soliton solutions where most of the formula have been derived from [9][10][11][26][29][30]. The soliton solution u(x, y, t) of the KP equation can be obtained by τ function

∂2 u(x, y, t) = 2 ln τ(x, y, t) (3.23) ∂x2

which depends on M k-parameters (k1, . . . , κM ) and an N × M matrix A, i.e.,

X τ(x, y, t) = A(J)E(J) (3.24) J⊂{1,...,M}

Here, J = {j1, . . . , jN } is a subset of {1,...,M} consisting of N unique integers

with ordering j1 < ··· < jN . A(J) is the N × N minor of the A-matrix with the Q column index in J and E(J) = l>m(kjl − kjm )Ej1 ··· EjN with Ei = exp(θi) where

2 3 θi = kix + ki y + ki t. The non-singular solution of u is obtained by imposing the non-negativity condition on the minors,

A(J) ≥ 0, for all J. (3.25)

We call a matrix A having the property (3.25) totally non-negative matrix, referred to as TNN matrix. It has been proven in the aforementioned articles that the τ function in (3.24) associated with the TNN irreducible N × MA matrix corresponds to an (N,M − N)-soliton solutions of the KP equation. The notation (N,M − N) indicates N asymptotic line solitons for y 0 and M −N asymptotic line solitons for y 0. Here, an irreducible matrix means no zero column and each row must contain at lease one nonzero element after the pivot column. Furthermore, if the row echelon

30 form of A-matrix has {i1, i2, . . . , iN } as the pivot columns and {j1, j2, . . . , jM−N } as the non-pivot ones, then there exists a permutation π such that the following two conditions are both satisﬁed

1. in < π(in) for n = 1,...,N which corresponds to N asymptotic line solitons for

y 0.

2. jn > π(jn) for n = 1,...,M − N which corresponds to M − N asymptotic line

solitons for y 0.

We deﬁne a chord diagram as follows. Draw a number line as the k-coordinate and mark the M k-parameters. Then, connect two k-parameters by drawing a chord above or below the k-coordinate. The permutation π ∈ SM can be displayed by a chord diagram where i and π(i) are connected by an upper chord if i < π(i) for i = 1, 2, ··· ,M and j and π(j) are connected by a lower chord if j > π(j) for j = 1, 2, ··· ,M. Therefore, a chord diagram associated with an N × M TNN irreducible A-matrix contains N upper chords [in, π(in)] for n = 1, 2, ··· ,N where

{i1, i2, ··· , iN } are the pivot column of the row echelon form of A-matrix and M − N lower chords [π(jn), jn] for n = 1, 2, ··· ,M − N where {j1, j2, ··· , jM−N } are the non-pivot column of the row echelon form of A-matrix. The values of the unknown parameters in A-matrix give the information of the interior solitons, such as the resonant patterns and the locations of the line solitons.

3.3.1 One-line soliton

For a one-line soliton solution, the k-parameters is given by (k1, k2) and the TNN irreducible A-matrix is a 1 × 2 matrix,

A = (1 a)

31 where a > 0. The correspond τ-function can be written as

τ(x, y, t) = E1 + a E2

2 3 where Ei = exp(θi) with θi = kix + ki y + ki t. Then, √ 1 (θ +θ ) 1 1 (θ −θ ) √ − 1 (θ −θ ) τ = a e 2 1 2 √ e 2 1 2 + ae 2 1 2 a √ 1 (θ +θ ) 1 1 = 2 a e 2 1 2 cosh θ − θ + ln . 2 1 2 a

The condition a > 0 insures non-singular and non-trivial solution and it determines the location of the line soliton. The KP soliton solution is given by,

1 1 1 u = (k − k )2sech2 θ − θ + ln . (3.26) 2 2 1 2 1 2 a

1 The crest of the soliton is localized along the line θ1 −θ2 +ln a = 0, hence it is referred as line-soliton solution, which has an explicit form

1 ln = θ − θ = (k − k )x + (k2 − k2)y + (k3 − k3)t, (3.27) a 2 1 2 1 2 1 2 1 or

1 1 2 2 x + (k1 + k2)y = ln − (k2 + k1k2 + k1)t. k2 − k1 a In general, there are two exponential terms in the τ-function (3.24) that dominate the region on each side of the soliton. They are E(I) and E(J) with the index subset satisfying J = I\{i} ∪ {j}, show in [i, j]. Therefore, we mark the one-line soliton as [i, j]-soliton(ﬁgure 3.4). All other terms are exponentially small in the τ-function

(3.24), hence

τ ≈ A(I)E(I) + A(J)E(J). (3.28)

Based on a similar deduction on τ-function, one can get " Q !# 1 1 A(I) (kl − km) u = (k − k )2sech2 θ − θ − ln − ln km

32 Then the crest of the line-soliton locates on the line

Q A(I) (kl − km) θ = ln + ln km

2 2 3 3 θij = (kj − ki)x + (kj − ki )y + (kj − ki )t. (3.31)

The above two expressions for θij indicate that if the A-matrix is known, then the

phase value θij is known by formula (3.30). Then, the location of the crest of the

[i, j]-soliton at any given time can be obtained from formula (3.31). On the other

hand, if we are provided with the location of the crest of the [i, j]-soliton, then we

can obtain θij by (3.31), hence formula (3.30) leads to the value of the parameters in

the A-matrix. We also deﬁne, based on the expression (3.27), that

2 2 3 3 K[i,j] = (kj − ki, kj − ki ) and Ω[i,j] = kj − ki given kj > ki. (3.32)

where K[i,j] gives the direction of propagation of the line soliton. Figure 3.4 illus-

trates the one-soliton solution with the corresponding chord diagram representing

the permutation i j π : . j i

33 Figure 3.4: The contour plot(left) and the chord diagram(right) of [i, j]-soliton.

If we put the soliton solution u in the form in terms of the amplitude and angle of the line soliton, we have " # rA u = A sech2 [i,j] x + tan Ψ y + c t − θ [i,j] 2 [i,j] [i,j] 0 where A[i,j] and Ψ[i,j] are the amplitude and the angle of the [i, j]-soliton. Compare with equation (3.29), the amplitude and the slope of the one-line soliton can be written in terms of ki and kj, which are

(k − k )2 A = j i , tan Ψ = k + k , (3.33) [i,j] 2 [i,j] i j and we have

2 2 1 3 2 θij c[i,j] = ki + kikj + kj = A[i,j] + tan Ψ[i,j], θ0 = . (3.34) 2 4 kj − ki

If provided the amplitude and the slope of a one-line soliton, we are able to get the value of k-parameters by solving (3.33),

1 1 k = (tan Ψ − p2A ) and k = (tan Ψ + p2A ). (3.35) i 2 [i,j] [i,j] j 2 [i,j] [i,j]

34 3.3.2 Y-shape soliton

There are two types of the Y-shape soliton solutions. One describes the soliton

pattern with two line solitons for y 0 and one line soliton for y 0(see the left

picture in ﬁgure 3.5). The other one is obtained by changing (x, y) → (−x, −y)(the

right picture in ﬁgure 3.5).

Figure 3.5: The three-dimensional view(left), the contour plot(middle) and the chord diagram(right) of two types of Y-shape soliton solution to the KP equation. The k-parameters are (k1, k2, k3) = (−1, −0.25, 0.75).

The total number of line solitons in the far ﬁeld is M = 3. Hence, there are three k-parameters, say k1 < k2 < k3. With formula (3.33), the sum of two k-parameters leads to the slope of the line soliton. Hence, [1, 2]-, [1, 3]- and [2, 3]-soliton are ordered

35 in the increasing slope. It is important to remember that the slope of the line here

is measured from the y axis in the counterclockwise direction. Therefore, we can

identify each line soliton according to their relative slopes, shown in the contour plot

in ﬁgure 3.5. The chord diagram of a soliton pattern can be constructed by the

far ﬁeld solitons, where the upper solitons correspond to the upper chords and the

lower solitons correspond to the lower chords. The chord diagrams of the two Y-shape

solitons are demonstrated below the contour plots in ﬁgure 3.5. Then the permutation

can be determined from the chord diagram which is

1 2 3 1 2 3 π : and π : , 2 3 1 3 1 2

respectively. The number of the line solitons in the upper half plane is N = 2 for the

ﬁrst case of the Y-shape soliton and N = 1 for the second case. Then the A-matrix

associated to the τ function which generates each of two Y-shape soliton solutions

has the following row echelon form

1 0 −b A = and A = (1 a b) 0 1 a

where a and b are positive parameters and may have diﬀerent values in the two

A-matrices.

In the following discussion, we consider the ﬁrst Y-shape soliton. Similar analysis

can be done to the second case. By formula (3.24),

τ = E(1, 2) + aE(1, 3) + bE(2, 3)

2 3 where E(i, j) = (kj − ki)EiEj with Ei = exp(kix + ki y + ki t) for i = 1, 2, 3. For a ﬁxed time value t, say t = 0, consider the line

x = −py where p = tan Ψ. (3.36)

36 Then, Ei = exp(ki(ki − p)y) := exp(hi(p)y) where we deﬁne

hi(p) = ki(ki − p) (3.37)

which is a line function in terms of p with x-intercept ki and slope −ki(ﬁgure 3.6).

Then, at a ﬁxed positive value of y, the dominant exponentials of the τ function is the greatest sum hi(p) + hj(p) for i, j ∈ {1, 2, 3}. And at a ﬁxed negative value of y, the least sum.

Figure 3.6: The plot of hi(p) versus p for diﬀerent k values k1, k2 and k3.

π π Therefore, as Ψ changes from − 2 to 2 on the upper half plane, the line x = −py sweeps from positive x-axis to negative x-axis and p = tan Ψ changes from −∞ to

+∞. As p changes from −∞ to +∞ in ﬁgure 3.6, the greatest sum changes from h2(p) + h3(p) to h1(p) + h3(p) when p passes the value at the [1, 2]-intersection, and then changes to h1(p)+h2(p) when p passes the value at the [2, 3]-intersection. There- fore, in the upper xy-plane, the dominant exponentials from right to left are in the order of E(2, 3), E(1, 3) and E(1, 2). The [1, 2]-soliton is formed as the dominant

37 exponential changes from E(2, 3) to E(1, 3). And [2, 3]-soliton is formed as the dom-

π inant exponential changes from E(1, 3) to E(1, 2). Similarly, as Ψ changes from − 2

π to 2 on the lower half plane, the line x = −py sweeps from negative x-axis to positive x-axis and p = tan Ψ changes from −∞ to +∞. Figure 3.6 indicates that the least sum changes from h1(p) + h2(p) to h2(p) + h3(p) when p passes the value at the

[1, 3]-intersection. Therefore, in the lower xy-plane, the dominant exponentials from left to right are in the order of E(1, 2) and E(2, 3). Hence, [1, 3]-soliton is formed as the dominant exponential changes from E(1, 2) to E(2, 3).

Near [2, 3]-soliton,

1 1 u = (k − k )2sech2 (θ − θ + θ− ) 2 3 2 2 2 3 23 where by formula (3.30) − 1 k2 − k1 θ23 = ln + ln (3.38) a k3 − k1 The subscript 23 and superscript − indicate that the [2, 3]-soliton appear in the

region where x < 0. Notice that rearranging the terms in (3.38), one can obtain an

− expression for a in terms of θ23,

k2 − k1 −θ− a = e 23 . (3.39) k3 − k1

− The location of the crest of the soliton satisﬁes θ2 − θ3 + θ23 = 0. This gives the phase value of the soliton as

− 2 2 3 3 θ23 = θ3 − θ2 = (k3 − k2)x + (k3 − k2)y + (k3 − k2)t. (3.40)

Similarly, near [1, 3]-soliton,

τ ≈ E(1, 2) + bE(2, 3).

38 The location of the crest line is given by

− 2 2 3 3 θ13 = θ3 − θ1 = (k3 − k1)x + (k3 − k1)y + (k3 − k1)t (3.41)

where − 1 k2 − k1 θ13 = ln + ln (3.42) b k3 − k2 Near [1, 2]-soliton, we have

τ ≈ aE(1, 3) + bE(2, 3).

The soliton line is located along

+ 2 2 3 3 θ12 = θ2 − θ1 = (k2 − k1)x + (k2 − k1)y + (k2 − k1)t (3.43)

where + a k3 − k1 θ12 = ln + ln . (3.44) b k3 − k2 The phases of the three solitons, formula (3.40), (3.41) and (3.43), indicate that

− − + θ13 = θ23 + θ12.

In fact, from formula (3.32), we always have

K[i,k] = K[i,j] + K[j,k] and Ω[i,k] = Ω[i,j] + Ω[j,k]. (3.45)

This is the resonant condition of the interaction among three line-solitons. Note that the formula (3.43) can be obtained from formula (3.40) and (3.41) by the resonant condition. This means once the phases of two line solitons are given, the third line- soliton’s phase is determined by the resonant condition. Therefore, there are two freedoms in the Y-shape soliton, i.e., the location of the intersection point, which are determined by the two unknown parameters in the A-matrix.

39 3.3.3 X-shape soliton: (2143)-soliton

Figure 3.7 are the contour plots of the X-shape soliton solution at three diﬀerent

times.

Figure 3.7: The X-shape soliton graph at t = −20(left), t = 0(middle) and t = 20(right). The k-parameters that used to generate this soliton solution are (k1, k2, k3, k4) = (−0.5, −0.25, 0.25, 1) and the intersection is placed at the origin at t = 0.

The X-shape soliton is associated with 2 × 4 A-matrix with

1 b 0 0 0 0 1 a

where a > 0 and b > 0. The corresponding τ function is

τ = E(1, 3) + aE(1, 4) + bE(2, 3) + abE(2, 4). (3.46)

Both [1,2]- and [3,4]-soliton appear for y 0 and y 0 in the counterclockwise

order. Therefore, the chord diagram is

40 Figure 3.8: Example of the chord diagram of X-shape soliton solution with π = (2143).

The permutation is given by 1 2 3 4 π : . 2 1 4 3

We also refer to a soliton solution by its permutation, therefore, the X-shape soliton is called (2143)-soliton.

The phase values are as follows

+ k4 − k1 1 − k3 − k1 1 θ12 = ln + ln and θ34 = ln + ln , (3.47) k4 − k2 b k4 − k1 a + k3 − k2 1 − k3 − k1 1 θ34 = ln + ln and θ12 = ln + ln . (3.48) k4 − k2 a k3 − k2 b We obtain that the phase shift of the [1,2]- and [3,4]-soliton are equal and invariant with respect to time, i.e.

− + (k3 − k1)(k4 − k2) − + θ12 − θ12 = ln = θ34 − θ34 (3.49) (k3 − k2)(k4 − k1) Notice that the formula (k − k )(k − k ) (k − k )(k − k ) 3 1 4 2 = 1 + 2 1 4 3 > 1 (k3 − k2)(k4 − k1) (k3 − k2)(k4 − k1)

− + − + which implies the total phase shift θ12 := θ12 − θ12 and θ34 := θ34 − θ34 are positive. If

+ − −θ12−θ34 + + we deﬁne s := e , then the phase values θ12 and θ34 can be used to determine the value of the parameters a and b in the A-matrix, i.e.,

k3 − k1 θ+ k3 − k1 θ+ a = se 12 , and b = se 34 . (3.50) k4 − k1 k3 − k2

41 3.3.4 (3142)-soliton

Figure 3.9 shows the contour plots of the (3142)-soliton solution at three diﬀerent

times where the permutation (3142) gives the chord diagram shown in ﬁgure 3.10.

Figure 3.9: The (3142)-soliton graph at t = −80, 0, 150 from left to right. The k- parameters are (k1, k2, k3, k4) = (−0.5, −0.3, 0.2, 1).

Figure 3.10: Example of the chord diagram of (3142)-soliton solution.

This type of solution also has a 2 × 4 A-matrix with

1 b 0 −c 0 0 1 a where a, b and c are all positive. The corresponding τ function is

τ = E(1, 3) + aE(1, 4) + bE(2, 3) + abE(2, 4) + cE(3, 4). (3.51)

42 For y 0, the two line solitons are [1,3]- and [3,4]-solitons. For y 0, there are

[1,2]- and [2,4]-solitons in counterclockwise order. The phase values are as follows

+ k4 − k1 a − k3 − k1 1 θ13 = ln + ln and θ34 = ln + ln , k4 − k3 c k4 − k1 a + k3 − k2 b − k3 − k1 1 θ24 = ln + ln and θ12 = ln + ln . k4 − k3 c k3 − k2 b

The two phase shifts are also equal, i.e.

− + (k3 − k1)(k4 − k3) − + θ12 − θ13 = ln = θ34 − θ24 (k3 − k2)(k4 − k1)

Therefore, three of the four line solitons have free phase values. Once their values

are assigned, the fourth one is determined by the above formula. If we deﬁne s :=

−θ+ −θ− e 13 24 , then the value of the parameters a, b and c can be determined by the phase

value

k3 − k1 θ+ k3 − k1 θ+ k4 − k1 −θ+ k3 − k1 a = se 13 , b = se 24 , and c = ae 13 = s. k4 − k1 k3 − k2 k4 − k3 k4 − k3

The middle plot in ﬁgure 3.9 shows that all solitons intersect at the origin at t = 0.

Note that [2, 3]-soliton appears on the right when t < 0 and [1, 4]-soliton shows up when t > 0. If we pick some special k values, the third plot in figure 3.9 will show a similar pattern of the Mach reflection as shown in figure 3.2. We will discuss this in the next section.

3.3.5 X-shape soliton versus (3142)-soliton

By choosing symmetric k-parameters, the X-shape solitons(figure 3.11) and (3142)- solitons(figure 3.12) have shown the similar pattern of the regular reflection(figure 3.1) and the Mach reflection phenomenon(figure 3.2), both of which are developed from the V-shape initial solitary waves with different incident angle Ψ0. Therefore, we fix

43 Figure 3.11: Contour plot of the X-shape Figure 3.12: Contour plot of the (3142)- soliton symmetric about the x-axis. soliton symmetric about the x-axis.

the amplitude and angle of the solitons in the positive x-axis so that it forms the

V-shape soliton pattern and symmetric with respect to the x-axis ( A[1,2] = A[3,4] (X-shape soliton) A0 = A[1,3] = A[2,4] ((3142)-soliton) (3.52) ( −Ψ[1,2] = Ψ[3,4] > 0 (X-shape soliton) Ψ0 = −Ψ[1,3] = Ψ[2,4] > 0 ((3142)-soliton)

Then we express the k-parameters in terms of A0 and tan Ψ0 by formula (3.33) with k1 = −k4 and k2 = −k3 due to the symmetry. Therefore, in the case of the X-shape soliton, we have

1 p 1 p k = − (tan Ψ + 2A ), k = − (tan Ψ − 2A ). (3.53) 1 2 0 0 2 2 0 0 √ The ordering k2 < k3, i.e. k2 < −k2 implies k2 < 0, i.e. tan Ψ0 > 2A0. Similarly, in

the case of (3142)-soliton, we have

1 p 1 p k = − (tan Ψ + 2A ), k = − (tan Ψ − 2A ). (3.54) 1 2 0 0 3 2 0 0 44 √ The ordering k2 < k3, i.e. −k3 < k3 implies k3 > 0, i.e. tan Ψ0 < 2A0. Thus, if

all the solitons in the positive x-axis have the same amplitude A0 for both X-shape √ and (3142)-solitons, then an X-shape soliton solution arises when tan Ψ0 > 2A0, √ and a (3142)-soliton arises when tan Ψ0 < 2A0. The limiting value at k2 = k3(= 0)

deﬁnes the critical angle Ψc, p tan Ψc := 2A0. (3.55)

Hence, we introduce a parameter kkp such that

 < 1, (3142)-soliton tan Ψ0  kkp = √ (3.56) 2A0  > 1, X-shape soliton

Note that at the critical angle, i.e. k2 = k3, the τ-function has only three exponential

terms, and this gives a Y-shape resonant solution.

One should note that for the (3142)-soliton solutions, the amplitude of the solitons

in the negative x-axis are smaller than those in the positive region, i.e. A[3,4] =

1 2 A[1,2] = 2 tan Ψ0 < A0. This may explain why the reﬂective wave was not observed in

Perroud’s experiment when the incident angle Ψ0 was small. Also note that the angles

of those solitons in the negative region do not depend on Ψ0, i.e., Ψ[3,4] = −Ψ[1,2] = Ψc.

Two sets of the Y-shape solitons {[1, 3], [3, 4], [1, 4]} on the upper half plane and

{[1, 2], [2, 4], [1, 4]} on the lower half plane are both in the soliton resonant state.

However the [1, 4]-soliton becomes a soliton solution only in an asymptotic sense

where the amplitude is suﬃciently developed.

3.4 Miles solitary waves in terms of the KP solitons

In this section, we consider Miles formula of the interaction of oblique solitary

waves in terms of the KP soliton solutions. We ﬁrst derive the formula (3.12) from

45 the X-shape soliton solution to the KP equation. For simplicity, we place the center

of the solitons at the origin which indicates the phases of each soliton on the region

x > 0 and x < 0 is symmetric about the origin

− + − + θ12 + θ12 = 0 = θ34 + θ34.

Together with formula (3.49), we can solve for the phases of each line soliton, q θ− = ln 1 = θ− 12 40 34 + √ + θ12 = ln 40 = θ34

(k3−k2)(k4−k1) − − where 40 = . Replace θ and θ in the second formula in both (3.47) (k3−k1)(k4−k2) 12 34 q and (3.48) with ln 1 , we obtain the expressions of a and b in the A-matrix as 40 follows,

k3 − k1 p k3 − k1 p a = 40 and b = 40. k4 − k1 k3 − k2 Then, τ-function in (3.46) can be written as p p τ = (k3 − k1)(E1E3 + 40E1E4 + 40E2E3 + E2E4) 1 1 1 1 1 1 1 1 !! 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 E1 E3 E2 E4 p E1 E4 E2 E3 ≡ E1 E2 E3 E4 1 1 + 1 1 + 40 1 1 + 1 1 (3.57) 2 2 2 2 2 2 2 2 E2 E4 E1 E3 E2 E3 E1 E4 −1 −1 p −1 −1 ≡ H1H2 + H1 H2 + 40 H1H2 + H1 H2

1 1 1 1 2 2 1 2 2 1 where H1 = E1 /E2 = exp( 2 (θ1 − θ2)) and H2 = E3 /E4 = exp( 2 (θ3 − θ4)). We

1 1 also denote d1 = 2 (k1 − k2) and d2 = 2 (k3 − k4). Then, a direct computation of the X-shape soliton solution leads to

ττ − τ 2 u = 2∂ ln τ = 2 xx x xx τ 2 h 2 2 2 p 2 4 2 2 2 4 2 2 2 2 (3.58) = 2 4(d1 + d2) H1 H2 + 40(4d2H1 H2 + 4d1H1 H2 + 4d1H1 + 4d2H2 ) 2 2 2 2 2 2 p 2 2 + 440(d1 − d2) H1 H2 ) / 1 + H1 H2 + 40(H1 + H2 )

46 Since KP [1, 2]- and [3, 4]-soliton are the two solitons η1 and η2 described in the Miles theory, the amplitudes of the two solitons in both KP and Miles setting implies

(k − k )2 √ 1 2 = l2 ⇒ 2d2 = l2 or d = l / 2 2 1 1 1 1 1 (k − k )2 √ 3 4 = l2 ⇒ 2d2 = l2 or d = l / 2 2 2 2 2 2 2 By imposing

p 2 p 2 2δ 1 G1 = 40H1 ,G2 = 40H2 , and e = , 40 formula (3.58) can be rewritten as

l2G + l2G + (l − l )2G G + e2δ[(l + l )2 + l2G + l2G ]G G u = 4 1 1 2 2 1 2 1 2 1 2 2 1 1 2 1 2 , (3.59) 2δ 2 (1 + G1 + G2 + e G1G2)

which has the same form as formula (3.12). Therefore, the X-shape soliton solution of

the KP equation can be used to describe the strong interaction of two solitary waves

in the Miles theory.

Remark: In section 3.2.2, we have seen that the solution breaks at κ = κ+, i.e.,

2δ e = ∞. It is equivalent to 40 = 0 which leads to k2 = k3. This is the critical

case between X-shape and (3142)-soliton discussed in section 3.3.5 which does give a

Y-shape resonant interaction discussed in section 3.2.3.

The maximum amplitude of the X-shape soliton can be obtained by evaluating

(3.57) at the origin √ (d + d )2 + 2 4 (d2 + d2) + 4 (d − d )2 u = 2 1 2 0 √1 2 0 1 2 max (1 + 4 )2 √ 0 (d + d )2 + 4 (d − d )2 = 2 1 2 √ 0 1 2 1 + 4 √0 . (3.60) 1 − 4 = 2d2 + 2d2 + 4 √ 0 d d 1 2 1 + 4 1 2 √0 1 − 40 p = A[1,2] + A[3,4] + 2 √ A[1,2]A[3,4] 1 + 40

47 If we consider [1, 2]- and [3, 4]-soliton are symmetric about the x-axis, then it is

expected to have the same amplitude ratio formula as in (3.13). Indeed, with (3.52)

and (3.53), we can express umax in terms of kkp, i.e.

4A0 4A0 umax = √ = q 1 + 40 (k2−k1)(k4−k3) 1 + 1 − (k −k )(k −k ) 3 1 4 2 (3.61) 4A0 4A0 = q = q 2A0 1 1 + 1 − 2 1 + 1 − 2 tan Ψ0 kkp

tan Ψ0 where kkp = √ is given by (3.56). This gives the same amplitude ampliﬁcation 2A0

factor as in formula (3.13) where kM is replaced with kkp.

Next, we derive the amplitude ratio formula (3.21) of the Mach reﬂection in Miles

theory from the (3142)-soliton solutions of the KP equation. We notice that the upper

half contour plot in ﬁgure 3.12 is in accord with the Mach reﬂection phenomenon

where [1, 3]-soliton corresponds to the incident wave, [3, 4]-soliton corresponds to the

reﬂective wave and [1, 4]-soliton corresponds to the Mach stem. Indeed, the amplitude

of the asymptotic [1, 4]-soliton depends on the value of k1 and k4(k4 = −k1) which is

given by (3.54). Then, it can be expressed with respect to kkp, i.e.,

1 u = A = (k − k )2 = A (1 + k )2. (3.62) max [1,4] 2 4 1 0 kp

where A0 is the amplitude of [1, 3]-soliton. This is identical with formula (3.21) by

replacing kM with kkp.

Overall, dividing A0 from formula (3.62) and (3.61), we obtain the formula of the

amplitude ratio in the KP theory,  (1 + k )2, for k < 1 ((3142)-soliton)  kp kp u  max = 4 (3.63) , for kkp > 1 (X-shape soliton) A0  q 1  1 + 1 − 2 kkp

tan Ψ0 where kkp = √ . 2A0 48 3.5 Miles vs. KP in the numerics

Recall that Ψ0 used in Miles theory is an approximation of sin Ψ0 based on the

small incident angle assumption. The approximation not only leads to inaccurate

theoretical prediction, but also fails to apply to the cases where the incident angles

are not so small. In the numerical simulation of water waves, the incident angles were

taken as large as 60o[54]. Then, one observes a prominent discrepancy between the

numerical results and Miles theory which are shown in the following sections where the

numerical simulation of the water waves is based on the incident amplitude ai = 0.3 and each incident angles from 10o to 60o with an increment of 5o. In addition, we compare the numerical results with those obtained by the KP solutions. Note that here and after, we use Ψi to denote the angle of the incident wave which is the same as Ψ0.

3.5.1 The amplitude ampliﬁcation factor aM ai

The formula of the maximum amplitude ratio aM in Miles theory is given by (3.22) ai that  3 2 + − 3 + 2 sin2 Ψ a Weak Interaction  2 0 i  2 sin Ψ0 aM  4 = q Regular Reﬂection kM > 1 ai 1 1 + 1 − k2  M  2 (1 + kM ) Mach Reﬂection kM < 1

Ψ0 where kM = √ . In KP theory, we obtain in (3.63) that 3ai  4  , for kkp > 1 aM  q 1 = 1 + 1 − k2 a kp i  2  (1 + kkp) , for kkp < 1

tan Ψ0 where kkp = √ . Note that the formula in the KP theory are the same as the 2A0 formula in Miles theory excluding the weak interaction case. The diﬀerence is that

49 the kM in Miles theory is replaced with kkp. Notice that 3ai = 2A0 according to the leading order term in formula (2.21). We show in ﬁgure 3.13 the comparisons between the numerical results and the two theoretical predictions of the amplitude ratio.

Figure 3.13: The amplitude ampliﬁcation factor aM . The left ﬁgure show the result ai obtained by Miles theory and the right one by the KP theory. The 3 are the numerical results with the incident amplitude ai = 0.3.

In the left plot in ﬁgure 3.13, the numerical data are extremely shifted to the left from the theoretical curve. Especially, it is concluded[54] that the numerical results at large incident angles are in accordance with the weak interaction formula in Miles’ theory.

The plot on the right in ﬁgure 3.13 shows clearly that the discrepancy between the numerical results and theoretical curve in the KP coordinate are smaller than that in

Miles theory. But there is still a gap between the numerical results and the theoretical predictions.

50 3.5.2 The angle Ψ∗ corresponding to the stem length

In the regular reﬂection, the interaction pattern is stationary with a constant phase shift, and this means Ψ∗ = 0. In the Mach reﬂection, Ψ∗ is given by (3.20), i.e.,  r ai  (1 − kM ), for kM < 1 Ψ∗ = 3 (3.64)  0, for kM > 1

Ψ0 where kM = √ . The comparison of the numerical results and the Miles prediction 3ai of Ψ∗ is shown on the left plot in ﬁgure 3.14.

Figure 3.14: The angle Ψ∗ corresponding to the stem. The left ﬁgure show the result obtained by Miles theory and the right one by the KP theory. The 3 are the numerical results with the incident amplitude ai = 0.3.

The theoretical curve in the right plot in ﬁgure 3.14 is based on the same formula

(3.64) with kM replaced by kkp. In this case, the critical angle of the transition between the regular reﬂection and the Mach reﬂection is between 40o and 45o. As a

o o result, the numerical results of Ψ∗ corresponding to Ψi = 45 and Ψi = 50 now agree with the theoretical prediction. The improvement can be observed at other incident angles as well.

51 3.5.3 The angle Ψr of the reﬂective wave

The reﬂective angle Ψr is the same as the incident angle Ψi in the case of regular

reﬂection. In the Mach reﬂection, Ψr, denoted by Ψ2 in section 3.2.3, is given by

(3.19). The Miles theory gives ( √ 3ai, for kM < 1 Ψr = (3.65) Ψi, for kM > 1

Ψ0 where kM = √ . The numerical results and Miles curve is shown in the left plot in 3ai ﬁgure 3.15.

Figure 3.15: The angle Ψr of the reﬂective wave. The left ﬁgure show the result obtained by Miles theory and the right one by the KP theory. The 3 are the numerical results with the incident amplitude ai = 0.3.

In KP theory, the (3142)-soliton describes the Mach reflection phenomenon. The reflective wave for y 0 corresponds to [3, 4]-soliton. Hence, by formula (3.54), the reflective angle Ψ34 satisfies

p tan(Ψ34) = k3 + k4 = k3 − k1 = 2A13

52 where A13 is considered as the amplitude of the incident wave in the KP coordinate

and 2A13 = 3ai from the leading terms in (2.21). The [3, 4]-soliton in the X-shape

soliton solution can be treated as the reﬂective wave in the regular reﬂection case.

Hence, by formula (3.53),

tan(Ψ34) = k3 + k4 = −k2 − k1 = tan Ψ0, i.e. Ψr = Ψ0.

Therefore, in the KP theory, the formula of the reﬂective angle Ψr is given by ( √ 2A13, for kkp < 1 tan Ψr = (3.66) tan Ψi, for kkp > 1

the same formula (3.65) with kM replaced by kkp and Ψr replaced by tanΨr in the

Mach reﬂection case. The former condition leads to a change of the critical angle

and the latter condition leads to a different reflective angle for the Mach reflection

case compared to Miles prediction. As can be seen, there is an improvement of the

discrepancy between the numerical results and the theoretical values.

3.5.4 The amplitude ar of the reﬂective wave

The amplitude ratio of the reflective wave and the incident wave ar is equal to ai one for the regular reflection. For the Mach reflection case, the Miles theory gives the

2 ar l2 2 formula (3.19), i.e., = 2 = kM according to formula (3.19). Hence, we have ai l1

( 2 a k , for kM < 1 r = M (3.67) ai 1, for kM > 1

Ψ0 where kM = √ . The plot on the left in ﬁgure 3.16 shows the comparison between 3ai the numerical results and Miles prediction given by formula (3.67)

53 Figure 3.16: The amplitude ar of the reﬂective wave. The left ﬁgure show the result obtained by Miles theory and the right one by the KP theory. The 3 are the numerical results with the incident amplitude ai = 0.3.

In the KP theory, we obtain the same formula (3.67) replacing kM with kkp. We plot the same numerical results versus kkp in the right plot in ﬁgure 3.16. As can be seen, the agreement between the numerical results and the theoretical curve has been improved.

Figures 3.13, 3.14, 3.15 and 3.16 have shown clearly that the numerical results of all four cases are in the better agreement with the results given by the KP theory.

To further eliminate the diﬀerence between the numerical results and the theoretical predictions, we consider the higher order terms in the KP equation which gives a better approximation to the Euler equation.

54 Chapter 4: Numerical study of the Euler equation

An exact solitary wave solution is not available for the Euler equation. Instead,

Grimshaw[19] has expressed the solution to any order of the Euler equation using the expansion method of Friedrichs and Hyers[16] and he explicitly derived the solitary wave solution up to the third order. We use Tanaka’s scheme[54] to simulate a higher order approximation to the Euler equation which admits Grimshaw’s third order solitary wave solution. His scheme is known to be one of the best numerical scheme to compute the Euler equation, better than those commonly used for water wave simulations based on the Boussinesq-type approximation. The numerical scheme uses the two dimensional pseudospectral method[13][54] where the linear terms and the derivatives are computed in the fourier domain and the nonlinear terms are computed in the physical domain. Runge-kutta method of order 4 is applied for the time variable. The diﬃculties in numerical computation have been solved and explained in the following sections.

4.1 Higher order scheme for the Euler equation

The last two equations in the Euler equation (2.1) are obtained at the free water surface which form a system of equations in terms of the velocity potential and the water surface elevation. We deﬁne the velocity potential at the free surface

55 φs(x, y, t) = φ(x, y, η(x, y, t), t), with η(x, y, t) the water surface elevation. Using the equation

s φt = φt + φzηt|z=η

we obtain

s 2 ηt = −∇φ · ∇η + W [1 + (∇η) ], (4.1) 1 1 φs = −η − (∇φs)2 + W 2[1 + (∇η)2], (4.2) t 2 2

where ∇ = (∂x, ∂y), and W (x, y, t) is the vertical component of velocity potential at

the free surface, namely φz(x, y, z, t)|z=η. Using the shallow water expansion method

of Friedrichs and Hyers for small amplitude of water elevation η with η ∼ O(), we

write M X φ(x, y, z, t) = φ(m)(x, y, z, t) m=1 where the order of φ(m) ∼ O(m−1). We apply Taylor expansion to every φ(m) at the

free surface around z = 0, the still water surface, and collect the same order, we have

M M−m X X ηk φs(x, y, t) = ∂kφ(m)(x, y, 0, t). (4.3) k! z m=1 k=0

By assigning φ(1)(x, y, 0, t) as the velocity potential at the free surface φs(x, y, t), one

can obtain φ(m) by deduction from (4.3), i.e.,

φ(1)(x, y, 0, t) := φs(x, y, t) (4.4) m−1 X ηk φ(m)(x, y, 0, t) = − ∂kφ(m−k)(x, y, 0, t)(m = 2, 3, ...M). (4.5) k! z k=1

With all φ(m) are known, the vertical velocity at free surface W can be written as

M M−m X X ηk W (x, y, t) = φ (x, y, z, t)| = ∂k+1φ(m)(x, y, 0, t). z z=η k! z m=1 k=0

56 For example, if M = 3, then

φ(x, y, z, t) = φ(1)(x, y, z, t) + φ(2)(x, y, z, t) + φ(3)(x, y, z, t),

s (1) (2) (1) φ (x, y, t) = φ (x, y, 0, t) + [φ (x, y, 0, t) + η(x, y, t)φz (x, y, 0, t)] + η2(x, y, t) [φ(3)(x, y, 0, t) + η(x, y, t)φ(2)(x, y, 0, t) + φ(1)(x, y, 0, t)] z 2 zz

Therefore, once φ(1) and η(x, y, t) are known, all other higher order expansions can be solved successively, i.e.,

φ(1)(x, y, 0, t) = φs(x, y, t)

(2) (1) φ (x, y, 0, t) = −η(x, y, t)φz (x, y, 0, t) η2(x, y, t) φ(3)(x, y, 0, t) = −η(x, y, t)φ(2)(x, y, 0, t) − φ(1)(x, y, 0, t) z 2 zz

Then,

(1) (2) (1) W (x, y, t) = φz (x, y, 0, t) + [φz (x, y, 0, t) + η(x, y, t)φzz (x, y, 0, t)] + η2(x, y, t) [φ(3)(x, y, 0, t) + η(x, y, t)φ(2)(x, y, 0, t) + φ(1) (x, y, 0, t)] z zz 2 zzz

We consider up to the third order steady solitary wave solution, which has the

following form for η and φs,

2 3 2 2 4 3 5 2 151 4 101 6 η = aS − 4 a (S − S ) + a 8 S − 80 S + 80 S (4.6)

s q a 5 1 2 3 2 2 2 1257 9 2 φ = 2 3 T + a 24 T − 3 S T + 4 (1 + η) S T + a − 3200 T + 200 S T +

6 4 2 9 2 3 4 4 3 2 9 4 25 S T + (1 + η) − 32 S T − 2 S T + (1 + η) − 16 S T + 16 S T (4.7)

57 where

S = sech[k(x cos Ψ0 + y sin Ψ0 − C t − x0)]

T = tanh[k(x cos Ψ0 + y sin Ψ0 − C t − x0)]

1 3 2 3 3 C = 1 + 2 a − 20 a + 56 a √ 1 k = 3a 1 − 5 a + 71 a2 = 2 8 128 D and D is the typical length scale of the incident solitary wave. Here and after in this chapter, the notation a = ai as was introduced in chapter 3.

4.2 Pseudospectral method and scheme accuracy

Spectral method is known as a best tool to solve diﬀerential equations to high accuracy if the data deﬁning the problem are smooth. Since the function η is smooth, spectral method can be applied on η and all its derivatives. This can be achieved by using a two dimensional fast fourier transformation, and we need to impose a periodic boundary condition in both x and y in the computational domain.

We, then, choose the computational domain to be {(x, y): −LX ≤ x ≤ LX , −LY ≤ y ≤ LY }, and let the solitary wave be symmetric with respect to y = 0. Thus, all

η, φs and W are periodic with respect to y coordinate automatically. For x coordinates, we simply choose LX large enough so that η becomes suﬃciently close to zero which guarantees the periodicity. However, the function φs behaves like the function tanh(kx) when y and t are held constant, which is a smoothed step function no matter how large LX is taken. To overcome this diﬃculty, we make an even extension of φ at

LX and then apply fast fourier transformation over a double domain in x with double

58 fourier modes. Once the derivatives in the frequency domain are computed in the double x domain, we simply use the left half of the data for the further calculation.

For 2π-periodic boundary condition in (x, y) and ﬁnite water depth h = 1, each

φ(m) can be expressed by a double Fourier series as

Nx −1 Ny −1 2 2 2πk y cosh[κk ,k (z + 1)] i 2πkxx +i y (m) X X ˆ(m) x y 2L 2L φ (x, y, z, t) = φ (kx, ky, t) e X Y (4.8) cosh(κkx,ky ) k =− Nx Ny x 2 ky=− 2 where Nx and Ny are the number of Fourier modes for x and y respectively, and

κkx,ky is given by 1 " 2 2# 2 2πkx 2πky κkx,ky = + . 2LX 2LY

ˆ(m) (m) The fourier coeﬃcients φ (kx, ky, t) can be obtained from FFT2 applying on φ , and the evaluation of the vertical velocity at the free surface follows as discussed in section (4.1).

Notice that in equation (4.1) and (4.2), nonlinear terms are present in the multiplication. We need to ﬁrst convert the derivatives from the frequency domain to the physical domain and then do the multiplication. Therefore, the numerical scheme is the so-called pseudospectral method. Since the FFT2 computation is fast if the mesh size is the product of powers of small primes, we choose both Nx and Ny as a power of 2. For the time evolution, we use Runge-Kutta formula of order 4.

We test the scheme accuracy on one soliton solution whose crest forms a line parallel to the y-axis, namely Ψ0 = 0. The exact solution to the perturbed shallow water equation is known from (4.6) and (4.7) at all times with an error term in the order of a4, where a is the maximum amplitude of the solitary wave. Hence, depending on the diﬀerent value of a, the error between the numerical and the approximated exact solution has a lower boundary in O(a4). Therefore, to check the accuracy of the

59 (a, time) dt (0.05, 1000) (0.1, 750) 1 2.9334e-03 3.5943e-02 0.5 1.5761e-04 1.8047e-03 0.25 8.6459e-06 8.2130e-05 0.125 5.0061e-07 4.0899e-06 0.0625 - - Order 4.1738 4.3762

Table 4.1: The L2 error between two successive time steps, dt and dt/2.

scheme, we calculate the error by comparing the two consecutive experiments where the time step of the later one is half of that of the preceding one. It is required that a needs to be small, hence we test the accuracy in two cases, a = 0.05 and a = 0.1.

D In both cases, we choose Nx = 1024, Ny = 256 and dx = dy = 4 . Notice that since we keep Nx and dx the same for all cases, LX is consequently ﬁxed. Since the speed of the solitary wave is positively related to the amplitude, the maximum propagating time allowed within the computational domain is diﬀerent for a = 0.05 and a = 0.1.

We calculate the L2 error between two successive steps along the x-direction

Z 2 1 error = ( |ηh − ηh/2| dx) 2

where ηh denotes the numerical result with dt = h. If the scheme is accurate of order

p h p p, then the error of ηh is of order h . Similarly, the error of ηh/2 is of order ( 2 ) . Thus,

p ||ηh − ηh/2||L2 = Ch−h/2h . Take logarithm function from both sides of the equation, we obtain

log(||ηh − ηh/2||L2 ) = log(Ch−h/2) + p log(h). (4.9)

60 Table (4.1) shows the error value ||ηh − ηh/2||L2 with h = dt. For instance, 2.9334e-03

means that the L2 pointwise error between the cases dt = 1 and dt = 0.5 with the

amplitude a = 0.05 at time t = 1000 is 0.0029334. Figure 4.1 and Figure 4.2 are

the plots of error versus dt with both coordinates in the logarithmic scale. Formula

(4.9) indicates that the data in those plots should form a line whose slope is the order

of accuracy of the scheme. From the picture, we observe that the log of L2 error does form a line against the time step. The slope we estimate is 4.1738 for the case a = 0.05 and 4.3762 for the case a = 0.1. Both have shown that our numerical scheme is accurate of order four.

Figure 4.1: a = 0.05 Figure 4.2: a = 0.1

4.3 Initial value problem and boundary condition of oblique incident waves

For the general case when Ψ0 6= 0, the location of the crest of the incident solitary

waves is parallel to the lines x = −y tan Ψ0 where Ψ0 is the angle measured coun-

terclockwise from the positive y-axis. To ensure the periodic boundary condition in

61 the y-direction, we replace the upper half plane containing a half soliton mentioned above by the mirror image of the lower half plane. Therefore, the soliton forms a

” < ” shape (Figure 4.3) and is symmetric about y = 0. The initial value is given by

(4.6) and (4.7) with

S = sech[k(x cos Ψ0 + |y| sin Ψ0 − x0)]

T = tanh[k(x cos Ψ0 + |y| sin Ψ0 − x0)]

Figure 4.3: The contour plot of the ini- Figure 4.4: The partial wave surface tial V-shape wave surface in the whole without boundary modiﬁcation around domain. y = ±LY .

For the following discussion about artiﬁcial modiﬁcation near the boundary y =

±LY and incident solitary waves, we mainly focus on the lower half plan case and the situation in the upper half plan should follow similarly.

∆x To ensure the crest line is captured by the mesh grids, we enforce ∆y = tan Ψ0. Therefore, any straight line that connects the mesh points (i, j) and (i + 1, j − 1) is parallel to the crest line of the incident wave and the number of subintervals in x

62 direction and that of y direction are the same. Then, dx is much smaller than dy

when Ψ0 is small. We also require that the distance between the two adjacent oblique

1 straight lines that parallel to the crest of the incident wave is 4 of the typical length scale D. Then we have  D  ∆x = 4 cos Ψ0 D  ∆y = 4 sin Ψ0 We choose Nx = 1024, Ny = 256 and dt = 0.05 for most of our numerical examples.

1 1 Then, LX = 2 Nx × dx and LY = 2 Ny × dy are determined.

◦ Figure 4.3 shows an initial condition where a = 0.3 and Ψ0 = 40 . As time

develops to t = 30, to show the details at the boundary, we simply demonstrate the

free surface in the lower half plan −LY ≤ y ≤ 0 in Figure 4.4. Notice that another

wave is formed near y = 0. Indeed, given enough time it ﬁnally will become the

reﬂective wave with the same amplitude and angle of the incident solitary wave. We

also notice that the amplitude of the wave near the boundary diminishes, and the

crest of the incident solitary wave is bent backward. Since we impose a periodic

boundary condition to use fast fourier transformation, the initial wave in an bounded

domain near y = −LY is considered to have ” > ” shape with the inner angle

π − 2Ψ0. The angular corner becomes more ﬂat and the amplitude of the crest of

the solitary wave diﬀuses as time elapses. However, because the original physical

space is unbounded and the incident solitary wave is inﬁnitely long with a constant

amplitude, it is necessary to modify the solitary wave near the boundary to keep its

original shape. We use the method given by Tanaka[54] and brieﬂy describe below.

1. Let r denote the last row of mesh points which are considered as the boundary

that may need to be modiﬁed.

63 2. Find the location of the crest of the incident solitary wave in row r. Determine

if the deviation of the amplitude from a becomes larger than 0.005a. If yes,

begin the artiﬁcial boundary modiﬁcation from row r as described below in 3.

If no, assign r − 1 to r and repeat this step until the answer is yes.

3. Assign the value on the mesh points in row r + 1 near the crest to every row

between 1 to r. Notice that our mesh points (i, j) are designed in the way that

when i + j = constant, they form a straight line that is parallel to the crest of

the incident solitary wave. Therefore, the artiﬁcial modiﬁcation of the data in

row between 1 to r is

s s η(r+1,j) → η(r+1−k,j+k), φ(r+1,j) → φ(r+1−k,j+k).

where k = 1 . . . r and j indicates those columns around the crest.

In order for this method to work properly, we need to carefully choose the following parameters. First, r: When the incident angle Ψ0 is close to the critical angle that separates the Mach reflection phenomenon and the regular reflection one, the length of the stem grows fast. Hence, row r needs to be small enough that it does not intersect with the stem. On the other hand, when the incident angle Ψ0 is large, the incident wave is long so that the amplitude diffuses and the crest line bends backward much faster than when Ψ0 is small. Hence, r can not be too small in order for the incident wave to keep its original shape and amplitude. Second, the time interval of the artificial boundary modification: The solitary wave near the boundary needs to be updated frequently in order not to lose the original shape and the amplitude as well as to maintain high accuracy of the scheme. Since dt = 0.05, we update the boundary every 10 time steps, namely the time interval of the boundary modification

64 is 0.5. Third, the number of columns around the crest to be used: Along each row of the mesh points, the wave surface forms a soliton in one-dimensional space, and the displacement reduces to zero rapidly when it is away from the crest. Hence, the number of columns around the crest needs to be moderately large so that it covers the main non-zero surface elevation. And ﬁnally, due to the number of columns around the peak for boundary modiﬁcation and the speed of the solitary wave depending on the amplitude, the maximum time allowed for the wave propagation in the computation domain may vary.

4.4 Numerical simulation of the Mach reﬂection vs. the regular reﬂection

We first use a = 0.3 to demonstrate the Mach reflection and regular reflection

o respectively. For the Mach reﬂection case, we use Ψi = 20 and let the maximum computing time tmax = 350. Figure 4.5 shows the water wave surface around the

Mach stem. Notice that a third wave is generated behind the stem and it has smaller amplitude than that of the incident wave. This wave is called a reﬂective wave and its

o amplitude and angle are denoted by ar and Ψr, respectively. We choose Ψi = 50 to demonstrate the case where the regular reflection occurs. At tmax = 350, the surface of the wave is shown in figure 4.6. As we can see, the reflective wave has the same amplitude and angle as those of the incident wave.

65 Figure 4.5: Surface of the Mach stem Figure 4.6: Surface of the regular reﬂec- o o with a = 0.3, Ψi = 20 and t = 350. tion with a = 0.3, Ψi = 50 and t = 350.

Figure 4.7: The crest line of the Mach Figure 4.8: The crest line of the regular reﬂection case with t = 0, 10, 20,..., 150. reﬂection case with t = 0, 10, 20,..., 100.

To show the details of the Mach stem development, we draw the crest lines of the water wave surfaces from t = 0 to t = 150 with an increment of t = 10. Only the

66 lower half domain in y is considered due to the symmetry of the waves. The straight

line in ﬁgure 4.7 indicates the incident wave at t = 0. As time increases, the stem is

getting longer and longer and the reﬂective wave is formed. The intersection of the

three waves is moving to the right along a straight line. The angle formed by this line

and x-axis is denoted by Ψ∗(see figure 3.2). We also draw the crest line of the water waves in the case of the regular reflection. Notice that in figure 4.8 it is not the stem but the phase shift that appears in the center and the reflective wave is symmetric with the incident wave from their central vertical line. These two figures also show that our boundary modification is satisfied. More numerical results will be used to compare with the higher order Miles theory in the next chapter.

67 Chapter 5: Higher order Miles theory

As we have shown in section 3.2, the Miles theory is based on the leading order approximation of the Euler equation. However, the numerical simulations of water waves used in [54], [17] and this thesis have considered up to the third order expansion of the Euler equation with respect to small amplitude. We now include higher order terms to the KP equation to discuss the numerical results obtained in the previous chapter. Note here that the higher order terms give the corrections to the assumptions of small amplitude, long wavelength and quasi-two-dimensinality.

In chapter 2, we have derived the KP equation including higher order terms,

∂ formula (2.20), from the expansion of the Euler equations. We notice that if ∂Y u = 0 in (2.20), one can get the KdV equation with higher order terms,

19 53 5 9 u = 6uu + u + u + u u + uu + O( 2 ). (5.1) T X XXX 60 XXXXX 6 X XX 3 XXX

The solution can be derived from the normal form theory[28][31][21] which has been discussed for weakly dispersive nonlinear wave equations where the leading phenomena can be described by the KdV equation. The KdV equation with higher order terms are asymptotically integrable up to the ﬁrst order. We shall brieﬂy discuss about the KdV case and then extend the normal form method to the perturbed KP equation (2.20).

68 5.1 The higher order KP equation

5.1.1 Normal form of the higher order KdV equation

As we have shown in chapter 2, the equations with one space variable x can be reduced to the higher oder KdV equation (5.1) under the assumptions of small amplitude and long wavelength. We assign a weight to u and its derivatives as follows,

1 3 1 W t(u) = 1, W t(∂ ) = , W t(∂ ) = 1, W t(∂ ) = and W t(D−1) = − x 2 y t 2 2

where the operator D−1 is the inverse of the derivative with respect to x and it reduces

1 the weight by 2 . That is, the weight represents the power of for each variable. For example, u ∼ O(1) means W t(u) = 1. In general, the higher order KdV equation

can be written as,

N+1+ 5 ut = F (u; ) + O( 2 ) (5.2) with F (u; ) = F (0)(u) + F (1)(u) + ··· + F (N)(u)

where F (n) can be found recursively by formula

(n) (n) X (n) (n1) −1 (n2) F = a1 u(2n+3)x + cij Qi D Qj (u). n1+n2=n−2 1≤i≤M(n1) 1≤j≤M(n2)

(n) (n) In the above formula, the coeﬃcients aj and cij are real constants determined by

(n) (n) the model equations of the physical problems. Qi (u) is a monomial in F (u) and M(n) is the total number of monomials in F (n)(u). As can be expected, non-local

terms will be produced when the operator D−1 acts on the monomials. The ﬁrst three

69 F (n)(u) are given by

(−1) (−1) F =a1 ux

(0) (0) (0) F =a1 u3x + a2 uux

(1) (1) (1) (1) (1) 2 F =a1 u5x + a2 uu3x + a3 uxu2x + a4 u ux

···

∂iu i (n) 5 where uix = ∂xi = ∂xu. Note that each term in F has a homogeneous weight n + 2 .

A polynomial P (u, ux, uxx,...) in terms of u and its x derivatives ux, uxx, etc is called a differential polynomial. We define a vector field operator set X as

( ∞ ) X i X = ∂x(P )∂uix (·): P is a diﬀerential polynomial . i=0

Then, an element XF ∈ X is deﬁned by

∞ X i XF = ∂x(F ) ∂uix (·) i=0 where F is a diﬀerential polynomial and XF is the corresponding operator acting on any diﬀerential polynomials. For example, for the standard KdV equation

(0) ut = uxxx + 6uux := F0 (u), (5.3) we have

(0) ∂ (0) ∂ (0) ∂ XF (0) · ux = [F0 (u)] ux + [F0 (u)]x ux + [F0 (u)]xx ux + ··· 0 ∂u ∂ux ∂uxx (5.4) 2 = [uxxx + 6uux]x = u4x + 6ux + 6uuxx.

Notice that XF acts on the u is simply F (u), hence for the time evolution equation in the form

ut = F (u), (5.5)

it can be put in the form of

ut = F (u) = XF · u. (5.6)

70 We introduce an adjoint operator of F on the space of diﬀerential polynomials

denoted by adF which is

adF · G(u) := [F,G](u) = XF · G(u) − XG · F (u).

Then, G(u) is said to be a symmetry of the system (5.5) if

adF · G(u) = 0.

In this case, we say G(u) commutes with F (u). For example, let G(u) = ux, then

(0) ∂ (0) ∂ (0) ∂ (0) XG · F0 (u) = [G(u)] F0 (u) + [G(u)]x F0 (u) + [G(u)]xxx F0 (u) ∂u ∂ux ∂uxxx 2 = [ux] (6ux) + [ux]x (6u) + [ux]xxx = u4x + 6ux + 6uuxx (5.7)

(−1) (0) Equation (5.4) and (5.7) imply that ux := F0 (u) is a symmetry of F0 (u). There are inﬁnitely many symmetries of the standard KdV equation. The ﬁrst few symme-

(j) tries, denoted by F0 for j ≥ −1, are

(−1) F0 =ux

(0) F0 =u3x + 6uux

(1) 2 F0 =u5x + 10uu3x + 20uxuxx + 30u ux. The idea of the normal form is to ﬁnd a near identity transformation

1 u = ψ(U) := eXψ U = U + X · U + (X · X ) · U + ··· (5.8) ψ 2 ψ ψ

so that by the change of variables, equation (5.2) is changed into

N+1+ 5 N+1+ 5 Ut = G(U) + O( 2 ) = XG · U + O( 2 ) (5.9)

where

(0) (1) (2) G(U) = F0 (U) + G (U) + G (U) + ··· +

71 and G(n)(U) can be put in the form

(n) (n) (n) (n) G (U) = a1 F0 (U) + R (U).

(n) (n) Here, a1 is given by the original perturbation equation and F0 is the symmetry of the n-th order KdV hierarchy, which is integrable. R(n) contains only the nonlinear terms that cannot be removed by the near identity transformation. To ﬁnd the explicit expression of the near identity transformation (5.8), ﬁrst notice that by (5.8)

ut = ψU Ut + ψUx (Ux)t + ψUxx (Uxx)t + ···

= ψU G + ψUx Gx + ψUxx Gxx + ···

Xψ = XG · ψ(U) = XG · e (U)

Also notice that,

∂ eXψ · X (U) = eXψ · U = (eXψ · U) = u . F t ∂t t

Equating the above two formulas implies the normal transformation satisﬁes

Xψ −Xψ XG = e · XF · e 1 1 = 1 + X + X2 + ··· · X · 1 − X + X2 − · · · ψ 2 ψ F ψ 2 ψ (5.10)

1 ad = X + [X ,X ] + [X , [X ,X ]] + ··· := e Xψ X F ψ F 2 ψ ψ F F

where adXψ XF = [Xψ,XF ]. The last line in (5.10) is the main formula for ﬁnding

Xψ.

Now, the problem can be summarized as follows. According to the notation in

(5.2), we have

0 1 9 F (u) = F + F + O( 2 )

72 where

0 F = 6uuX + uXXX 19 5 53 F 1 = u + uu + u u 60 5X 3 XXX 6 X XX

Here, factor is removed by the scaling variable change (2.19). The near identity transformation under consideration has the form

u = ψ(U) = U + ψ1(U) + ··· (5.11) with

1 2 −1 ψ (U) = α1Uxx + α2U + α3UxD U where ψn is a homogenous diﬀerential polynomial with weight W t(ψn) = n+1. Using this transformation, ut = F (u) can be transformed to Ut = G(U) where G(U) =

0 1 9 G + G + O( 2 ) with

0 G = 6UUX + UXXX

1 2 G = α0(U5X + 10UUXXX + 20UX UXX + 30U UX )

Applying transformation (5.10) on u (or U) gives an explicit form

1 1 G = F + [ψ, F ] + [ψ, [ψ, F ]] + [ψ, [ψ, [ψ, F ]]] + ··· (5.12) 2! 3! where [ψ, F ] = Xψ(F )−XF (ψ) is the lie bracket of the vector ﬁeld. A list of equating functions can be obtained shown below by picking the terms with the same weight from equation (5.12),

5 (W t = ): G0 = F 0 2 7 (W t = ): G1 = F 1 + [ψ1,F 0] 2 ···

73 The ﬁrst equation is trivial since F 0 = G0. The second implies

[F 0, ψ1] = F 1 − G1

where

0 1 2 [F , ψ ] = (−3α3)uuxxx + (12α1 − 6α2 − 3α3)uxuxx + (−6α2 − 3α3)u ux 19 5 53 F 1 − G1 = ( − α )u + ( − 10α )uu + ( − 20α )u u − 30α u2u 60 0 xxxxx 3 0 xxx 6 0 x xx 0 x

Compare the corresponding coeﬃcients of the above equations, we ﬁnd

19 4 1 α = , α = 1, α = and α = . 0 60 1 2 3 3 2

i.e. the normal transformation is given by

4 1 u = U + ψ1(U) = U + U + U 2 + U D−1U. (5.13) xx 3 2 x

By this normal form, the equation for U is

19 2 9 U = 6UU + U + U + 10UU + 20U U + 30U U + O( 2 ). T X XXX 60 XXXXX XXX X XX X

(1) (1) We note that there are four terms in F where the coeﬃcient a1 of the linear

1 term gives the coeﬃcient α0 in G . The rest of the three coeﬃcients provide exactly

the same number of conditions to solve for the three unknown coeﬃcients {α1, α2, α3}

in ψ1. Therefore, the higher order KdV equation has exact soliton solutions up to

the ﬁrst order. Also note that the above equation admits a one-line soliton solution

in the form, r A U = A sech2 0 [X + X (T )] , 0 2 0

where X0(T ) is determined by

d X (T ) 19 0 := C = 2A + A2 + O(3). dT hKdV 0 15 0 74 Therefore, the solution of the original higher order KdV equation (5.1) can be obtained from (5.13) 2 u = A S2 + A2S2 − A2S4 + O(3) 0 0 3 0 q A0 where S = sech 2 (X + X0(T )). In the non-dimensional physical coordinate, η in (2.21) for the KdV case is given by

2 2 1 αη = A S2 + A2S2 + A2S4 + O(3) (5.14) 3 0 9 0 3 0 q A0 ChKdV The augment of sech-function becomes 2β x − 1 + 6 t in non-dimensional coordinate(see (2.22). Hence, the velocity of the one-line soliton is corrected to

1 19 1 19 1 + 2A + A2 + O(3) = 1 + A + A2 + O(3). (5.15) 6 0 15 0 3 0 90 0

5.1.2 Normal form of the higher order KP equation

The normal form of the higher order KP equation can be found in [31] where the non-integrable eﬀect, called obstacles to integrability, has been discussed. It turns out that the obstacles in the higher order correction provide suﬃcient conditions for the asymptotic expansion of the wave equation to be integrable up to a certain order. However, the higher order KP equation given in (2.20) does not satisfy the obstacles given in Proposition 5.2 in [31]. Hence, unlike KdV case, the KP equation is not asymptotic integrable in the next order. Instead, we look for a near identity transformation

u = U + ψ1(U) + ··· (5.16)

∂ satisfying (i) formula (5.16) becomes formula (5.13) when ∂y u = 0 and (ii) with the near identity transformation (5.16), the higher order KP equation (2.20) becomes an

75 evolution equation in terms of U

(0) 9 (1) 2 Ut = KP0 (U) + G (U) + O( )

which admits a line soliton solution exact up to the next order. Here, ψn(U) has

a homogenous weight of n + 1 and the max order of D−1 in ψn(U) is 2n when the

variable y is present. Hence, the monomials in ψ1(U) with weight 2 are

1 2 −1 −2 ψ = β1UXX + β2U + β3UX D U + β4D UYY . (5.17)

The KP equation is

−1 (0) ut = uxxx + 6uux + 3D uyy := KP0 (5.18)

5 (1) 7 which itself is nonlocal with weight 2 . G (U) contains the terms with weight 2 which has more non-local terms in the higher order expansion of the KP equation.

Now, replace u in the higher order KP equation (2.20) with the near identity transformation (5.16) with (5.17), we obtain an equation in terms of U, 19 3 3 U = 6UU + U + 3D−1U + U + U − D−3U T X XXX YY 60 XXXXX 2 XYY 4 YYYY 53 5 + ( − 12β + 6β + 3β )U U + ( + 3β )UU + (6β + 3β )U 2U 6 1 2 3 X XX 3 3 XXX 2 3 X 5 + (1 − 3β + 6β )U D−2U + ( − 6β + 3β + 6β )UD−1U 3 4 X YY 2 2 3 4 YY −1 3 −1 2 9 +(4 + 6β )U D U + (− + 3β − 3β − 3β )D (U ) + O( 2 ) 3 Y Y 4 2 3 4 YY (5.19)

Condition (i) implies that we take

4 1 β = 1, β = and β = 1 2 3 3 2

from the near identity transformation (5.13) of the KdV case. Condition (ii) implies

that we further assume equation (5.19) admits a solitary wave in the form of a one-line

76 soliton solution, which can be written as r A U = A sech2 0 [X + Y tan Ψ + C T ]. (5.20) 0 2 0 hKP

Therefore, replacing U in formula (5.19) by solution (5.20), one can obtain the fol-

√ 3 2 lowing equation(after cancelling a common factor − 2A0 ), 2 2 19 2 2 3 4 2 ChKP S T a = 2A0 + 3 tan Ψ0 + A0 + 3A0 tan Ψ0 − tan Ψ0 S T a 15 4 (5.21) 2 4 + 3A0 tan Ψ0(2β4 − 1)S T a q q A0 A0 where S = sech 2 (X + Y tan Ψ0 + ChKP T ) and T a = tanh 2 (X + Y tan Ψ0 +

ChKP T ). The higher order term being zero implies that

1 β = . 4 2

Thus, we obtain the near identity transformation for the higher order KP equation

(2.20), 4 1 1 u = U + U + U 2 + U D−1U + D−2U + O(3) (5.22) XX 3 2 X 2 YY

If we plug in U with (5.20), then u can be written as follows

1 2 u = A S2 + A2S2 + A tan2 Ψ S2 − A2S4 + O(3). 0 0 2 0 0 3 0

o Maintaining the same maximum amplitude u = 0.1 as in section 2.3 for Ψ0 = 0 ,

o o Ψ0 = 15 and Ψ0 = 30 , one obtain A0=0.097, 0.094 and 0.084, respectively. The

q 2 width of the line soliton is proportional to cos Ψ0 which are 4.54, 4.46 and 4.23, A0 respectively. The contour plot of u is generated in ﬁgure 5.1.

77 o Figure 5.1: Contour plot of line soliton with higher order correction for Ψ0 = 0 (left), o o Ψ0 = 15 (middle) and Ψ0 = 30 (right). The widths of the soliton in the latter two cases are 98.2% and 93.2% of the original soliton.

Compare with ﬁgure 2.2 obtained without considering higher order term in u, the width of the line soliton here remains almost constant as the angle increases. This implies that the rotational symmetry of the real solitary wave is corrected up to

O(2). Therefore, the KP soliton solutions with a higher order correction improves the approximation to the water waves. In the non-dimensional physical coordinate,

η can be found by (2.21), up to the second order,

2 2 2 2 2 2 2 2 1 2 4 3 αη = A0S + A0 tan Ψ0S + A0S + A0S + O( ) 3 3 9 3 (5.23) 2 2 1 = [A ]S2 + [A ]2S2 + [A ]2S4 + O(3) 3 0 9 0 3 0

2 A0 where [A0] := A0(1 + tan Ψ0) = 2 . And the augment of sech-function in the cos Ψ0 q [A0] ChKP non-dimensional coordinate is 2β x cos Ψ0 + y sin Ψ0 − 1 + 6 cos Ψ0 t . The higher order velocity of the line soliton can be obtained from the leading order terms in (5.21),

19 3 C = 2A + 3 tan2 Ψ + A2 + 3A tan2 Ψ − tan4 Ψ + O(3). (5.24) hKP 0 0 15 0 0 0 4 0

78 In the non-dimensional physical coordinate, the velocity of the one-line soliton has direction (cos Ψ0, sin Ψ0) with magnitude

1 19 3 1 + 2A + 3 tan2 Ψ + A2 + 3A tan2 Ψ − tan4 Ψ + O(3) cos Ψ 6 0 0 15 0 0 0 4 0 0 1 19 1 = 1 + A + A2 + A tan2 Ψ + O(3) 3 0 90 0 3 0 0 1 19 = 1 + [A ] + [A ]2 + O(3) 3 0 90 0 (5.25)

Here, we have used

2 − 1 1 2 3 4 3 cos Ψ = (1 + tan Ψ ) 2 = 1 − tan Ψ + tan Ψ + O( ). 0 0 2 0 8 0

Notice that the correction to the quasi-two dimensional approximation can be absorbed into both the amplitude (5.14) and the velocity (5.15) of the KdV equation by replacing A0 with [A0] which contains the angle adjustment. Especially, formula

(5.23) gives the relation between the observed amplitude a of the solitary wave from the numerical simulation (or experiment) of the shallow water wave system and the

KP amplitude A0. The maximum value of the amplitude of the solitary wave can be obtained when S = 1. Therefore, we have

2 5 a = [A ] + [A ]2. (5.26) 3 0 9 0

Solving A0 in terms of a, we have the formula to convert the observed amplitude of the water wave a into KP amplitude A0, i.e.,

2 3a cos Ψ0 A0 = √ for small a > 0. (5.27) 1 + 1 + 5a

Remark: The soliton solutions to the higher order KP equation do not satisfy the resonant condition. Consider the Y-shape solitons we mentioned in section 3.3.2.

79 A direct calculation with higher order velocity (5.24) gives

rA rA Ω + Ω = [1,2] C + [2,3] C [1,2] [2,3] 2 hKP[1,2] 2 hKP[2,3] 8 8 = 2(k2 − k2) + (k5 − k5) + (k4k + k4(k − k ) − k4k ) (5.28) 3 1 15 3 1 3 1 2 2 3 1 3 2 2 + (k3k2 + k3(k2 − k2) − k3k2) 3 1 2 2 3 1 3 2 and r A[1,3] Ω[1,3] = ChKP[1,3] 2 (5.29) 8 8 2 = 2(k2 − k2) + (k5 − k5) + (k4k − k4k ) + (k3k2 − k3k2). 3 1 15 3 1 3 1 3 3 1 3 1 3 3 1

The resonant condition (3.45) requires that Ω[1,2] + Ω[2,3] − Ω[1,3] = 0. However, from

(5.28) and (5.29) we know that

1 Ω + Ω − Ω = (k − k )(k − k )(k − k )(3(k2 + k2 + k2) + 5(k + k + k )2). [1,2] [2,3] [1,3] 3 1 2 1 3 2 3 1 2 3 1 2 3

Thus we expect that the resonant phenomenon occurs only at the leading order, and

Y-shape wave may not be stable for large time.

5.2 Higher order KP solution from numerical results by minimization

Using the formula (5.22), we can transform a solution U to the solution u of the

higher order KP equation. We approximate U with a soliton solution of the KP

equation. We then use the corresponding u to match a given numerical data at a

speciﬁc time. Recall that the KP soliton solution U can be expressed in terms of

the τ function which is determined by k-parameters and A-matrix. The value of

k-parameters can be found by solving (3.33) if the amplitude and angle of a line

soliton is given. To ﬁnd A-matrix, we ﬁrst determine the form of the A-matrix and

express the entries in terms of the locations of the line solitons in the solution based

80 on the formula (3.30) and (3.31). However, this may not give the correct values of

the entities in the A-matrix if the locations of line solitons are not accurate from the

measurement. Hence, we impose a minimization method to ﬁnd the best match of

the higher order KP solution u and the numerical data un.

The idea of the minimization method is to focus on a bounded region in which

the soliton pattern is stable and well covered, and then to minimize the diﬀerence

between the solution u of the higher order KP equation and the given numerical data

within the region. We can set up the scheme of the minimization as follows: Let the

domain under consideration be an elliptic disc given by

2 2 2 (x − xc) (y − yc) D(xc,yc)(rx, ry) = (x, y) ∈ R : 2 + 2 < 1 . rx ry

Here the center (xc, yc) is chosen so that the main interaction part of the wave pattern

is well covered. The goal is to compute

2 min kuA − unkL2(D) (5.30) A where ZZ 2 2 kuA − unkL2(D) := |uA − un| dxdy.

(x,y)∈D(xc,yc)(rx,ry)

Here uA is the higher order solution u in formula (5.22) with the exact KP soliton

solution U associated with the A-matrix. The symbol min stands for minimization A over the A-matrices. In other words, we vary the values of the entries of the A-matrix

to obtain the minimum of kuA − unkL2(D). The procedure has the following steps.

1. Change the value of one of the entries in the A-matrix and keep the other

entries at their current values which are either the initial values or those from

the preceding minimization.

81 2. Compute the KP solution UA associated with each A-matrix from last step by

formula (3.23) and (3.24), and convert it to the higher order solution uA by

formula (5.22).

3. Record the diﬀerence between uA and un for each A-matrix. Update the value

of the entry with the one that gives the minimum diﬀerence.

4. Repeat steps 1-3 on every other entries of the A-matrix.

5. Repeat steps 1-4 until either the diﬀerence arrives at the tolerance level or the

desired number of iteration where the diﬀerence stays approximately the same.

One problem of this default minimization procedure is that it is not very informative to vary the value of the entries in the A-matrix. In fact, the initial value of those entries obtained by the location of the line solitons (based on formula (3.30) and

(3.31)) can be exponentially large or small. How large or how small to change the value is indeterminate. To solve this diﬃculty, it is the location of the line soliton instead of the parameters in the A-matrix that is chosen to be the unknown variable to change in the minimization procedure. Intuitively, we know the range of the soliton location, and we can express the entries in the A-matrix in terms of the location of the line soliton.

We consider a numerical simulation of the Mach reﬂection with an amplitude

o a = 0.05 and angle Ψ0 = 15 for the incident wave at a computing time t = 900.

Figure 5.2 is a contour plot around the center of the waves.

82 Figure 5.2: A contour plot of the water wave pattern of the numerical simulation. The incident soliton has amplitude 0.05 and angle 15o.

We observe that the solitary waves for x 0 have larger amplitudes than those for x 0. Also from their angles, we expect that such kind of wave pattern corresponds to the Mach reﬂection phenomenon which may be described by a (3142)-soliton solution of the KP equation. The symmetric shape of the wave pattern about the x-axis

is the case we discussed in section 3.3. To ﬁnd the k-parameters, since we imple- ment a boundary-patching scheme for the numerical computation and the incident amplitude and angle are maintained at their original values, we use formula (3.33) to

ﬁnd the k-parameters to the incident soliton, namely [1, 3]-soliton for y > 0. We ﬁrst

apply the higher order correction formula (5.27) to obtain the amplitude in the KP

coordinate and then use formula (3.54) to ﬁnd

k1 = −0.3157, and k3 = 0.0478.

By symmetry, we have

k1 = −0.3157, k2 = −0.0478, k3 = 0.0478, and k4 = 0.3157.

83 The A-matrix of the (3142)-soliton has the form

1 b 0 −c 0 0 1 a

where a, b and c are positive. To ﬁnd the values of the three parameters, we ﬁrst note that the symmetry gives one condition and there are two other freedoms to determine the parameters. The subsets I and J in formula (3.28) of the (3142)-soliton can be denoted by I = {i, l} and J = {j, l} where i, j, and l are distinct integers from

{1, 2, 3, 4}. Applying formula (3.30) in the (3142)-soliton, we have

|kl − ki| A(jl) = A(il) e(ki−kj )(x0+(ki+kj )y0+xij ). (5.31) |kl − kj|

where xij in the exponent is an auxiliary term to control the location of the crest of the

line soliton. Then, we can change the value of xij instead of the unknown parameters

in the A-matrix to compute the diﬀerence between uA and the numerical data un.

Pick a point on the stem, say (xc0, yc0), and note that I = {1, 3} and J = {3, 4}, we

have

k3 − k1 c = A(34) = A(13) exp[(k1 − k4)(xc0 + (k1 + k4)yc0 + x14)] k4 − k3

where A(13) = 1. Similarly, pick a point (xi0, yi0) on the incident wave where I =

{3, 4} and J = {1, 4}, we have

k4 − k3 a = A(14) = c exp[(k3 − k1)(xi0 + (k1 + k3)yi0 + x13)]. k4 − k1

By symmetry, point (xi0, −yi0) is on the [2, 4]-soliton and x13 = x24. Hence, with

I = {3, 4} and J = {2, 3},

k4 − k3 b = A(23) = c exp[(k4 − k2)(xi0 + (k2 + k4)(−yi0) + x24)] k3 − k2 k4 − k3 k4 − k1 = c exp[(k3 − k1)(xi0 + (k1 + k3)yi0 + x13)] = a k3 − k2 k3 − k2 84 Therefore, there are two free parameters x14 and x13 to control the phase of [1, 4]-

and [1, 3]-soliton respectively.

We focus on an elliptic area D(x0,y0)(rx, ry) around the center where ry equals

three times of the length of the stem above x axis and rx covers the same number of grids as ry does. As is known, the [1, 4]-soliton has the maximum amplitude among all the line solitons in the (3142)-soliton solutions. It contributes the most to the difference between the higher order KP solution and the given wave data. Therefore, the minimization procedure can put into the following steps: 1. Keep the phase of [1, 3]-soliton unchanged and minimize the difference by varying x14; 2. Once the minimum is obtained, keep the value of x14 obtained from step 1 and change of the value of x13 until the difference is minimized; 3. Repeat 1 and 2 sufficient number of times to derive the minimization over both factors.

Figure 5.3 demonstrates the first few minimization steps. The errors are shown in Table 5.1. We can see that the error decreases fast in the first five iterations and gradually slow down to a saturated value. The corresponding A-matrix from minimization is 1 0.1530 0 −1.3453 A = . 0 0 1 0.0232

iteration (1) (2) (3) (4) (5) (6) (7) (8)

minimization x14 x13 x14 x13 x14 x13 x14 x13 variable L2 error 8.5705 4.5692 0.1418 0.0186 0.0123 0.0121 0.0120 0.0120

Table 5.1: The minimum L2 error between the higher order (3142)-soliton solution of the KP equation and the numerical results of the Mach reﬂection.

85 (1) (2)

(3) (4)

(5) (7)

Figure 5.3: Plots (1)-(7) show the differences between the higher order (3142)-soliton solution of the KP equation and the numerical results of the Mach reflection. Plot (1) is obtained by minimization on x14 with initial fixed value of x13. Plot (2) is obtained by minimization on x13 with fixed value of x14 from last step. (3),(4),(5),(6) and (7) repeat the same previous procedures where plot (6) is omitted due to the similarity with plot (7).

86 Figure 5.4: Contour plot of the leading order and the higher order KP (3142)-soliton solution(left column) and their diﬀerences to the numerical results(right column) of the Mach reﬂection.

Using the k-parameters and the A-matrix, we obtain the KP soliton solutions

UA, and use formula (5.22) to obtain the higher order (3142)-soliton solution uA.

The plots on the left in ﬁgure 5.4 are the contour plots of (3142)-solitons UA and

higher order (3142)-solitons uA. On the right are their diﬀerences to the numerical

results. We conﬁrm that the higher order KP soliton solution uA gives a much better

approximation to the numerical results than the KP exact solution UA.

We perform a similar procedure to ﬁnd a higher order X-shape soliton solution

o from the numerical data of the regular reﬂection with a = 0.05, Ψ0 = 30 and t = 900.

The contour plot around the intersection is given in ﬁgure 5.5.

87 Figure 5.5: A contour plot of the simulation of the water waves. The incident soliton has amplitude 0.05 and angle 30o.

With higher order correction (5.27), we ﬁrst obtain the amplitude in the KP coordinate and then ﬁnd the k values using formula (3.53)

k1 = −0.4516, and k2 = −0.1257.

By symmetry, we have

k1 = −0.4516, k2 = −0.1257, k3 = 0.1257, and k4 = 0.4516.

The A-matrix has the form 1 b 0 0 0 0 1 a where a and b are free parameters to be determined. Based on formula (5.31), one can derive  k4 − k1 +  b = exp[(k1 − k2)(xb0 + (k1 + k2)yb0 + x12)], for [1, 2] -soliton,  k4 − k2  k3 − k2 +  a = exp[(k3 − k4)(xa0 + (k3 + k4)ya0 + x34)], for [3, 4] -soliton, k4 − k2

88 where (xa0, ya0), (xb0, yb0) are two points on the crest of each soliton. Notice that due to the symmetry of the two line soliton, we may consider xa0 = xb0 and ya0 = −yb0

+ in this case. x12, x34 are the two parameters which control the location of [1, 2] -

and [3, 4]+-solitons. They are adjusted to obtain the minimum diﬀerence between

the higher order X-shape soliton solution and the numerical data. First, we keep the

value of x34 and minimize the diﬀerence by varying x12. Once we have arrived at

the minimum value, we keep x12 and change of the value of x34 until the diﬀerence is

minimized. We repeat the above two procedures suﬃcient number of times to derive

the minimization over both factors.

We show the minimum L2 error of each iteration in table 5.2. Clearly, the dif-

ference is removed gradually by varying the value of x12 and x34 alternately. It is noticeable that the minimization is achieved since iteration (3). Figure 5.6 is the first three steps of the minimization procedure where the contour plot of the minimum difference at each step is given. The final A-matrix from the minimization is

1 1.8853 0 0 A = 0 0 1 0.5248

iteration (1) (2) (3) (4) (5) (6) (7) (8)

minimization x12 x34 x12 x34 x12 x34 x12 x34 variable L2 error 2.8330 0.0042 0.0011 0.0011 0.0011 0.0011 0.0011 0.0011

Table 5.2: The minimum L2 error between the higher order X-shape soliton solution of the KP equation and the numerical results of the regular reﬂection.

89 (1) (2)

(3)

Figure 5.6: Plots of the difference between the higher order X-shape soliton solution of the KP equation and the numerical results of the regular reflection. Plot (1) is obtained by minimization on x12 with initial fixed value of x34. Plot (2) is obtained by minimization on x34 with fixed value of x12 from last step. (3) and later iterations repeat the same previous procedures.

We plot the KP solution UA and its higher order solution uA obtained from this

A-matrix and k-parameters in figure 5.7(left column). The corresponding difference to the numerical results which is shown in figure 5.5 is demonstrated in the right column in figure 5.7. As can be seen, the higher order X-shape soliton solution of the

KP equation has a closer approximation to the numerical results.

90 Figure 5.7: Contour plot of the leading order and the higher order X-shape soliton solution(left column) and their diﬀerences to the numerical results(right column) of regular reﬂection.

In this section, we begin with the solitary wave patterns obtained from the numerical simulation with initial V-shape solitary waves discussed in chapter 4. De- pending on the angle of the incident wave, the V-shape solitary waves may develop into the Mach reflection phenomenon(small incident angle) or the regular reflection pattern(large incident angle). The numerical simulations confirm that (3142)- and

X-shape soliton can be used to describe the Mach reﬂection phenomenon and the regular reﬂection, respectively. Moreover, the above two numerical examples demonstrate that the higher order KP soliton solutions u are better approximations to the given numerical results than the KP solitons solutions U.

91 5.3 Numerical results reconsidered with higher order correction

We have obtained formula (3.63) of the amplitude ampliﬁcation factor aM from the ai

KP theory and formula (5.27) converts ai(the non-dimensional maximum water elevation from the experiment or the real physical phenomenon) to A0(the non-dimensional maximum amplitude of the soliton solution to the KP equation) under the consideration of the higher order corrections. Therefore, we derive the maximum amplitude ratio  (1 + K)2 if K < 1 ((3142)-soliton or the Mach stem)   aM 4 = if K > 1 (X-shape soliton or the regular reﬂection) ai q  1  1 + 1 − K2 (5.32)

where K is the higher order KP parameter given by

tan(Ψi) K = q √ . (5.33) 2 6a cos (Ψi)/(1 + 1 + 5a)

Since K = 1 corresponds to the critical angle that separates the aforementioned two

types of soliton solution to the KP equation, we then solve a quadratic equation to

obtain the critical angle between the Mach stem and the regular reﬂection. Utilizing

a trigonometric identity, one has s ! 2 1 24a tan (Ψch) = −1 + 1 + √ 2 1 + 1 + 5a

where Ψch is the critical angle under higher order correction. Table 5.3 shows the

critical angle for diﬀerent water wave amplitude. The Mach stem occurs when the

incident angle is smaller than the critical angle and the regular reﬂection occurs when

the incident angle is larger than the critical angle.

92 a tan Ψch critical angle Ψch(in degree) 0.05 0.354700 19.53 0.1 0.469998 25.17 0.2 0.603593 31.11 0.3 0.687988 34.53

Table 5.3: Critical angle for the diﬀerent amplitude of the incident wave.

5.3.1 The amplitude ampliﬁcation factor aM ai

Here we compare the amplitude factor aM given by (5.32) and (5.33) with the ai numerical results. Since the independent variable in the numerical computation is the angle of the incident wave Ψi, it can be converted to K directly with higher order correction formula (5.33). For the ratio of the amplitude, we can apply the higher order correction formula to obtain the ratio in KP coordinate when the angle of the incident wave is smaller than the critical angle. When the incident angle is larger than the critical angle, we simply use the numerical ratio of the amplitude because the higher order correction is based on a small incident angle.

In the following discussion, we ﬁnd a formula to convert the ratio of the amplitude from the numerical results to the KP coordinates when the angle of the incident wave is smaller than the critical angle. A variable with ˜ indicates the observed value. A variable without ˜ indicates the value in KP coordinate. Using formula (5.27) with higher order correction, we have

2 2ãM cos (Ψstem) 2ãM aM = √ = √ 1 + 1 + 5ãM 1 + 1 + 5ãM 2 2ãi cos (Ψi) ai = √ 1 + 1 + 5ãi

93 where Ψstem is the angle of the stem which is 0. The ratio of the amplitude from the

a˜M numerical computation, denoted by αobs, is . Therefore, the ratio in KP coordinate a˜i is √ √ a 2α a˜ 1 + 1 + 5ã α 1 + 1 + 5ã M √ obs i i obs √ i = 2 = 2 ai 1 + 1 + 5αobsa˜i 2ãi cos (Ψi) cos (Ψi) 1 + 1 + 5αobsa˜i

In ﬁgure 5.8, the * show the numerical result in the case thata ˜i = 0.3 and

o o o o t = 350 for Ψi = 10 , 15 , 20 ,..., 55 . The curve is the theoretical prediction of the ampliﬁcation factor in formula (5.32). Figure 5.9 shows the ampliﬁcation factor in

o o o o o o o o o o the case thata ˜i = 0.1 at t = 500 for Ψi = 10 , 15 , 20 , 24 , 25 , 26 , 27 , 30 , 35 , 40 .

Figure 5.8: aM vs. K with a = 0.3. Figure 5.9: aM vs. K with a = 0.1. ai ai

The numerical results shown in ﬁgure 5.8 and 5.9 indicate an excellent agreement

with the theoretical prediction using the higher order KP equation. Recall that in

ﬁgure 3.13, the numerical results for Ψi > Ψch seems to ﬁt the curve that represents

the weak interaction case. One should note that in ﬁgure 5.8 and 5.9 that the formula

(5.32) for K > 1 also gives the Miles curve for the weak interaction case. This can

94 be shown as follows. Consider the amplitude ratio formula (5.32) in the case of the

X-shape solitons, i.e., for K > 1, we can expand

aM 2 1 ∼ 1 ∼ 2(1 + 2 ). ai 1 − 4K2 4K

Substituting K by formula (5.33) derives the following,

a 3a cos2 Ψ 3a cos2 Ψ M = 2 1 + √i ∼ 2 + i 2 2 ai 2 tan Ψi(1 + 1 + 5a) 2 tan Ψi 2 2 3a (1 − sin Ψi) 3 3 2 = 2 + 2 = 2 + 2 − 3 + sin Ψi ai 2 sin Ψi 2 sin Ψi 2 This formula is close to the formula (3.8). Indeed, the only diﬀerence comes from the

3 coeﬃcient of the last term, 2 instead of 2 in Miles weak interaction formula (3.8).

The quasi-two dimensionality in the KP theory assumes that the incident angle Ψi is small which leads to the last term in the order of O(2). This means that with the next order correction in O(), the maximum amplitude ratio of the X-shape soliton solutions agree with the Miles weak interaction case up to this order.

One should also note that when the incident angle Ψi approaches to the critical angle, the closer it gets to the critical angle, the greater the diﬀerence from the numerical result to the theoretical value. This may be due to the fact that the closer

Ψi gets to the critical angle, the smaller Ψ∗ becomes and the more time for the Mach stem to achieve its asymptotic state, when a is held constant.

5.3.2 The angle Ψ∗ corresponding to the stem length

For the regular reﬂection case, the stem is not formed hence Ψ∗ = 0. For the Mach reﬂection case, the asymptotic state of the wave can be described by (3142)-soliton solution of the KP equation. The coordinates of the intersection point of the triplet

95 [1, 3]-, [3, 4]-, [1, 4]-soliton are given in [30], A X = 0 (1 + K)2T ∗ 2 r A Y = 0 (1 − K)T ∗ 2 We convert it into the physical coordinate and then replace the KP amplitude by the

higher order correction formula (5.27)

2 2 A0 2 a cos (Ψi) 2 x˜∗ = 1 + · (1 + K) C0t˜= 1 + √ (1 + K) C0t˜ 3 2 1 + 1 + 5a

r s 2 2 A0 2a cos (Ψi) y˜∗ = · (1 − K)C0t˜= √ (1 − K)C0t˜ 3 2 3(1 + 1 + 5a)

where K is given by (5.33). Hence, we obtain the angle Ψ∗ by taking the inverse y˜ tangent function of ∗ . x˜∗

Figure 5.10: Ψ∗ vs. Ψi with a = 0.3. Figure 5.11: Ψ∗ vs. Ψi with a = 0.1.

In ﬁgure 5.10 and 5.11, the numerical results are marked by o and the curves are the theoretical value of Ψ∗. Figure 5.10 shows the case that a = 0.3, t = 350 and

o o o o Ψi = 10 , 15 , 20 ,..., 60 . The stem angle Ψ∗ is 0 for regular reﬂection which has

96 a good agreement with the theoretical value. It is also indicated by ﬁgure 4.8 that

there is no stem around the line y = 0. For the Mach reﬂection cases, the numerical results of Ψ∗ appear to be larger than the theoretical values. Figure 5.11 shows

o o o o o o o o o o the case that a = 0.1, t = 500 and Ψi = 10 , 15 , 20 , 24 , 25 , 26 , 27 , 30 , 35 , 40 .

As in ﬁgure 5.10, the numerical results of Ψ∗ are quite similar to the theoretical

prediction. Compared with the result in section 3.5.2, the theoretical curve given by

the KP solution(right picture in ﬁgure 3.14) is above the numerical results whereas

the theoretical curve given by the higher order KP solution(ﬁgure 5.10) is below the

o numerical results. Note that Ψ∗ at the incident angle Ψi = 40 is in agreement

with the theoretical curve given by higher order KP solution. The diﬀerences at the

incident angle Ψi < Ψch, namely, in the Mach reﬂection case may be because the

Mach stem has not fully developed for t = 350. We will consider an asymptotic

simulation to demonstrate the tendency of the Ψ∗.

5.3.3 The angle Ψr of the reﬂective wave

The formula for the angle Ψr of the reﬂective wave has been derived in formula

(3.65). With higher order correction (5.27), we obtain the theoretical formula of the

angle of the reﬂective wave, s 2 ! 6a cos (Ψi) Ψr = arctan √ 1 + 1 + 5a

97 Figure 5.12: Ψr vs. Ψi with a = 0.3. Figure 5.13: Ψr vs. Ψi with a = 0.1.

Figures 5.12 and 5.13 show the reflective angle Ψr versus the incident angle Ψi for a = 0.3 and a = 0.1 respectively. Compare figure 5.12 with figure 3.15 on the

o right, the angle of the reflective wave at Ψi = 40 has a good agreement with the theoretical prediction given by higher order KP solutions. Although it appears that some Ψ∗ at small incident angles agree with the theoretical curve given by KP solution, the slightly negative correlation between Ψ∗ and Ψi in the Mach reflection case, i.e., Ψi < Ψch is identified by the theoretical curve from higher order KP solution.

Compare figure 5.12 and figure 5.13, a smaller incident amplitude leads to a smaller difference between the numerical and theoretical values for the Mach reflection case.

5.3.4 The amplitude ar of the reﬂective wave

We denote the maximum amplitude of the reﬂective wave as ar and keep the ratio

a˜r αobs = in the numerical result. In order to compare with the theoretical values, we a˜i follow the same idea as described in section 3.5.2 for aM to convert the ratio into the ai

98 KP coordinates. With the higher order correction, we have

2 2ãr cos (Ψr) ar = √ 1 + 1 + 5ãr 2 2ãi cos (Ψi) ai = √ 1 + 1 + 5ãi

where Ψr is the angle of the reﬂective wave obtained in the previous section. Calculate

the ratio, we obtain, in the KP coordinates, √ a cos2(Ψ ) 1 + 1 + 5˜a r r √ i = αobs · 2 · ai cos (Ψi) 1 + 1 + 5αobsa˜i

This is the formula in the Mach reﬂection case. For regular reﬂection, the ratio is

obviously ar = 1. ai

Figure 5.14: ar vs. K with a = 0.3. Figure 5.15: ar vs. K with a = 0.1. ai ai

In figure 5.14, we show ar for a = 0.3 and t = 350. Compare with the right plot in ai figure 3.16, the numerical data at Ψ = 40o is in accordance with the theoretical curve given by the higher order KP solution. The numerical results at the other incident angles for the Mach reflection case are in a better agreement with the theoretical

99 prediction of the higher order KP equation. For the case when a = 0.1 in ﬁgure 5.15, the numerical data denoted by o are even closer to the theoretical curve which again indicates the small incident amplitude is more preferable in order to compare with the theoretical prediction.

In section 3.5, we have seen that the numerical results in the case of a = 0.3 have a better agreement with the KP theoretical curves than with the Miles theoretical curves. Now, we have shown in ﬁgures 5.8, 5.10, 5.12 and 5.14 that the numerical results are in a better agreement with the theoretical curves obtained from the higher order KP equation. This indeed conﬁrms that the numerical simulations in the previous articles[17][54] are correct and they do have a good agreement with the theoretical prediction if considering the higher order correction and using the KP coordinates.

5.3.5 Asymptotic simulations

From the numerical simulations, we see that the simulated data are distributed near the theoretical KP curves. However, the differences tend to be larger as the incident angle is closer to the critical point between the Mach reflection and the regular reflection. From those results between the case a = 0.3 and a = 0.1, we can see that the numerical data in the case of a = 0.1 is closer to the theoretical curve than in the case of a = 0.3. One problem here is that the KP equation is the leading order approximation of the Euler equation under the assumptions of small amplitude, long wave and week dependence on y. Therefore, a = 0.3 may be too large for the amplitude of the incident wave. Another problem is that the KP soliton solution represents an asymptotic state of the solution of the initial value problem with a V-shape initial wave. In other words, we need to compute for a large amount

100 of time before measuring the parameters of interest to compare with the theoretical values from the higher order Miles theory.

For this purpose, we perform the following test on two cases: the incident amplitude is a = 0.1 and a = 0.05. For the case a = 0.1, we consider the incident angle Ψi equals 10o, 15o, 20o, 24o, 25o, 26o, 27o, 30o, 35o and 40o respectively. In each situation,

aM ar we collect our numerical result about ,Ψ∗,Ψr and at time t = 300 and t = 600. ai ai And we plot the numerical data at the two different times in one figure with green star or circle indicating an earlier time and blue star or circle a later one. Notice that we use star for the numerical result updated by the higher order corrections, and circle is used for observed angle without modification. The theoretical KP curve is colored in red. As we can see in figures 5.16, 5.17, 5.18 and 5.19, the blue stars(data at a later time) do get closer to the KP curve. In fact, there is a lot of improvement in the case where the incident angle is around the critical angle.

Figure 5.16: aM vs. K with a = 0.1. ai Figure 5.17: Ψ∗ vs. Ψi with a = 0.1.

101 Figure 5.18: ar vs. K with a = 0.1. ai Figure 5.19: Ψr vs. Ψi with a = 0.1.

A similar test is performed for the case that the incident amplitude is a = 0.05.

The incident angle used for this case is 5o, 10o, 15o, 18o, 20o, 22o, 25o, 30o and 35o.

aM ar The two diﬀerent times where we record ,Ψ∗,Ψr and are t = 900 and t = 1800. ai ai The numerical results and the KP curve are shown in ﬁgure 5.20, 5.21, 5.22 and 5.23.

Notice that the setting of the color and shape of the data are the same as described in

the case of a = 0.1. One may notice a signiﬁcant movement towards the theoretical

curve for the data at a later time, especially for aM and ar . ai ai

102 Figure 5.20: aM vs. K with a = 0.05. ai Figure 5.21: Ψ∗ vs. Ψi with a = 0.05.

Figure 5.22: ar vs. K with a = 0.05. ai Figure 5.23: Ψr vs. Ψi with a = 0.05.

Summary: For the regular reflections where K > 1, the numerical data are distributed near the theoretical curves closely and they can not be distinguished at the two different times. For the Mach reflection when K < 1, the numerical results

103 at a later time are much closer to the theoretical curves than those at an earlier time.

For those incident angles near the critical angle, the numerical results are getting closer toward the theoretical curve as time increases. Thus the numerical results tend to approach the theoretical predictions given by the higher order KP equation.

104 Chapter 6: Stability of some KP solitons

The previous results of the construction of the KP solution from the numerical simulation and the robustness of line soliton solutions suggest that the solution u(x, y, t) of the initial value problem of the KP equation with certain class of initial data may converge to an exact KP soliton solution u0(x, y, t) in certain compact region D of

2 2 R . That is, we expect that there exists a compact region Dt ⊂ R , so that for any

(x, y) ∈ Dt, we have

u(x, y, t) → u0(x, y, t) as t → ∞.

That is, one may state that

ZZ 2 |u(x, y, t) − u0(x, y, t)| dxdy → 0 as t → ∞. Dt

Here a compact domain Dt may depend on the time t and covers the main part of the interaction pattern of the solution. In this chapter, we choose Dt as a rectangular region (section 6.1),

2 Dt = {(x, y) ∈ R : |x − x0(t)| ≤ dx, |y| ≤ dy},

with some x0(t), or a circular disk (section 6.2),

2 2 2 2 Dt := {(x, y) ∈ R : |x − x0(t)| + |y − y0(t)| ≤ r }

105 with appropriate choice of (x0(t), y0(t)).

In this chapter, we perform numerical simulations of the initial value problem of the KP equation to conﬁrm this type of stability. In section 6.1, we consider the stability of one-soliton solution under small localized perturbations, and in section

6.2, we demonstrate a convergence of the solution for the initial value problem. We choose a particular class of initial data, so that the solution converges to a Y-soliton solution.

6.1 Stability of one-line soliton with transverse perturbations

Recall that the one-line soliton solution of the KdV equation can be written as

2 √ φc(θ, x¯) = 2c sech c (θ − x¯) with θ = x − ct, (6.1) wherex ¯ is the initial position in the x-coordinate.

To study the stability of this solution under small transverse perturbation, we consider an initial value problem of the KP equation with the initial data,

2 2 u(x, y, 0) = 2[1 + m0(y)] sech [(1 + m0(y))(θ − ξ0(y) − x¯)] + w0(x, y). (6.2)

where m0(y) and ξ0(y) represent transverse perturbations to the line soliton solution

φc(θ, x¯). We impose an additional term w0(x, y) in (6.2) so that the following condition is satisﬁed, Z +∞ uyy dx = 0. (6.3) −∞ We then expect that for a large time, the solution may be expressed in the form,

u(x, y, t) = 2[1 + m(y, t)]2 sech2[(1 + m(y, t))(θ − ξ(y, t) − x¯)] + radiations (6.4)

106 with m(y, 0) = m0(y) and ξ(y, 0) = ξ0(y). Here the term “radiations” expresses the

dispersive radiations which decay as t increases. In the following sections, we discuss

the behavior of those functions.

6.1.1 Phase shift perturbation

We ﬁrst consider a local phase shift ξ0(y) of the one-line soliton in a small region

of y while the amplitude of the soliton is the same everywhere, i.e. m0(y) = 0.

The constraint (6.3) is satisﬁed without the add-on term w0. We then consider the

following initial data,  d  ξ0(y) = {tanh[10(y + 2)] − tanh[10(y − 2)]}  2  (6.5) m0(y) = 0     w0(x, y) = 0 where d is a small constant. Figure 6.1 shows the solitary wave crest location with d = 0.5 andx ¯ = −5 at t = 0.

Figure 6.1: Initial solitary wave with a local phase shift.

107 We set our computation domain large enough so that the perturbation at the

central area does not reach the boundary at the maximum computing time, t = 10.

The computation domain is [−32, 32] × [−64, 64] with 32 grids in one unit. Our

observation window is a moving domain, [¯x − 10 + t, x¯ + 10 + t] × [−20, 20]. We

2 calculate the L (Dt) norm of the diﬀerence between u(x, y, t) and φ(θ, x¯) with a

moving rectangular domain Dt = [¯x − 3 + t, x¯ + 3 + t] × [−20, 20]. We also record the maximum local phase shift of the crest to estimate the velocity of the propagation in the y direction. Here we take d = 0.1, 0.2, 0.3, 0.4, 0.5.

Figure 6.2: d = 0.1

Figure 6.3: d = 0.2

108 Figure 6.4: d = 0.3

Figure 6.5: d = 0.4

Figure 6.6: d = 0.5

109 The pictures in the left column shows the phase shift (z axis) of the crest of the soliton from t = 0 to t = 10. y is between -20 and 20, our observation window. It is clear that the curve at t = 0 indicates the phase shift of d from y = −2 to y = 2(see

ﬁgure 6.2). The curve at t = 10 takes strictly 0 at y = 0 for the case d = 0.1, 0.2 and one grid distance from 0 for the case d = 0.3, 0.4, 0.5. We have found the tendency that as the initial local phase shift increases, the center of the phase shift grows for a large time. We also notice that the initial proﬁle of the phase shift separates into two humps as time evolves. This is the front of the phase shift moving in y direction.

We record the location of the maximum phase shift and show in the middle column of figure 6.2-6.6. A straight line is fitted to the location data and we note that the slope is close to 1 for all the cases with d = 0.1,..., 0.5. Figures in the right column are the L2 norm of the difference between the soliton with perturbation and the exact line soliton solution for each time, i.e.,

ZZ 2 2 2 ku − uekL (Dt) := |u(x, y, t) − φc(θ, x¯)| dxdy Dt where Dt = [¯x − 3 + t, x¯ + 3 + t] × [−20, 20].

We observe that for d = 0.1 and 0.2 cases, the middle phase shift drops to 0 and two humps spread out in the opposite direction. The decay of the L2-norm shown in the right graphs in these ﬁgures implies that the solution u(x, y, t) for each d value approaches to the corresponding line-soliton solution in the domain Dt of the

2 L -measure, i.e., the phase shifts appear locally and escape from Dt.

110 Figure 6.7: d = 0.1 phase shift to right.

Figure 6.8: d = 0.2 phase shift to right.

As a comparison, we perform a test with the local phase shift of the small per-

turbation ahead of the exact line soliton, i.e., ξ0(y) in (6.2)is replaced with −ξ0(y).

The result is shown in ﬁgure 6.7-6.8. Notice that in the ﬁrst column, the negative

direction in z axis indicates a phase shift to the right. The center is stable to zero

value with two humps propagating in the ±y direction with speed near 1(shown in the middle column of ﬁgure 6.7 and 6.8). The error decreases in a similar manner as the phase shift to the left. These ﬁgures clearly show that the phase shifts propagate along the y-direction and escape from the domain of the L2-norm.

111 6.1.2 Amplitude perturbations

We consider the second type of transverse perturbation with a small amplitude dent in the middle given by m0(y). In this case, we need w0(x, y) to satisfy the constraint (6.3). We take the following form of m0(y) and w0(x, y)  ξ0(y) = 0    β  m0(y) = {tanh[10(Y − 1)] − tanh[10(Y + 1)]}  2 (6.6) 2 2  w0(x, y) = 2[1 − m0(y)] sech [(1 − m0(y))(θ − ξ0(y) − x¯w)]    2  − 2sech [θ − ξ0(y) − x¯w] where β controls the height of the dent. We choose three diﬀerent locations for the function w0(x, y),x ¯w = −35, −40, −45 where in this case the domain is [−64, 64] ×

[−64, 64] with 8 grids in one unit. The window for error check is the same as described in the last case.

Figure 6.9: An amplitude dent in the middle of the soliton crest and an artiﬁcial amplitude hump is placed atx ¯w = −35.

112 Figure 6.10: An amplitude dent in the middle of the soliton crest and an artiﬁcial amplitude hump is placed atx ¯w = −40.

Figure 6.11: An amplitude dent in the middle of the soliton crest and an artiﬁcial amplitude hump is placed atx ¯w = −45.

The numerical results of β = 0.05 are shown in ﬁgure 6.9, 6.10 and 6.11. The eﬀect of the small amplitude dent leads to a local phase shift of the soliton crest to the left because of the fact that a smaller amplitude corresponds to a lower travelling speed.

The plots on the ﬁrst column in ﬁgure 6.9 6.10 and 6.11 are the phase shift ξ(0, t)

at y = 0 for time t up to 10. The positive phase shift indicates a leftward shift from

the main soliton as expected. It saturates to a ﬁxed value 0.125 after a small interval

of time. The pictures in the middle column show the phase shift at all y values.

113 The pictures at the right column describe the errors. The constant rate of increasing errors implies that the phase shift propagates towards y = ±∞ at a constant speed.

The results of the phase shifts for the diﬀerent locationx ¯ of the function w0(x, y) are almost the same. We shall usex ¯w = −40 for the following numerical tests.

Figure 6.12: Initial soliton has a smaller amplitude in the middle.

Figure 6.13: Initial soliton has a larger amplitude in the middle.

For an amplitude dent m0(y) in the middle, we choose diﬀerent values β =

0.05, 0.1, 0.2, and the numerical results are shown in ﬁgure 6.12. We also consider an amplitude hump in the middle of the crest of the main testing soliton and place

114 a corresponding artificial soliton behind it atx ¯w = −40. The numerical results on different amplitude β = −0.05, −0.1, −0.2 are shown in figure 6.13. The amplitude dent creates a left phase shift while the amplitude hump leads to a right one. As the absolute amount of the amplitude change increases, so does the phase shift for both cases. The time before the threshold of the phase shift gets shorter. The stable absolute phase shift is 0.125 for the case |β| = 0.05, 0.1 and 0.25 for |β| = 0.2.

From the above numerical evidence, we observe that this type of the small disturbance causes a non-vanishing phase shift propagating toward y = ±∞. This implies that the KP line soliton solution is unstable in the L2(R2) sense, i.e., ZZ 2 |u(x, y, t) − φc(θ, x¯)| dydx → ∞, as t → ∞ 2 R for any choice of constant shiftx ¯. However, we observe that for each y0, there exists a large time T such that for all t > T , the phase shift passes beyond y0 and

Z +∞ 2 |u(x, y0, t) − φc(θ, x¯)| dx → 0, as t → ∞ −∞

Mizumachi and Tzvetkov[43] have proven the stability of a two dimensional solitary wave under a transverse disturbance in an inﬁnite domain of x and a periodic domain in y. They found that if the initial data u(x, y, 0) are suﬃciently close to an

2 1 exact soliton solution φc(θ, x¯) in the sense of L norm for (x, y) = Lx × Ly ∈ R × S , then the solution u(x, y, t) corresponding to the initial data stays arbitrarily close to the exact soliton solution up to a phase shift. Mizumachi[42] has extended the problem on an inﬁnite two dimensional space and has proven that (1) the amplitude of the line soliton converges to that of the line soliton initially and (2) a jump of the local phase shift ξ(y, t) of the crest propagates with a ﬁnite speed toward y = ±∞.

From his results, the local amplitude and the phase shift of the line soliton satisfy a

115 system of one dimensional wave equations with diﬀraction terms, whose linear terms

have the following form

 4b ∼ 3b + 8ξ + 6(bξ )  t yy yy y y 1 (6.7)  4ξ ∼ 2b − γb + ξ + 3(ξ )2 − b2 t yy yy y 4

4 3 2 2 where b(y, t) = 3 c(y, t) − 1 , and γ is a constant. Setting c(y, t) = (1 + m(y, t)) with a small variance m(y, t), we have b(y, t) ' 4m(y, t). Also, we consider a smooth

∂ perturbation, i.e., ∂y is small. Then, Mizumachi’s equation becomes

1 m ' ξ , ξ ' 2m. t 2 yy t

Then, ξ satisﬁes the initial value problem ( ξ(y, 0) = ξ0(y) ξtt ' ξyy with ξt(y, 0) = 2m0(y)

The solution of the initial value problem is

1 Z x+t ξ(y, t) = (ξ0(y − t) + ξ0(y + t)) + m0(s)ds. 2 x−t

This formula explains roughly the behavior of the above numerical solutions.

6.2 Stability of Y-shape soliton

Here, we discuss a generation of the resonant solution, i.e., Y-shape solitons, and

demonstrate a convergence of the solution of the initial value problem for a Y-shape

soliton. We consider an initial data u(x, y, 0) which consists of [1, 2]-soliton for y > 0

and [1, 3]-soliton for y < 0, part of the Y-shape soliton solution without [2, 3]-soliton.

Figure 6.14 shows the corresponding chord diagram of this initial value.

116 Figure 6.14: The initial(partial) chord diagram corresponding to (6.8).

The upper chord indicates the [1, 2]-soliton in y > 0, and the lower chord indicates

the [1, 3]-soliton in y < 0. Then initial value is given by

( 1 2 2 1 (k2 − k1) sech (θ2 − θ1 − θ12) for y > 0 u(x, y, 0) = 2 2 (6.8) 1 2 2 1 2 (k3 − k1) sech 2 (θ3 − θ1 − θ13) for y < 0

2 where θj = kjx + kj y. Then, we expect that the solution u(x, y, t) converges to a Y-shape soliton solution with π = (2, 3, 1), i.e., the chord diagram will be completed by adding [2, 3]-soliton in y > 0.

We first perform the simulation with this initial condition to t = 12 and obtain un, the numerical result at every time t. Then, we use the data at t = 10 and apply the minimization method introduced in section 5.2 to find the A-matrix that generates the exact Y-shape soliton solution ue. Compare the numerical data un and the exact solution ue, we expect that the difference between the two decreases as time evolves.

The computing domain is [−Lx,Lx] × [−Ly,Ly] with Lx = 64 and Ly = 64. There are four mesh grids in one unit for both x-axis and y-axis, therefore the number of modes for two-dimensional FFT is 512 for both x- and y-direction. We pick the values

of k-parameters as {−0.75, 0.25, 1.25}. Therefore, the slopes and the amplitudes of

117 [1, 2]- and [1, 3]-soliton are given by (3.33)  1  A = 2  A12 =  13 2 and . 1 1  tan Ψ = −  tan Ψ13 = 12 2 2

We consider the intersection of the two solitons at the origin, i.e., θ12 = 0 and θ13 = 0,

hence we can ﬁnd the initial A-matrix

1 0 −1 A = . 0 1 0.5

Notice that the different amplitude and angle of the two line solitons on the upper and lower boundaries break the periodicity condition for the numerical computation. This R will create a disturbance near y = ±Ly. Also, note that uyydx = 0 is not satisfied, due to the amplitude difference, and it will lead to a transverse disturbance near x = ±Lx. Both disturbances near the boundaries at the beginning will propagate to the center which will cause the the error in the numerical simulation. To prevent this, we use a patching scheme at the boundaries y = ±Ly in order to keep the original amplitudes after every small time step. We also apply a smooth masking scheme at the boundaries x = ±Lx which maintains the solution to vanish near x = ±Lx.

118 Figure 6.15: Numerical simulation of KP equation with initial value u(x, y, 0) given by (6.8) and the exact Y-shape soliton solution by minimization.

The ﬁrst three pictures in ﬁgure 6.15 are the contour plots of the numerical simulation of the KP equation with initial value u(x, y, 0) given in the formula (6.8) at t = 0, t = 6 and t = 12. Note that the [2, 3]-soliton gradually develops as time evolves. In order to test if it approaches to an exact Y-shape soliton solution, we consider the numerical data within a bounded region Dt around the intersection point

(x0(t), y0(t)) of the three line solitons at t = 10,

2 2 2 Dt = {(x, y):(x − x0(t)) + (y − y0(t)) ≤ r }

where we choose r = 10. The A-matrix by minimization method is

1 0 −1.1761 A = 0 1 0.5591

119 and the corresponding phase shifts of the two solitons are θ12 = 0.0505 and θ13 =

−0.0811. Since the [i, j] line-soliton has the form

1 1 u = (k − k )2sech2 (θ − θ − θ ), 2 j i 2 j i ij then θij > 0(< 0) implies a positive(negative) phase shift. Therefore, the [1, 2]-soliton

has a positive phase shift and the [1, 3]-soliton has a negative phase shift, and each phase shift propagates along the crest of the corresponding soliton. We show the exact

Y-shape soliton solution of the KP equation associated with the above A-matrix at

t = 0, t = 6 and t = 12 in the bottom three pictures in ﬁgure 6.15, respectively.

2 Figure 6.16: The average L2 error between the exact solution and the numerical results versus time.

In ﬁgure 6.16, we plot the error function deﬁned by

RR |u − u |2 dA Dt e n E(Dt) = Area of Dt

where ue is the exact solution obtained from the k-parameters and the A-matrix

obtained by minimization. As we can see, the error keeps decreasing beyond the time

120 t = 10 at which the minimization is conducted to ﬁnd the exact KP solution. This numerical study implies that the solution of the initial value problem with the initial data (6.8) converges to the Y-soliton associated to the completed chord diagram of

1 2 3 π = 2 3 1 .

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