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

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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 By 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 propa- gation 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 so- lution 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.
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