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Regularizations for shock and rarefaction waves in the perturbed solitons of the KP equation

Guangfu Han, Yuji Kodama, Chuanzhong Li, Lin Sun

TL;DR

This work analyzes the perturbed soliton dynamics of the KP equation by coupling adiabatic soliton parameter evolution to slow variables $Y=\epsilon y$ and $T=\epsilon t$, yielding a quasi-linear κ-system for the pair $(\kappa_1,\kappa_2)$. It demonstrates that shocks in the κ-system trigger resonant generation of new solitons (Y-solitons) while rarefactions produce parabolic-solitons; a dispersive regularization inspired by KdV-Whitham theory ensures global solutions for initial data featuring half-lines and V-shapes. The authors derive simple-wave behavior, provide explicit peak-trajectory descriptions (including parabolic segments), and perform numerical simulations that agree with the perturbative predictions. Collectively, the results offer an analytic mechanism for local stability and the asymptotic soliton composition under perturbations, clarifying how resonant interactions regularize singularities and produce complex KP soliton networks with practical relevance to Mach reflection phenomena.

Abstract

By means of an asymptotic perturbation method, we study the initial value problem of the KP equation with initial data consisting of parts of exact line-soliton solutions of the equation. We consider a slow modulation of the soliton parameters, which is described by a dynamical system obtained by the perturbation method. The system is given by a quasi-linear system, and in particular, we show that a singular solution ({shock wave}) leads to a generation of new soliton as a result of resonant interaction of solitons. We also show that a regular solution corresponding to a rarefaction wave can be described by a parabola (we call it {parabolic}-soliton). We then perform numerical simulations of the initial value problem and show that they are in excellent agreement with the results obtained by the perturbation method.

Regularizations for shock and rarefaction waves in the perturbed solitons of the KP equation

TL;DR

This work analyzes the perturbed soliton dynamics of the KP equation by coupling adiabatic soliton parameter evolution to slow variables and , yielding a quasi-linear κ-system for the pair . It demonstrates that shocks in the κ-system trigger resonant generation of new solitons (Y-solitons) while rarefactions produce parabolic-solitons; a dispersive regularization inspired by KdV-Whitham theory ensures global solutions for initial data featuring half-lines and V-shapes. The authors derive simple-wave behavior, provide explicit peak-trajectory descriptions (including parabolic segments), and perform numerical simulations that agree with the perturbative predictions. Collectively, the results offer an analytic mechanism for local stability and the asymptotic soliton composition under perturbations, clarifying how resonant interactions regularize singularities and produce complex KP soliton networks with practical relevance to Mach reflection phenomena.

Abstract

By means of an asymptotic perturbation method, we study the initial value problem of the KP equation with initial data consisting of parts of exact line-soliton solutions of the equation. We consider a slow modulation of the soliton parameters, which is described by a dynamical system obtained by the perturbation method. The system is given by a quasi-linear system, and in particular, we show that a singular solution ({shock wave}) leads to a generation of new soliton as a result of resonant interaction of solitons. We also show that a regular solution corresponding to a rarefaction wave can be described by a parabola (we call it {parabolic}-soliton). We then perform numerical simulations of the initial value problem and show that they are in excellent agreement with the results obtained by the perturbation method.

Paper Structure

This paper contains 21 sections, 5 theorems, 109 equations, 35 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Background
  3. One line-soliton
  4. O-type soliton and Y-soliton
  5. O-type soliton
  6. Y-soliton: resonant solution
  7. The $\kappa$-system for soliton perturbations
  8. The initial value problems for the $\kappa$-system
  9. Half-line initial data
  10. The initial value problem with V-shaped initial data
  11. The cases (a) and (b)
  12. The case (c)
  13. The cases (d) and (e)
  14. The case (f)
  15. The cases (g) and (h)
  16. ...and 6 more sections

Key Result

Lemma 2.2

The intersection point $(x_{0}(t),y_{0}(t))$ of those Y-solitons is given by

Figures (35)

  • Figure 1: The left panel shows the contour plot of the $[i,j]$-soliton solution (\ref{['92']}) with $\kappa_{i}=-1$, $\kappa_{j}=2$ at $t=0$. The dotted line is the crest of the soliton. The middle panel shows the corresponding permutation (transposition $i\leftrightarrow j$), which we call the chord diagram of the soliton. The diagram indicates the asymptotic structure of KP soliton, that is, the upper (lower) part of the diagram shows the $[i,j]$-soliton for $y\gg0$ ($y\ll 0$). The right panel shows the corresponding colored $\kappa$-graphs.
  • Figure 2: The parameters in the matrix $A$ are $a=\frac{1}{a'}=\sqrt{\frac{3\cdot 10^5}{2}}$, $b=\frac{1}{b'}=\sqrt{\frac{2}{3\cdot 10^{5}}}$ (i.e., $ab=1$). The $\kappa$-parameters are given by $(\kappa_{1},\kappa_{2},\kappa_{3},\kappa_{4})=(-\frac{3}{2},-10^{-5},10^{-5},\frac{3}{2})$. The left panel shows the contour plot of the solution $u(x,y,0)$. The middle panel is the chord diagram for the O-type soliton. The right panel shows the colored $\kappa$-graph in the slow scale $Y=\epsilon y$, and note that the phase shift in the left figure is ignored in this scale.
  • Figure 3: Y-solitons as the result of the resonant interactions of two line-solitons of O-type in the limit $\kappa_3\to\kappa_2$. In this limit, the chord diagram becomes singular, and the middle panel shows that the singular chord diagram splits into two asymptotic diagrams for $y\gg 0$ and $y\ll 0$. Then the singularity can be represented by three wave resonant interaction i.e., the generation of $[1,4]$-soliton. The colored $\kappa$-graphs around the points $A$ and $B$ are on the shifted coordinates $y-y_+$ and $y+y_+$, respectively.
  • Figure 4: Y-soliton and the colored $\kappa$-graph with the singular point $C$.
  • Figure 5: The initial data \ref{['45']}. Each bold face line shows a semi-infinite line-soliton (half-soliton). In the right panel, the amplitude of soliton 2 ($u_0^-$) is fixed to be 2, and that of soliton 1 ($u_0^+$) is a variable $A_0$.
  • ...and 30 more figures

Theorems & Definitions (15)

  • Definition 2.1
  • Lemma 2.2
  • Example 2.3
  • Lemma 3.1
  • proof
  • Proposition 4.1
  • proof
  • Definition 4.2
  • Example 4.3
  • Definition 5.1
  • ...and 5 more