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arXiv · 2503.01093

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

Guangfu Han·Yuji Kodama·Chuanzhong Li·Lin Sun
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Abstract

Using an asymptotic perturbation method, we study the initial value problem for the KP equation with initial data consisting of parts of exact line-soliton solutions. We consider a slow modulation of the soliton parameters, described by a dynamical system obtained via the perturbation method. {The dynamical system is given by a $2$-component quasi-linear system.} In particular, we show that a singular solution (\emph{shock wave}) {of the system} leads to the generation of a new soliton as a result of the resonant interaction of solitons. We also show that a regular solution corresponding to a rarefaction wave {of the system} can be described by a parabola (which we call a \emph{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.

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Guangfu Han, Yuji Kodama, Chuanzhong Li, Lin Sun. 2025-03-03. Regularizations for shock and rarefaction waves in the perturbed solitons of the KP equation. https://arxiv.org/abs/2503.01093

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