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<p>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 (<i>shock wave</i>) 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 <i>parabolic-soliton</i>). 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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年5月18日

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  • Guangfu Han,
  • Yuji Kodama,
  • Chuanzhong Li,
  • Lin Sun

摘要

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 (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 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.