Interactions among different types of nonlinear waves described by the Kadomtsev-Petviashvili Equation

TL;DR

This paper explores the interactions among various nonlinear wave types described by the KP equation, using Darboux transformations to find solutions that include interactions between solitons, cnoidal waves, and Painlevé waves.

Contribution

It introduces a method to derive interaction solutions among different nonlinear waves via localized nonlocal symmetries and Darboux transformations for the KP equation.

Findings

Interaction solutions among different wave types are obtained.

Darboux transformations add an additional soliton to basic wave solutions.

The method reveals the role of DT in wave interactions.

Abstract

In nonlinear physics, the interactions among solitons are well studied thanks to the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of nonlinear waves such as the solitons (or solitary waves), the cnoidal periodic waves and Painlev\'e waves. In this paper, the nonlocal symmetries related to the Darboux transformations (DT) of the Kadomtsev-Petviashvili (KP) equation is localized after imbedding the original system to an enlarged one. Then the DT is used to find the corresponding group invariant solutions such that interaction solutions among different types of nonlinear waves can be found. It is shown that starting from a Boussinesq wave or a KdV-type wave, which are two basic reductions of the KP equation, the essential and unique role of the DT is to add an additional soliton.