Lump chains in the KP-I equation

TL;DR

This paper constructs a broad class of solutions for the KP-I equation, including lump chains and their interactions, using a Grammian-based approach, expanding the understanding of soliton-like structures in KP-I.

Contribution

It introduces a new method to generate diverse lump chain solutions for KP-I, including degenerate and interacting configurations, using a reduced Grammian $ au$-function.

Findings

Constructed linear arrangements of lump chains with distinct velocities.

Allowed degenerate configurations like parallel or superimposed chains.

Described interactions between lump chains and individual lumps.

Abstract

We construct a broad class of solutions of the KP-I equation by using a reduced version of the Grammian form of the $\tau$-function. The basic solution is a linear periodic chain of lumps propagating with distinct group and wave velocities. More generally, our solutions are evolving linear arrangements of lump chains, and can be viewed as the KP-I analogues of the family of line-soliton solutions of KP-II. However, the linear arrangements that we construct for KP-I are more general, and allow degenerate configurations such as parallel or superimposed lump chains. We also construct solutions describing interactions between lump chains and individual lumps, and discuss the relationship between the solutions obtained using the reduced and regular Grammian forms.