Lump chains in the KP-I equation

We construct a broad class of solutions of the Kadomtsev-Petviashvili (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.

Automated summary

In this study, a class of solutions for the Kadomtsev-Petviashvili (KP)-I equation is constructed using a reduced version of the Grammian form of the tau-function. The solutions consist of linear periodic lump chains with distinct group and wave velocities, evolving into linear arrangements of lump chains. These solutions can be seen as the KP-I analogues of line-soliton solutions in KP-II, but with more general linear arrangements that allow for degenerate configurations such as parallel or superimposed lump chains. The interactions between lump chains and individual lumps are also discussed, along with the relationship between solutions obtained using reduced and regular Grammian forms.