Nonlinearly dispersive KP equations with new compacton solutions

TL;DR

This paper classifies and constructs various explicit compacton solutions for a generalized KP equation with nonlinear dispersion, expanding understanding of wave profiles and their properties in higher dimensions.

Contribution

It provides a complete classification of compacton solutions for a generalized KP equation with nonlinear dispersion, including explicit examples and their properties.

Findings

Derived explicit compacton solutions with various profiles

Identified conditions for compacton existence based on nonlinearity powers

Discussed kinematic properties and conservation laws of solutions

Abstract

A complete classification of compacton solutions is carried out for a generalization of the Kadomtsev-Petviashvili (KP) equation involving nonlinear dispersion in two and higher spatial dimensions. In particular, precise conditions are given on the nonlinearity powers in this equation under which a travelling wave can be cut off to obtain a compacton. Numerous explicit examples having various profiles are derived, including a quadratic function, powers of a cosine, and powers of Jacobi $\cn$ functions, all of which are symmetric. The cosine and $\cn$ symmetric compactons have an anti-symmetric counterpart. In comparison, explicit solitary waves of the generalized KP equation are found to have profiles given by a power of a sech and a reciprocal quadratic function. Kinematic properties of all of the different types of compactons and solitary waves are discussed, along with conservation laws of the generalized KP equation.