Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with p-power nonlinearities in two dimensions

TL;DR

This paper explores nonlinear generalizations of the KP and Boussinesq equations with p-power nonlinearities in two dimensions, deriving their symmetries, conservation laws, and explicit line soliton solutions.

Contribution

It provides a Hamiltonian framework, symmetry analysis, conservation laws, and explicit soliton solutions for these generalized equations, extending understanding of nonlinear PDEs in two dimensions.

Findings

Hamiltonian formulations for all p≠0

Complete Lie symmetry classifications including special p values

Explicit line soliton solutions for p>0

Abstract

Nonlinear generalizations of integrable equations in one dimension, such as the KdV and Boussinesq equations with $p$-power nonlinearities, arise in many physical applications and are interesting in analysis due to critical behaviour. This paper studies analogous nonlinear $p$-power generalizations of the integrable KP equation and the Boussinesq equation in two dimensions. Several results are obtained. First, for all $p\neq 0$, a Hamiltonian formulation of both generalized equations is given. Second, all Lie symmetries are derived, including any that exist for special powers $p\neq0$. Third, Noether's theorem is applied to obtain the conservation laws arising from the Lie symmetries that are variational. Finally, explicit line soliton solutions are derived for all powers $p>0$, and some of their properties are discussed.