Symmetry Analysis for a Generalized Kadomtsev-Petviashvili Equation

TL;DR

This paper performs a symmetry analysis of a generalized Kadomtsev-Petviashvili equation, revealing an infinite-dimensional Lie group of symmetries and deriving solutions and subalgebras, including physically meaningful transformations.

Contribution

It identifies the symmetry structure of the GKPE with arbitrary functions, constructs relevant subalgebras, and finds explicit solutions involving arbitrary functions of time.

Findings

Infinite-dimensional Lie symmetry group with arbitrary functions.

Derived low-dimensional and physically meaningful subalgebras.

Obtained explicit solutions involving arbitrary functions of time.

Abstract

A generalized Kadomtsev-Petviashvili equation (GKPE) $(u_t+u u_x + \beta(t)u +\gamma(t)u_{xxx})_x+\sigma(t)u_{yy}\ = \ 0$ is shown to admit an infinite-dimensional Lie group of symmetries when $\bt(t), \ga(t)$ and $\si(t)$ are arbitrary. The Lie algebra of this symmetry group contains two arbitrary functions $f(t)$ and $g(t)$. Further, low-dimensional subalgebras and physically meaningful five dimensional Lie algebra containing translation and Galilei transformation are derived. A solution of GKPE involving two arbitrary functions of time $t$, in addition to $f(t)$ and $g(t)$, is obtained using an one-dimensional subalgebra.