Lie symmetries and similarity solutions for the generalized Zakharov equations

TL;DR

This paper applies Lie symmetry analysis to the generalized Zakharov system, deriving similarity solutions and reducing the system to solvable ordinary differential equations, including the Ermakov-Pinney equation.

Contribution

It introduces a symmetry-based reduction method for the generalized Zakharov system and finds explicit travel-wave solutions.

Findings

The system admits five Lie point symmetries.

Travel-wave similarity solutions are explicitly constructed.

The system reduces to the Ermakov-Pinney equation.

Abstract

The theory of Lie point symmetries is applied to study the generalized Zakharov system with two unknown parameters. The system reduces into a three-dimensional real value functions system, where we find that admits five Lie point symmetries. From the resulting point, we focus on these which provide travel-wave similarity transformation. The reduced system can be integrated while we remain with a system of two second-order nonlinear ordinary differential equations. The parameters of the latter system are classified in order the equations to admit Lie point symmetries. Exact travel-wave solutions are found, while the generalized Zakharov system can be described by the one-dimensional Ermakov-Pinney equation.