New Exact Solutions of a Generalized Shallow Water Wave Equation

TL;DR

This paper introduces an extended elliptic function method to find new exact solutions to a generalized shallow water wave equation, including rational, periodic, and solitary waves, with analysis of their singularities.

Contribution

It presents a novel extended elliptic function approach for classifying and deriving new exact solutions of the generalized shallow water wave equation.

Findings

Derived new elliptic solutions including rational, periodic, and solitary waves.

Identified regions where solutions are free from singularities.

Compared the new solutions with canonical methods.

Abstract

In this work an extended elliptic function method is proposed and applied to the generalized shallow water wave equation. We systematically investigate to classify new exact travelling wave solutions expressible in terms of quasi-periodic elliptic integral function and doubly-periodic Jacobian elliptic functions. The derived new solutions include rational, periodic, singular and solitary wave solutions. An interesting comparison with the canonical procedure is provided. In some cases the obtained elliptic solution has singularity at certain region in the whole space. For such solutions we have computed the effective region where the obtained solution is free from such a singularity.