Functional separation solutions of the Sinh-Gordon type equations

TL;DR

This paper classifies and constructs all functional separable solutions for sinh-Gordon type equations, providing a comprehensive understanding of their solution structure and introducing new solution families.

Contribution

It offers a complete classification of functional separable solutions for sinh-Gordon equations and constructs new solution families in a unified framework.

Findings

Identified all families with functional separation property.

Constructed new solutions for hyperbolic and elliptic sinh-Gordon equations.

Provided a unified approach for sine and sinh-Gordon equations.

Abstract

In this paper, we address the following question: Which hyperbolic or elliptic PDEs admit functional separable solutions. We shall focus on the study of a sinh-Gordon type equation. We construct solutions to this equation via the method of functional separation. We prove that these are the only families that have the property of functional separation and so we obtain a classification. To this end, we construct new families of solutions for the hyperbolic and elliptic versions of both sine and sinh Gordon equations in a unified way.