Representations and classification of traveling wave solutions to Sinh-G{\"o}rdon equation

TL;DR

This paper classifies all single traveling wave atom solutions to the Sinh-Gordon equation, discusses their qualitative properties, and demonstrates how these solutions encompass previous results and lead to new identities in elliptic functions.

Contribution

It provides a complete classification of atom solutions to the Sinh-Gordon equation and clarifies their properties, extending and unifying earlier solutions.

Findings

All solutions from the paper are included in the classification.

Some qualitative properties from dynamic system methods are not valid.

New identities on Jacobian elliptic functions are derived.

Abstract

Two concepts named atom solution and combinatory solution are defined. The classification of all single traveling wave atom solutions to Sinh-G{\"o}rdon equation is obtained, and qualitative properties of solutions are discussed. In particular, we point out that some qualitative properties derived intuitively from dynamic system method aren't true. In final, we prove that our solutions to Sinh-G{\"o}rdon equation include all solutions obtained in the paper[Fu Z T et al, Commu. in Theor. Phys.(Beijing) 2006 45 55]. Through an example, we show how to give some new identities on Jacobian elliptic functions.