A note on the three dimensional sine--Gordon equation

TL;DR

This paper explores solutions to the three-dimensional sine-Gordon equation using an elementary ansatz, revealing non-uniqueness in initial value problem solutions and contributing to understanding higher-dimensional integrable models.

Contribution

Introduces a simple ansatz for solutions to the 3D sine-Gordon equation and demonstrates non-uniqueness of solutions for the initial value problem.

Findings

A class of solutions is found using elementary methods.

The equations are shown not to be evolution equations with unique solutions.

Solution non-uniqueness indicates complex solution structure.

Abstract

Using a simple ansatz for the solutions of the three dimensional generalization of the sine--Gordon and Toda model introduced by Konopelchenko and Rogers, a class of solutions is found by elementary methods. It is also shown that these equations are not evolution equations in the sense that solution to the initial value problem is not unique.