Well-posedness of a highly nonlinear shallow water equation on the circle

TL;DR

This paper introduces a new model for large amplitude shallow water waves, reviews existing studies, and proves local well-posedness of the Cauchy problem for periodic solutions in Sobolev spaces.

Contribution

It establishes the first local well-posedness result for the nonlinear shallow water equation on the circle with Sobolev initial data.

Findings

Proves local well-posedness for s > 3/2 in Sobolev space

Provides a comprehensive literature review of the model

Introduces a new model equation for large amplitude waves

Abstract

We present a comprehensive introduction and overview of a recently derived model equation for waves of large amplitude in the context of shallow water waves and provide a literature review of all the available studies on this equation. Furthermore, we establish a novel result concerning the local well-posedness of the corresponding Cauchy problem for space-periodic solutions with initial data from the Sobolev space $H^s$ on the circle for $s>3/2$.