Asymptotics in shallow water waves

TL;DR

This paper investigates the initial value problem for shallow water equations on the real line, focusing on solutions with specific asymptotic behaviors at infinity and demonstrating their invariance under the solution map.

Contribution

It constructs solutions with prescribed asymptotics at infinity and proves their invariance, advancing understanding of asymptotic behaviors in shallow water wave models.

Findings

Solutions with prescribed asymptotics are constructed.

Such solutions are invariant under the solution map.

The paper provides a framework for analyzing asymptotic behaviors in shallow water equations.

Abstract

In this paper we consider the initial value problem for a family of shallow water equations on the line $\R$ with various asymptotic conditions at infinity. In particular we construct solutions with prescribed asymptotic expansion as $x\to\pm\infty$ and prove their invariance with respect to the solution map.