Standing wave solutions of $abcd$-systems for water waves

TL;DR

This paper investigates the existence of standing wave solutions in $abcd$-systems for water waves, analyzing the applicability of the Lyapunov-Schmidt method and proving bifurcation results for specific systems.

Contribution

It extends previous work by assessing the feasibility of the Lyapunov-Schmidt method for $abcd$-systems and establishes the existence of bifurcating standing waves for the Bona-Smith system.

Findings

Classified $abcd$-systems into feasible, infeasible, and uncertain categories.

Proved existence of nontrivial bifurcating standing waves for the Bona-Smith system.

Analyzed the applicability of the Lyapunov-Schmidt method to these systems.

Abstract

We continue the study for standing wave solutions of $abcd$-systems which was started by Chen and Iooss \cite{chen2005standing} for the BBM system via the Lyapunov-Schmidt method. In this paper, we will first discuss the feasibility of the Lyapunov-Schmidt method for bifurcating standing wave solutions of $abcd$-systems. These systems will be characterized into three categories: feasible, infeasible and uncertain feasible ones. In particular, we prove the existence of nontrivial bifurcating standing waves for the Bona-Smith system.