Hamiltonian studies on counter-propagating water waves

TL;DR

This paper employs a Hamiltonian normal form approach to analyze the dynamics of small amplitude long water waves, revealing decoupled right and left propagating waves at order μ^5 and their coupling at order μ^7.

Contribution

It introduces a Hamiltonian normal form analysis for water waves, showing decoupling at order μ^5 and coupling at order μ^7, advancing understanding of wave interactions.

Findings

Decoupled equations for right and left waves at order μ^5

Conjugation to KdV hierarchy equations

Coupling of counter-propagating waves at order μ^7

Abstract

We use a Hamiltonian normal form approach to study the dynamics of the water wave problem in the small amplitude long wave regime (KdV regime). If $\mu$ is the small parameter corresponding to the inverse of the wave length, we show that the normal form at order $\mu^5$ consists of two decoupled equation, one describing right going waves and the other describing left going waves. Performing a further non Hamiltonian transformation we conjugate each of these equations to a linear combination of the first three equations in the KdV hierarchy. At order $\mu^7$ we find nontrivial terms coupling the two counter-propagating waves.