Collision of two breathers at surface of deep water

TL;DR

This paper simplifies the water wave Hamiltonian using a canonical transformation, leading to a more manageable equation that enables the discovery of breather solutions and supports the integrability hypothesis of 2D free surface hydrodynamics.

Contribution

The authors introduce a novel canonical transformation that simplifies the Zakharov equation for water waves, facilitating analysis and simulation of breather interactions.

Findings

Found localized breather solutions in water waves.

Numerical simulations support the integrability of 2D free surface hydrodynamics.

Simplified equations are suitable for both analytical and numerical studies.

Abstract

We applied canonical transformation to water wave equation not only to remove cubic nonlinear terms but to simplify drastically fourth order terms in Hamiltonian. This transformation explicitly uses the fact of vanishing exact four waves interaction for water gravity waves for 2D potential fluid. After the transformation well-known but cumbersome Zakharov equation is drastically simplified and can be written in X-space in compact way. This new equation is very suitable as for analytic study as for numerical simulation. Localized in space breather-type solution was found. Numerical simulation of collision of two such breathers strongly supports hypothesis of integrability of 2-D free surface hydrodynamics.