Higher-order Hamiltonian Model for Unidirectional Water Waves

TL;DR

This paper derives second-order accurate Hamiltonian models for unidirectional water waves, providing longer-term solutions and establishing well-posedness, with some models exhibiting indefinite continuation due to Hamiltonian structure.

Contribution

It introduces higher-order Hamiltonian equations for water waves that improve long-term accuracy and analyzes their well-posedness and Hamiltonian properties.

Findings

Models are formally second-order correct and more accurate over longer times.

A local well-posedness theory is established for the class of equations.

Some models have a Hamiltonian structure allowing indefinite continuation.

Abstract

Formally second-order correct, mathematical descriptions of long-crested water waves propagating mainly in one direction are derived. These equations are analogous to the first-order approximations of KdV- or BBM-type. The advantage of these more complex equations is that their solutions corresponding to physically relevant initial perturbations of the rest state may be accurate on a much longer time scale. The initial-value problem for the class of equations that emerges from our derivation is then considered. A local well-posedness theory is straightforwardly established by way of a contraction mapping argument. A subclass of these equations possess a special Hamiltonian structure that implies the local theory can be continued indefinitely.