A resonant Lyapunov centre theorem with an application to doubly periodic travelling hydroelastic waves

TL;DR

This paper develops a Lyapunov centre theorem for complex Hamiltonian systems with resonances, enabling the construction of doubly periodic hydroelastic waves on water surfaces beneath ice sheets.

Contribution

It introduces a Lyapunov centre theorem applicable to finite and infinite-dimensional Hamiltonian systems with resonances, extending the mathematical framework for wave solutions.

Findings

Established a Lyapunov centre theorem for resonant Hamiltonian systems.

Applied the theorem to construct doubly periodic hydroelastic waves.

Demonstrated the approach using spatial dynamics for water waves beneath ice sheets.

Abstract

We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate $1:1$ or $1:-1$ semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a "trivial" eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application we show how Kirchg\"{a}ssner's spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of water (of finite or infinite depth) beneath a thin ice sheet (hydroelastic waves). The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable and the infinite-dimensional phase space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. Applying our Lyapunov centre theorem at a point in parameter space associated with a $1:1$ or $1:-1$ semisimple resonance yields a periodic solution of the spatial Hamiltonian system corresponding to a doubly periodic hydroelastic wave.