Variational existence theory for hydroelastic solitary waves

TL;DR

This paper develops a variational existence theory for hydroelastic solitary waves at the interface of a thin ice sheet and an ideal fluid, demonstrating convergence to nonlinear Schrödinger solutions as a key parameter diminishes.

Contribution

It introduces a novel variational framework for proving the existence of hydroelastic solitary waves and links these solutions to nonlinear Schrödinger equation limits.

Findings

Existence of solitary waves established via minimization of energy under impulse constraint.

Detected waves converge to nonlinear Schrödinger equation solutions as a parameter approaches zero.

Theoretical results apply to models involving hyperelastic ice sheets and ideal fluids.

Abstract

This paper presents an existence theory for solitary waves at the interface between a thin ice sheet (modelled using the Cosserat theory of hyperelastic shells) and an ideal fluid (of finite depth and in irrotational motion) for sufficiently large values of a dimensionless parameter $\gamma$. We establish the existence of a minimiser of the wave energy ${\mathcal E}$ subject to the constraint ${\mathcal I}=2\mu$, where ${\mathcal I}$ is the horizontal impulse and $0< \mu \ll 1$, and show that the solitary waves detected by our variational method converge (after an appropriate rescaling) to solutions of he nonlinear Schr\"{o}dinger equation with cubic focussing nonlinearity as $\mu \downarrow 0$.