Nonlinear modulational instability of two-dimensional deep hydroelastic Stokes waves

TL;DR

This paper demonstrates that two-dimensional deep hydroelastic Stokes waves are nonlinearly unstable to long-wave perturbations, using a cubic NLS approximation to analyze the instability mechanism.

Contribution

It provides a rigorous justification of the cubic NLS approximation for 2D deep hydroelastic waves and links this to the nonlinear modulational instability of Stokes waves.

Findings

Stokes waves are nonlinearly unstable under long-wave perturbations.

A focusing cubic NLS approximation accurately describes the wave system.

The instability mechanism of the cubic NLS explains the instability of Stokes waves.

Abstract

In this paper, we study the nonlinear modulational instability of two-dimensional hydroelastic Stokes waves in infinite depth. We first justify a focusing cubic nonlinear Schr\"odinger (NLS) approximation result for 2D deep hydroelastic wave system in the spirit of Ifrim-Tataru [22]. Then we exploit the instability mechanism of the cubic NLS to prove that the Stokes waves are nonlinearly unstable under long-wave perturbations.