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Nonlinear modulational instability of two-dimensional deep hydroelastic Stokes waves

Lizhe Wan, Jiaqi Yang

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 Schrödinger (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.

Nonlinear modulational instability of two-dimensional deep hydroelastic 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 Schrödinger (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.

Paper Structure

This paper contains 13 sections, 12 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Existence of the pure hydroelastic Stokes waves
  4. Derivation of the Babenko equation
  5. Local bifurcation of the Babenko equation
  6. The NLS approximation of the hydroelastic waves
  7. The NLS truncation
  8. Normal form analysis and NLS approximation
  9. Replacing approximate solutions by exact solutions
  10. The proof of nonlinear modulational instability of the Stokes waves
  11. Local bifurcation theorem
  12. Modified energy estimate for hydroelastic waves
  13. Instability of cubic NLS equation

Key Result

Theorem 1.1

Let $U_0 \in H^3$, and $U$ be the corresponding solution to the cubic NLS equation and let $Y^\epsilon$ be defined in YURelation, and $T>0$. Then there exists a constant $\epsilon_0(\|U\|_{H^3}, T)$ so that for each $0<\epsilon< \epsilon_0$, there exists a pair of solution $(W, Q)$ to the system HF14 with $\sigma = 1$ for $t$ in the time interval $[0, T\epsilon^{-2}]$ with the fol

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Proposition 3.1: ifrim2019nls
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • Theorem A.1: Local bifurcation, MR288640
  • ...and 4 more