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Invariant Manifolds for Capillary Waves and a Class of Quasilinear PDEs

Jalal Shatah, Chongchun Zeng

TL;DR

This work develops an energy-estimate-based framework to construct local stable and unstable manifolds for equilibria of broad classes of quasilinear and fully nonlinear PDEs, with a key focus on irrotational water waves with surface tension. By embedding the nonlinear dynamics into an invariant splitting and enforcing dissipativity with energy forms, the authors establish existence, uniqueness, and smoothness of local invariant manifolds via a Lyapunov–Perron approach, even in the presence of loss of regularity. The methods apply to Hamiltonian PDEs such as nonlinear Schrödinger and nonlinear waves, as well as to gradient-type and other PDEs, and yield concrete manifolds for water-wave models including the Dirichlet–Neumann operator. In the water-wave context, the framework produces finite-dimensional, smoothly varying unstable manifolds around spectrally unstable equilibria, and clarifies how nonlocal operators and free interfaces fit into a unified invariant-manifold theory with consequences for nonlinear instability phenomena. The results advance the local dynamical picture of capillary gravity waves and related PDEs, offering a robust toolkit for analyzing homoclinic/heteroclinic structures, traveling waves, and interface dynamics.

Abstract

This paper studies the local stable and unstable manifolds of equilibria for quasilinear and fully nonlinear PDEs. These manifolds are fundamental objects in the analysis of local dynamics. While their existence is well understood for ODEs, semilinear PDEs, and certain parabolic-type quasilinear PDEs, invariant manifold theorems are often unavailable for quasilinear PDEs whose nonlinearities involve a loss of regularity and whose linear parts do not provide sufficient smoothing. Our main results establish the existence, uniqueness, and smoothness of local stable and unstable manifolds for nonlinear PDEs that satisfy suitable energy estimates. With the main focus on irrotational water waves with surface tension, this framework applies to a broad class of PDEs, including nonlinear Schrödinger equations, nonlinear wave equations, and the MMT model, as well as to certain gradient-type PDEs.

Invariant Manifolds for Capillary Waves and a Class of Quasilinear PDEs

TL;DR

This work develops an energy-estimate-based framework to construct local stable and unstable manifolds for equilibria of broad classes of quasilinear and fully nonlinear PDEs, with a key focus on irrotational water waves with surface tension. By embedding the nonlinear dynamics into an invariant splitting and enforcing dissipativity with energy forms, the authors establish existence, uniqueness, and smoothness of local invariant manifolds via a Lyapunov–Perron approach, even in the presence of loss of regularity. The methods apply to Hamiltonian PDEs such as nonlinear Schrödinger and nonlinear waves, as well as to gradient-type and other PDEs, and yield concrete manifolds for water-wave models including the Dirichlet–Neumann operator. In the water-wave context, the framework produces finite-dimensional, smoothly varying unstable manifolds around spectrally unstable equilibria, and clarifies how nonlocal operators and free interfaces fit into a unified invariant-manifold theory with consequences for nonlinear instability phenomena. The results advance the local dynamical picture of capillary gravity waves and related PDEs, offering a robust toolkit for analyzing homoclinic/heteroclinic structures, traveling waves, and interface dynamics.

Abstract

This paper studies the local stable and unstable manifolds of equilibria for quasilinear and fully nonlinear PDEs. These manifolds are fundamental objects in the analysis of local dynamics. While their existence is well understood for ODEs, semilinear PDEs, and certain parabolic-type quasilinear PDEs, invariant manifold theorems are often unavailable for quasilinear PDEs whose nonlinearities involve a loss of regularity and whose linear parts do not provide sufficient smoothing. Our main results establish the existence, uniqueness, and smoothness of local stable and unstable manifolds for nonlinear PDEs that satisfy suitable energy estimates. With the main focus on irrotational water waves with surface tension, this framework applies to a broad class of PDEs, including nonlinear Schrödinger equations, nonlinear wave equations, and the MMT model, as well as to certain gradient-type PDEs.
Paper Structure (21 sections, 34 theorems, 448 equations)

This paper contains 21 sections, 34 theorems, 448 equations.

Table of Contents

  1. Introduction
  2. Local stable and unstable manifolds
  3. Stable and unstable manifolds of a model quasilinear PDE
  4. Stable and unstable manifolds of a class of nonlinear PDEs
  5. Smoothness of stable and unstable manifolds
  6. Some concrete nonlinear PDEs
  7. Hamiltonian PDEs where $\Omega^*=-\Omega$
  8. Other PDE systems
  9. Irrotational waver waves with surface tension
  10. Set-up and main results
  11. Preliminary analysis
  12. Proof of Theorem \ref{['T:CWW-LInMa-main']}
  13. Fluid interface problem
  14. Hamiltonian formulation
  15. Function spaces and preliminary analysis
  16. ...and 6 more sections

Key Result

Theorem 2.1

Assume (C-1)--(C.4). For any $\lambda \in \Sigma_+$, $M_0 > \frac{2 C_0^{2(k+1)}}{1-L_1(\lambda)}$, and $l \in (L_1(\lambda), 1)$, there exists $\varepsilon\in (0, \frac{R_0}{M_0})$ determined by $k, R_0, \lambda_\pm, C_f$, and $C_0$ (see E:temp-18 and Remark R:size-1) such that there exists $h_+ :X

Theorems & Definitions (87)

  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.2
  • Remark 2.3
  • proof
  • Corollary 2.2.1
  • proof : Proof of Theorem \ref{['T:QLPDE-UM']}.
  • Lemma 2.3
  • proof
  • ...and 77 more