Invariant Manifolds for Capillary Waves and a Class of Quasilinear PDEs

TL;DR

This paper establishes the existence, uniqueness, and smoothness of local stable and unstable manifolds for a broad class of quasilinear and fully nonlinear PDEs, including water waves and Schrödinger equations, under suitable energy estimates.

Contribution

It provides new invariant manifold theorems for quasilinear PDEs with nonlinearities involving loss of regularity, extending the understanding beyond classical semilinear and parabolic PDEs.

Findings

Proves existence of invariant manifolds for nonlinear PDEs with energy estimates.

Applies the framework to water waves, Schrödinger, and wave equations.

Establishes smoothness and uniqueness of these manifolds.

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 Schr\"odinger equations, nonlinear wave equations, and the MMT model, as well as to certain gradient-type PDEs.