Invariant Manifolds for Capillary Waves and a Class of Quasilinear PDEs

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.