Analysis Seminar

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Stable and unstable manifolds for capillary gravity water waves and a class of nonlinear PDEs

Speaker: Chongchun Zeng, Georgia Tech

Location: Warren Weaver Hall 1302

Date: Thursday, February 19, 2026, 11 a.m.

Synopsis:

 

Invariant manifold theory is a fundamental tool in the study of local dynamics near invariant structures in smooth evolution systems. It provides a mechanism for promoting linear invariant structures to nonlinear ones. The theory is well developed for diffeomorphisms, ODEs, semilinear PDEs, and certain quasilinear parabolic equations. However, it becomes considerably more subtle for genuinely quasilinear or fully nonlinear PDEs, particularly in the absence of smoothing effects, where regularity issues play a central role. In this talk, we consider a class of nonlinear PDEs whose linearizations satisfy suitable energy estimates. We show that a linear exponential dichotomy implies the existence of local stable and unstable manifolds near equilibria. Our results apply, in particular, to a class of nonlinear Hamiltonian PDEs, including capillary–gravity water waves (for one or two fluids), quasilinear wave and Schrödinger equations, and KdV-type equations, for which we also discuss the relevant linear analysis. For such systems and under appropriate conditions, spectral instability implies the existence of stable and unstable manifolds. This yields nonlinear instability in rough Sobolev norms and, in some regimes, the existence of solutions that decay in higher Sobolev norms. This is joint work with Jalal Shatah.