An a posteriori KAM theorem for whiskered tori in Hamiltonian partial differential equations with applications to some ill-posed equations

TL;DR

This paper develops an a posteriori KAM theorem for Hamiltonian PDEs with hyperbolic tori, applicable even to ill-posed equations, providing a method to verify true invariant tori from approximate solutions.

Contribution

It introduces a novel a posteriori KAM framework for Hamiltonian PDEs with hyperbolic directions, applicable to ill-posed equations, without relying on near-integrability or transformation theory.

Findings

Constructed small amplitude tori for scalar Boussinesq equation

Extended KAM theory to ill-posed Hamiltonian PDEs

Provided an iterative method for invariant tori without action-angle variables

Abstract

The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result has an \emph{a-posteriori} format, i.e., we show that if there is an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, then there is a true solution nearby. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). The method of proof is based on an iterative method to solve a functional equation for the parameterization of the torus satisfying the invariance equations and for parametrization of directions invariant under the linearizatation. The iterative method does not use transformation theory or action-angle variables. It does not assume that the system is close to integrable. We first develop an abstract theorem. Then, we show how this abstract result applies to some concrete examples, including the scalar Boussinesq equation and the Boussinesq system so that we construct {\sl small amplitude} tori for the equations, which are even in the spatial variable. Note that the equations we use as examples are ill-posed.