Asymptotic stability of ground states in some Hamiltonian PDEs with symmetry

TL;DR

This paper proves that under certain conditions, ground states (solitons) in specific Hamiltonian PDEs are not only orbitally stable but also asymptotically stable, extending the understanding of their long-term behavior.

Contribution

It establishes asymptotic stability of ground states in Hamiltonian PDEs with symmetry, using advanced Hamiltonian system tools and accommodating multiple eigenvalues.

Findings

Orbitally stable solitons are asymptotically stable under specified conditions.

The theory applies to the 3D translationally invariant nonlinear Schrödinger equation.

The approach extends Hamiltonian system techniques to symmetric PDEs with unbounded generators.

Abstract

We consider a ground state (soliton) of a Hamiltonian PDE. We prove that if the soliton is orbitally stable, then it is also asymptotically stable. The main assumptions are transversal nondegeneracy of the manifold of the ground states, linear dispersion (in the form of Strichartz estimates) and nonlinear Fermi Golden Rule. We allow the linearization of the equation at the soliton to have an arbitrary number of eigenvalues. The theory is tailor made for the application to the translational invariant NLS in space dimension 3. The proof is based on the extension of some tools of the theory of Hamiltonian systems (reduction theory, Darboux theorem, normal form) to the case of systems invariant under a symmetry group with unbounded generators.