Stability of bound states of Hamiltonian PDEs in the degenerate cases

TL;DR

This paper establishes criteria for the stability and instability of bound states in Hamiltonian PDEs with degeneracy, applying the results to nonlinear Klein-Gordon and Schrödinger equations.

Contribution

It provides a new criterion for stability analysis of bound states in degenerate Hamiltonian PDEs, extending previous results to more complex cases.

Findings

Derived a stability criterion for degenerate Hamiltonian systems

Applied the criterion to nonlinear Klein-Gordon and Schrödinger equations

Identified conditions leading to stability or instability of bound states

Abstract

We consider a Hamiltonian systems which is invariant under a one-parameter unitary group. We give a criterion for the stability and instability of bound states for the degenerate case. We apply our theorem to the single power nonlinear Klein-Gordon equation and the double power nonlinear Schr\"odinger equation.