Energy transfer and radiation in Hamiltonian nonlinear Klein-Gordon equations: general case

TL;DR

This paper analyzes energy transfer and decay in nonlinear Klein-Gordon equations with multiple eigenvalues, extending previous results to the most general case using advanced mathematical techniques.

Contribution

It provides a comprehensive solution for energy transfer in Klein-Gordon equations with multiple, possibly degenerate eigenvalues, generalizing prior work to the full spectrum case.

Findings

Established a pseudo-one-dimensional cancellation structure.

Developed a refined Birkhoff normal form transformation.

Proved an enhanced damping effect and generalized Fermi's Golden Rule.

Abstract

In this paper, we consider Klein-Gordon equations with cubic nonlinearity in three spatial dimensions, which are Hamiltonian perturbations of the linear one with potential. It is assumed that the corresponding Klein-Gordon operator $B = \sqrt{-\Delta + V(x) + m^2} $ admits an arbitrary number of possibly degenerate eigenvalues in $(0, m)$, and hence the unperturbed linear equation has multiple time-periodic solutions known as bound states. In \cite{SW1999}, Soffer and Weinstein discovered a mechanism called Fermi's Golden Rule for this nonlinear system in the case of one simple but relatively large eigenvalue $\Omega\in (\frac{m}{3}, m)$, by which energy is transferred from discrete to continuum modes and the solution still decays in time. In particular, the exact energy transfer rate is given. In \cite{LLY22}, we solved the general one simple eigenvalue case. In this paper, we solve this problem in full generality: multiple and simple or degenerate eigenvalues in $(0, m)$. The proof is based on a kind of pseudo-one-dimensional cancellation structure in each eigenspace, a renormalized damping mechanism, and an enhanced damping effect. It also relies on a refined Birkhoff normal form transformation and an accurate generalized Fermi's Golden Rule over those of Bambusi--Cuccagna \cite{BC}.