On small energy stabilization in the NLKG with a trapping potential

TL;DR

This paper demonstrates that small standing waves act as attractors for small solutions in a complex-valued nonlinear Klein-Gordon equation with a trapping potential, extending previous results from related equations and real-valued solutions.

Contribution

It extends the understanding of energy stabilization in NLKG to complex solutions and broadens the scope of prior results from Schrödinger and Dirac equations.

Findings

Small standing waves are attractors for small solutions.

Extension of stabilization results to complex-valued NLKG solutions.

Builds on and generalizes previous real-valued solution results.

Abstract

We consider a nonlinear Klein Gordon equation (NLKG) with short range potential with eigenvalues and show that in the contest of complex valued solutions the small standing waves are attractors for small solutions of the NLKG. This extends the results already known for the nonlinear Schr\"odinger equation and for the nonlinear Dirac equation. In addition, this extends a result of Bambusi and Cuccagna (which in turn was an extension of a result by Soffer and Weinstein) which considered only real valued solutions of the NLKG.