Stability Properties of Periodic Standing Waves for the Klein-Gordon-Schrodinger System

TL;DR

This paper investigates the existence and stability properties of periodic standing wave solutions in the Klein-Gordon-Schrodinger system, employing elliptic functions and classical stability theories to establish conditions for stability and instability.

Contribution

It introduces new periodic wave solutions using Jacobian elliptic functions and applies classical and generalized stability criteria to analyze their stability in the system.

Findings

Existence of periodic waves depending on Jacobian elliptic functions.

Application of Grillakis, Shatah, and Strauss theory for stability analysis.

Use of a general instability criterion from linear instability results.

Abstract

We study the existence and orbital stability/instability of periodic standing wave solutions for the Klein-Gordon-Schr\"odinger system with Yukawa and cubic interactions. We prove the existence of periodic waves depending on the Jacobian elliptic functions. For one hand, the approach used to obtain the stability results is the classical Grillakis, Shatah and Strauss theory in the periodic context. On the other hand, to show the instability results we employ a general criterium introduced by Grillakis, which get orbital instability from linear instability.