Cnoidal Waves for the cubic nonlinear Klein-Gordon and Schr\"odinger Equations

TL;DR

This paper investigates the stability of cnoidal periodic waves in cubic nonlinear Klein-Gordon and Schr"odinger equations, showing instability in Klein-Gordon and stability in Schr"odinger through spectral analysis and Lyapunov functionals.

Contribution

It establishes the orbital stability of cnoidal waves for the Schr"odinger equation and their instability for the Klein-Gordon equation within the energy space.

Findings

Klein-Gordon cnoidal waves are orbitally unstable.

Schr"odinger cnoidal waves are orbitally stable.

Spectral analysis uses Floquet theory and Morse Index Theorem.

Abstract

In this paper, we establish orbital stability results for \textit{cnoidal} periodic waves of the cubic nonlinear Klein-Gordon and Schr\"odinger equations in the energy space restricted to zero mean periodic functions. More precisely, for one hand, we prove that the cnoidal waves of the cubic Klein-Gordon equation are orbitally unstable as a direct application of the theory developed by Grillakis, Shatah, and Strauss. On the other hand, we show that the cnoidal waves for the Schr\"odinger equation are orbitally stable by constructing a suitable Lyapunov functional restricted to the associated zero mean energy space. The spectral analysis of the corresponding linearized operators, restricted to the periodic Sobolev space consisting of zero mean periodic functions, is performed using the Floquet theory and a Morse Index Theorem.