Cnoidal waves for the quintic Klein-Gordon and Schr\"odinger equations: Existence and orbital instability

TL;DR

This paper proves the existence and orbital instability of cnoidal periodic waves in the quintic Klein-Gordon and nonlinear Schrödinger equations, using spectral analysis and abstract instability criteria.

Contribution

It introduces new results on the existence and instability of cnoidal waves for these equations, employing Floquet theory and abstract instability theorems.

Findings

Existence of cnoidal periodic waves established.

Orbital instability of these waves demonstrated.

Spectral analysis via Floquet theory conducted.

Abstract

In the present paper, we establish the existence and orbital instability results of cnoidal periodic waves for the quintic Klein-Gordon and nonlinear Schr\"odinger equations. The spectral analysis for the corresponding linearized operator is established by using the Floquet theory. The orbital instability is determined by applying an abstract result due to Shatah and Strauss.