Breather solutions for a semilinear Klein-Gordon equation on a periodic metric graph

TL;DR

This paper proves the existence of infinitely many localized, time-periodic solutions (breathers) for a nonlinear Klein-Gordon equation on a periodic graph, using variational methods and improved embedding properties.

Contribution

It introduces a variational approach that overcomes previous restrictions on the nonlinearity exponent, ensuring higher regularity and broader applicability for breather solutions on metric graphs.

Findings

Existence of infinitely many breather solutions on a periodic graph.

Enhanced embedding properties allowing for higher regularity.

Solutions are weak solutions of the initial value problem.

Abstract

We consider the nonlinear Klein-Gordon equation $\partial_t^2u(x,t)-\partial_x^2u(x,t)+\alpha u(x,t)=\pm|u(x,t)|^{p-1}u(x,t)$ on a periodic metric graph (necklace graph) for $p>1$ with Kirchhoff conditions at the vertices. Under suitable assumptions on the frequency we prove the existence and regularity of infinitely many spatially localized time-periodic solutions (breathers) by variational methods. We compare our results with previous results obtained via spatial dynamics and center manifold techniques. Moreover, we deduce regularity properties of the solutions and show that they are weak solutions of the corresponding initial value problem. Our approach relies on the existence of critical points for indefinite functionals, the concentration compactness principle, and the proper set-up of a functional analytic framework. Compared to earlier work for breathers using variational techniques, a major improvement of embedding properties has been achieved. This allows in particular to avoid all restrictions on the exponent $p>1$ and to achieve higher regularity.