The nonlinear Klein-Gordon equation on metric graphs: Modeling reflectionless transmission of the kink soliton

TL;DR

This paper models reflectionless transmission of kink solitons in the nonlinear Klein-Gordon equation on metric graphs, demonstrating conditions for reflectionless propagation and conservation laws.

Contribution

It introduces a framework for solving the nonlinear Klein-Gordon equation on metric star graphs with specific boundary conditions, highlighting reflectionless soliton transmission.

Findings

Reflectionless propagation of kink solitons demonstrated

Reflection coefficient plotted and analyzed

Extended analysis to various graph topologies

Abstract

In this paper we consider the nonlinear Klein-Gordon equation on the metric star graph with tree semi-infinite bonds. At the branched point we put two types of vertex boundary conditions: the weight continuity and the condition for derivatives of wave functions as the generalized Kirchhoff rule. We solve this equation satisfying vertex boundary conditions and the energy, momentum conservation laws. We also show reflectionsless propagations of the kink soliton solution, plot the reflection coefficient and extend to other topologies.