Kink-antikink soliton solutions of the nonlinear Klein-Gordon equation on branched structures

This paper studies soliton propagation governed by the nonlinear Klein-Gordon equation on a simple network consisting of three connected branches. We focus on how waves behave at the junction, where physically motivated continuity and conservation conditions are imposed. Using a combination of analytical and numerical approaches, we construct soliton solutions that pass through the junction without reflection and conserve key physical quantities. Numerical simulations confirm the analytical predictions and illustrate how soliton behavior depends on system parameters. These results improve the understanding of reflectionless soliton propagation in branched structures and can be extended to more complex graph topologies.