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

¹ TMC Institute Tashkent - 41-b Mukimi street, Tashkent, Uzbekistan
² Tashkent University of Information Technologies - 108 Amir Temus street, 100200, Tashkent, Uzbekistan
³ Tashkent State University of Economics - 49 Islam Karimov street, 100066, Tashkent, Uzbekistan
⁴ Tashkent Perfect University - Zarqaynar street, 6/1, Tashkent, Uzbekistan
⁵ Kimyo International University in Tashkent - Shota Rustaveli street 156, Tashkent 100121, Uzbekistan

Received: 11 November 2025
Accepted: 6 March 2026

Abstract

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.