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

TL;DR

The paper addresses NLKG dynamics on branched networks and aims to achieve reflectionless kink transmission at graph vertices. It derives vertex boundary conditions from energy conservation, yielding weighted continuity and weighted Kirchhoff laws, and identifies a central sum rule $1/\beta_1 = \sum_{j>1} 1/\beta_j$ (extensible to $N$ bonds) that guarantees reflectionless propagation. The authors construct exact kink soliton solutions on a star graph and validate the theory with numerical experiments, showing energy and momentum conservation under the rule and contrasting reflection when the rule is violated. They further generalize the approach to tree and loop topologies, presenting corresponding sum rules and demonstrating reflectionless transmission, thereby offering a framework for controlled wave propagation on branched networks.

Abstract

In this paper, we investigate the nonlinear Klein-Gordon equation on a metric star graph with three semi-infinite bonds. At the branching point, we impose a weighted continuity condition and a generalized weighted Kirchhoff condition for the derivatives of the wave function. By employing both analytical methods and numerical techniques, we construct exact and numerical soliton solutions that satisfy the vertex conditions and conserve energy and momentum. The results of analytic calculations are confirmed through numerical experiments, which demonstrate reflectionless propagation of kink-antikink soliton solutions. We compute and analyze the reflection coefficient, study the impact of various nonlinearity parameters, and further extend the formulation to other graph topologies, such as tree and loop graphs.

Figures (7)

• Figure 1: The metric star graph
• Figure 2: Propagation of the kink soliton, the center of which is initially located at $l=-5$ in the first bond with velocity $v=0.9$ for the three time moments a) t=0; b) t=5; c) t=10. For the chosen nonlinearity coefficients, $\beta_1=1.2$, $\beta_2=2$, $\beta_3=3$ (the sum rule is satisfied), the kink propagates without reflection through the vertex.
• Figure 3: (Color online) The energies on the each bonds and the total energy
• Figure 4: Propagation of the kink with the same initial conditions as in Fig. \ref{['pic2']}, but for the nonlinearity coefficients $\beta_1=0.5$, $\beta_2=2$, $\beta_3=3$ (the sum rule is not satisfied). The kink propagation is not reflectionless anymore.
• Figure 5: (Color online) Time dependence of the reflection coefficient for the cases when the sum rule is satisfied (blue line) and broken (red line).