Dynamics of the sine-Gordon equation on tadpole graphs

TL;DR

This work studies the sine-Gordon dynamics on a tadpole graph with δ-type vertex coupling, focusing on stationary single-lobe kink profiles composed of a periodic loop component and a decaying tail. Using extension theory for symmetric operators, Sturm–Liouville oscillation theory, and analytic perturbation theory, the authors establish local well-posedness in the natural energy space and derive a linear instability criterion for stationary states. They construct explicit positive single-lobe states on the loop via snoidal elliptic functions and glue them to a kink on the half-line, obtaining existence conditions in terms of the coupling Z and graph parameters; they show the linearized operator has Morse index 1 and trivial kernel, leading to spectral and nonlinear instability of these kinks (including a degenerate boundary case). The paper then proves that instability persists nonlinearly, providing the first stability result for stationary sine-Gordon states on a tadpole graph, with rigorous spectral, perturbative, and variational underpinnings that illuminate soliton transport on networks.

Abstract

This work studies the dynamics of solutions to the sine-Gordon equation posed on a tadpole graph $G$ and endowed with boundary conditions at the vertex of $δ$-type. The latter generalize conditions of Neumann-Kirchhoff type. The purpose of this analysis is to establish an instability result for a certain family of stationary solutions known as \emph{single-lobe kink state profiles}, which consist of a periodic, symmetric, concave stationary solution in the finite (periodic) lasso of the tadpole, coupled with a decaying kink at the infinite edge of the graph. It is proved that such stationary profile solutions are linearly (and nonlinearly) unstable under the flow of the sine-Gordon model on the graph. The extension theory of symmetric operators, Sturm-Liouville oscillation results and analytic perturbation theory of operators are fundamental ingredients in the stability analysis. The local well-posedness of the sine-Gordon model in an appropriate energy space is also established. The theory developed in this investigation constitutes the first stability result of stationary solutions to the sine-Gordon equation on a tadpole graph.

Figures (14)

• Figure 1: Tadpole graph $\mathcal{G}$ with vertex at $\nu = L$.
• Figure 2: Graph of the kink-soliton profile (in solid blue line) for $x \in [L, \infty)$ for different values of $a$. Here $L =2$ and $c_2 = 1$ (color online).
• Figure 3: An example of the graph a single-lobe kink state profile for the sine-Gordon model according to Definition \ref{['singlelobe']}.
• Figure 4: Single-lobe solution $\phi_{1, k}$ in \ref{['formula1a']} with $L=\pi$, $c_1=1$ and $k^2 = 0.98$.
• Figure 5: Phase-plane of a positive single-lobe kink state for \ref{['trav21']}. Here we take $c_1 = c_2 = 1$ and the translation variable $\xi = x - ct$ with $c = 0$ (subluminal waves). The picture shows the phase plane of the solutions $\phi = \phi(\xi)$ for \ref{['trav21']}. The separatrices are represented in red solid lines and the librations are the periodic waves inside the separatrices (in light gray). The single-lobe kink profile (depicted with blue thick arrow lines) is composed of a periodic libration, symmetric in $[-L,L]$ and connecting $\phi = \pi$ with itself, glued with a decaying kink connecting $\phi = \pi$ to $\phi = 0$ along a separatrix (color online).

Theorems & Definitions (70)

• Definition 1.1
• Theorem 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Proposition 2.4
• proof
• Theorem 2.5
• proof