Dynamics of the sine-Gordon equation on tadpole graphs

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 $\delta$-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.