2025/02/23 by Jaime Angulo Pava, Pava, Jaime Angulo, Ramón G. Plaza +1
Mathematics · Physics and Astronomy · #35B35 #35J61 #35Q51 #47E05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2502.16726
openalex publication_date 2025/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.