Instability theory of kink and anti-kink profiles for the sine–Gordon equation on Josephson tricrystal boundaries

Abstract

The aim of this work is to establish an instability study for stationary kink and antikink/kink profiles solutions for the sine–Gordon equation on a metric graph with a structure represented by a Y-junction so-called a Josephson tricrystal junction. By considering boundary conditions at the graph-vertex of δ′-interaction type, it is shown that kink profiles which are continuous at the vertex, as well as anti-kink/kink profiles possibly discontinuous at the vertex, are linearly (and nonlinearly) unstable. 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 H1(Y)×L2(Y) is also established. The theory developed in this investigation has prospects for the study of the (in)-stability of stationary wave solutions of other configurations for kink-solitons profiles.