Instability of Static Solutions of the sine-Gordon Equation on a Y-Junction Graph with δ-Interaction

The aim of this work is to establish a linear instability result of static, kink and kink/anti-kink soliton profile solutions for the sine-Gordon equation on a metric graph with a structure represented by a Y -junction. The model considers boundary conditions at the graph-vertex of δ -interaction type. It is shown that kink and kink/anti-kink soliton type static profiles are linearly (and nonlinearly) unstable. For that purpose, a linear instability criterion that provides the sufficient conditions on the linearized operator around the wave to have a pair of real positive/negative eigenvalues, is established. As a result, the linear stability analysis depends upon the spectral study of this linear operator and of its Morse index. The extension theory of symmetric operators, Sturm–Liouville oscillation results and analytic perturbation theory of operators are fundamental ingredients in the stability analysis. A comprehensive study of the local well-posedness of the sine-Gordon model in E ( Y ) × L 2 ( Y ) where E ( Y ) ⊂ H 1 ( Y ) is an appropriate energy space, is also established. The theory developed in this investigation has prospects for the study of the instability of static wave solutions of other nonlinear evolution equations on metric graphs.