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The aim of this work is to establish a linear instability result ofstationary, kink and kink/anti-kink soliton profile solutions for thesine-Gordon equation on a metric graph with a structure represented by a$\mathcal Y$-junction. The model considers boundary conditions at thegraph-vertex of $\delta$-interaction type, or in other words, continuity of thewave functions at the vertex plus a law of Kirchhoff-type for the flux. It isshown that kink and kink/anti-kink soliton type stationary profiles arelinearly (and nonlinearly) unstable. For that purpose, a linear instabilitycriterion that provides the sufficient conditions on the linearized operatoraround the wave to have a pair of real positive/negative eigenvalues, isestablished. As a result, the linear stability analysis depends upon of thespectral study of this linear operator and of its Morse index. The extensiontheory of symmetric operators, Sturm-Liouville oscillation results and analyticperturbation theory of operators are fundamental ingredients in the stabilityanalysis. A comprehensive study of the local well-posedness of the sine-Gordonmodel in $\mathcal E(\mathcal Y) \times L^2(\mathcal{Y})$ where $\mathcalE(\mathcal Y) \subset H^1(\mathcal{Y})$ is an appropriate energy space, is alsoestablished. The theory developed in this investigation has prospects for thestudy of the instability of stationary wave solutions of other nonlinearevolution equations on metric graphs.

Unstable kink-soliton profiles for the sine-Gordon equation on a $\mathcal{Y}$-junction graph with $\delta$-interaction