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The sine-Gordon equation on a metric graph with a structure represented by a$\mathcal{Y}$-junction, is considered. The model is endowed with boundaryconditions at the graph-vertex of $\delta'$-interaction type, expressingcontinuity of the derivatives of the wave functions plus a Kirchhoff-type rulefor the self-induced magnetic flux. It is shown that particular stationary,kink and kink/anti-kink soliton profile solutions to the model are linearly(and nonlinearly) unstable. To that end, a recently developed linearinstability criterion for evolution models on metric graphs by Angulo andCavalcante (2020), which provides the sufficient conditions on the linearizedoperator around the wave to have a pair of real positive/negative eigenvalues,is applied. This leads to the spectral study to the linearize operator and ofits Morse index. The analysis is based on analytic perturbation theory,Sturm-Liouville oscillation results and the extension theory of symmetricoperators. The methods presented in this manuscript have prospect for the studyof the dynamic of solutions for the sine-Gordon model on metric graphs withfinite bounds or on metric tree graphs and/or loop graphs.

Unstable kink and anti-kink profile for the sine-Gordon equation on a $\mathcal{Y}$-junction graph with $\delta'$-interaction at the vertex