Instability theory of stationary kink and anti-kink profiles for the sine-Gordon equation on a Y-junction graph

The purpose of this talk is to communicate recent results regarding the (in)stability theory of static solutions of kink and anti-kink type for the sine-Gordon equation posed on a Y-junction graph. The boundary conditions at the vertex are assumed to be of delta- or delta'-type. Applications of the model include the study of tri-crystal boundaries of Josephson junctions in superconductivity theory. It is shown that kink and kink/anti-kink soliton type stationary profiles are linearly (and nonlinearly) unstable. 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. 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. This is joint work with J. Angulo Pava (Univ. of Sao Paulo).