Jaime Angulo

Universidade de São Paulo, Brazil

Title: The cubic Schr\"odinger equation on a looping-edge graph with $\delta'$-interaction at the vertex

Abstract: The aim of this lecture is to provide  novel results   in the mathematical studies associated to  the existence and orbital stability   of standing wave solutions for the cubic nonlinear Schr\"odinger equation (NLS) on a looping edge graph $\mathcal G_N$, namely,  a graph consisting of a circle and a finite amount $N$ of infinite half-lines attached to a common vertex. By considering interactions of $\delta'$-type (where continuity of the profiles at the vertex is not required), we study the dynamics of standing wave solutions with a periodic-profile  on the circle and soliton  tail-profiles  on the half-lines. The existence and (in)stability of these profiles will depend on the relative size of  the phase-velocity. The theory developed in this investigation has prospects for the study of other standing wave profiles of the NLS on a looping edge graph. This work was done in collaboration with Alexander Munoz/IME-USP.