In this work, we study the existence and orbital (in)stability of certain standing-wave solutions for the cubic nonlinear Schr\"odinger equation (NLS) posed on a looping-edge graph $\mathcal{ G}$, consisting of a circle and a finite number $N$ of infinite half-lines attached to a common vertex. Our main goal is to take the first steps toward understanding the dynamics of the NLS under $\delta'$-type interactions. Here we consider a negative $Z$-strength at the vertex, where continuity of the wave function is not mandatory. On the circle, we propose Jacobi elliptic profiles of dnoidal type combined with soliton tail profiles on the half-lines, and we establish the existence and (in)stability of these solutions depending on the relative size of $N$, $Z$, and the phase velocity of the standing wave. Tools from Krein--von Neumann extension theory for symmetric operators play a fundamental role in our stability analysis. We also develop a local and global well-posedness theory for the NLS in the energy space $H^1(\mathcal{ G})$. Finally, we present an approach to characterize the domains of self-adjoint extensions of the Laplace operator on a looping-edge graph, which incorporate the continuity of derivatives at the vertex.
Categories: math.AP