Existence and (in)stability of standing waves for the nonlinear Schr\"odinger Equations on looping-edge graphs with $\delta'$-type interactions

In this work, we investigate the existence and orbital (in)stability of several branches of 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. The model is endowed with $\delta'$--type interaction boundary conditions at the vertex, which enforce continuity of the derivatives of the wave functions, while continuity of the wave function itself is not required. By means of the Implicit Function Theorem, we establish the existence of families of standing--wave profiles that converge, on the circular component of the graph, to Jacobi elliptic solutions of dnoidal type, coupled with soliton--type tail profiles on the half--lines. Tools from perturbation theory and Kre\u{\i}n--von Neumann extension theory for symmetric operators play a central role in the (in)stability analysis of such standing wave solutions. Our approach may be extended to other bound states for the NLS on looping graphs or more general non--compact metric graphs.