Nonlinear Schr\"odinger Equations on looping-edge graphs with $\delta'$-type interactions

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. We consider the self-adjoint realization $(\mathcal{H}_Z, D(\mathcal{H}_Z))$ of the Laplacian, where the domain $D(\mathcal{H}_Z)$ encodes on the half-lines a $\delta'$-type vertex conditions (continuity of derivatives at the vertex, without requiring continuity of the wave function) and $Z \in \mathbb{R}\setminus\{0\}$. On the circle, we propose Jacobian elliptic profiles of dnoidal type combined with either trivial (zero) or soliton tail profiles on the half-lines with full derivative matching at the boundary. For the trivial tail case we establish orbital stability for all $Z \in \mathbb{R}\setminus\{0\}$, while for the non-trivial tail case (which requires $Z < 0$) we establish both existence and orbital (in)stability depending on the relative size of $N$, $Z$, and the phase velocity of the standing wave.