Classification and stability of positive solutions to the NLS equation on the $\mathcal{T}$-metric graph

Given $\lambda>0$ and $p>2$, we present a complete classification of the positive $H^1$-solutions of the equation $-u''+\lambda u=|u|^{p-2}u$ on the $\mathcal{T}$-metric graph (consisting of two unbounded edges and a terminal edge of length $\ell>0$, all joined together at a single vertex). This study implies, in particular, the uniqueness of action ground states. Moreover, for $p\sim 6^-$, the notions of action and energy ground states do not coincide and energy ground states are not unique. In the $L^2$-supercritical case $p>6$, we prove that, for $\lambda\sim 0^+$ and $\lambda\sim +\infty$, action ground states are orbitally unstable for the flow generated by the associated time-dependent NLS equation $i\partial_tu + \partial^2_{xx} u + |u|^{p-2}u=0$. Finally, we provide numerical evidence of the uniqueness of energy ground states for $p\sim 2^+$ and of the existence of both stable and unstable action ground states for $p\sim6$.