Existence and stability of Schrödinger solitons on noncompact manifolds

We consider the focusing nonlinear Schrödinger equation on a large class of rotationally symmetric, noncompact manifolds. We prove the existence of a solitary wave by perturbing off the flat Euclidean case. Furthermore, we study the stability of the solitary wave under radial perturbations by analyzing spectral properties of the associated linearized operator. Finally, in the L2-critical case, by considering the Vakhitov--Kolokolov criterion (see also results of Grillakis--Shatah--Strauss), we provide numerical evidence showing that the introduction of a nontrivial geometry destabilizes the solitary wave in a wide variety of cases, regardless of the curvature of the manifold. In particular, the parameters of the metric corresponding to standard hyperbolic space will lead to instability consistent with the blow-up results of Banica--Duyckaerts (2015). We also provide numerical evidence for geometries under which it would be possible for the Vakhitov--Kolokolov condition to suggest stability, provided certain spectral properties hold in these spaces.