Wave instabilities in the presence of non vanishing background in nonlinear Schrodinger systems

We investigate wave collapse ruled by the generalized nonlinear Schroedinger (NLS) equation in 1+1 dimensions, for localized excitations with non-zero background, establishing through virial identities a new criterion for blow-up. When collapse is arrested, a semiclassical approach allows us to show that the system can favor the formation of dispersive shock waves. The general findings are illustrated with a model of interest to both classical and quantum physics (cubic-quintic NLS equation), demonstrating a radically novel scenario of instability, where solitons identify a marginal condition between blow-up and occurrence of shock waves, triggered by arbitrarily small mass perturbations of different sign

Orbital stability of the black soliton for the quintic Gross-Pitaevskii equation

In this work, a rigorous proof of the orbital stability of the black soliton solution of the quintic Gross-Pitaevskii equation in one spatial dimension is obtained. We first build and show explicitly black and dark soliton solutions and we prove that the corresponding Ginzburg-Landau energy is coercive around them by using some orthogonality conditions related to perturbations of the black and dark solitons. The existence of suitable perturbations around black and dark solitons satisfying the required orthogonality conditions is deduced from an Implicit Function Theorem. In fact, these perturbations involve dark solitons with sufficiently small speeds and some proportionality factors arising from the explicit expression of their spatial derivative. We are also able to control the evolution of the modulation parameters along the quintic Gross-Pitaevskii flow by estimating their growth in time. Finally by using a low order conservation law (momentum), we prove that the speed of the perturbation is bounded and use that control to finish the proof of the orbital stability of black solitons. As a direct consequence, we also prove the orbital stability of the dark soliton in a small speed interval.Comment: 46 pages, 3 figures. Introduction extende

Scattering for the non-radial inhomogeneous NLS

We extend the result of Farah and Guzm\'an on scattering for the $3d$ cubic inhomogeneous NLS to the non-radial setting. The key new ingredient is a construction of scattering solutions corresponding to initial data living far from the origin.Comment: 16 page