On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation

2015/06/11 by Isabella Ianni, Ianni, Isabella, Stefan Le Coz +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1506.03761

Abstract

We consider the one-dimensional Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions with modulus 1 at infinity. Then we show the persistence of the stationary black soliton of the unperturbed problem as a solution. We also prove the existence of another branch of non-trivial stationary waves. Depending on the attractive or repulsive nature of the Dirac perturbation and of the type of stationary solutions, we prove orbital stability via a variational approach, or linear instability via a bifurcation argument.

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