On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2015

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

Résumé

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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Dates et versions

hal-01163000 , version 1 (11-06-2015)
hal-01163000 , version 2 (13-06-2016)
hal-01163000 , version 3 (14-12-2016)

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Isabella Ianni, Stefan Le Coz, Julien Royer. On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation. 2015. ⟨hal-01163000v1⟩
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