Variational approach to the orbital stability of standing waves of the Gross-Pitaevskii equation. (English) Zbl 1304.35655
Summary: This paper is concerned with the mathematical analysis of a masssubcritical nonlinear Schrödinger equation arising from fiber optic applications. We show the existence and symmetry of minimizers of the associated constrained variational problem. We also prove the orbital stability of such solutions referred to as standing waves and characterize the associated orbit. In the last section, we illustrate our results with few numerical simulations.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35J60 | Nonlinear elliptic equations |
| 37J10 | Symplectic mappings, fixed points (dynamical systems) (MSC2010) |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 35B35 | Stability in context of PDEs |
| 35B32 | Bifurcations in context of PDEs |
| 35A15 | Variational methods applied to PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 90C52 | Methods of reduced gradient type |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
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