Universal dynamics for the defocusing logarithmic nonlinear Schrodinger equation
Résumé
We consider the nonlinear Schrödinger equation with a logarithmic nonlinearity, whose sign is such that no non-trivial stationary solution exists. Explicit computations show that in the case of Gaussian initial data, the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time and the H^1-norm of the solution grows logarithmically in time. Moreover after rescal-ing in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile (whose variance is independent of the initial variance). In the case of general initial data, we show that these properties remain, in a weaker sense. One of the key steps of the proof consists in using the Madelung transform to reduce the equation to a variant of the isother-mal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker–Planck operator.
Origine | Fichiers produits par l'(les) auteur(s) |
---|