Universal dynamics for the defocusing logarithmic Schrodinger equation

Rémi Carles (1) , Isabelle Gallagher (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 IMJ - Institut de Mathématiques de Jussieu

We consider the nonlinear Schrodinger 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 positive Sobolev norms of the solution grow logarithmically in time. Moreover after rescaling 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 isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker--Planck operator.

Domaines: Equations aux dérivées partielles [math.AP]

Rémi Carles, Isabelle Gallagher. Universal dynamics for the defocusing logarithmic Schrodinger equation. 2016. ⟨hal-01398526v2⟩