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Pré-Publication, Document De Travail Année : 2016

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.
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Dates et versions

hal-01398526 , version 1 (17-11-2016)
hal-01398526 , version 2 (05-12-2016)
hal-01398526 , version 3 (24-04-2017)
hal-01398526 , version 4 (25-01-2018)

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Rémi Carles, Isabelle Gallagher. Universal dynamics for the defocusing logarithmic nonlinear Schrodinger equation. 2016. ⟨hal-01398526v1⟩
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