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Chapitre D'ouvrage Année : 2019

Universal Dynamics for the Logarithmic Schrödinger Equations

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 behaviour of the solution, on at least three aspects. The dispersion is faster than usual by a logarithmic factor in time. The positive Sobolev norms of the solution grow logarithmically in time. Finally, 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, up to weakening the third point (weak convergence instead of strong convergence). One of the key steps of the proof for the last point consists in using the Madelung transform. It reduces the equation to a variant of the isothermal compressible Euler equation, whose large time behaviour turns out to be governed by a parabolic equation involving a Fokker-Planck operator.
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Dates et versions

hal-03347609 , version 1 (17-09-2021)

Identifiants

Citer

Rémi Carles. Universal Dynamics for the Logarithmic Schrödinger Equations. Jean-Michel Coron; Tatsien Li; Wei-Min Wang. Series in Contemporary Applied Mathematics, 23, World Scientific, pp.177-212, 2019, Harmonic Analysis and Wave Equations, 978-981-120-836-2. ⟨10.1142/9789811208379_0004⟩. ⟨hal-03347609⟩
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