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Equation de schrödinger non-linéaire avec non-linéarité logarithmique

Abstract : The topic of this thesis is the analysis of the nonlinear Schrödinger equation with logarithmic nonlinearity (logNLS).Firstly, we study the long-time behavior in defocusing regime. A universal behavior has already been observed in this context. We specify this behavior by an optimal convergence rate in Wasserstein distance (also called Kantorovich-Rubinstein). In parallel, we show that the properties of this equation allows us to obtain, via the Wigner Transform, a limit object in the semiclassical limit satisfying the the same long-time behavior. This commutation between semiclassical limit and long-time behavior is an unusual feature for a Schrödinger equation.Then, we are interested in the focusing regime, and specifically in the interactions between solitons (which are in this case Gaussian functions called Gaussons), and more generally between the explicit gaussian solutions that this equation admits. We first show that a solution to logNLS with a sum of gaussian functions which are far away from each other for initial data stays close to the sum of corresponding gaussian solutions until at least a time of order the square of the minimal distance between the Gaussian.Then, we prove the existence of multi-Gaussons and multi-gaussian (solution which behaves in large time like a sum of several gaussian solutions) with a convergence rate faster than exponential, and their uniqueness for such a convergence rate.Last, a WKB analysis is performed for this equation. The limit equations of this analysis are the isotherm Euler system, whose solution have been done thanks to the Riemann variables. We establish a new Cauchy theory by using new variables similar as the Riemann variables, corresponding to solutions of logNLS of the form [...] in a semiclassical regime, under an assumption of analycity. We also show that these variables converge in the semiclassical limit, and that the limit functions are solutions to the isothermal Euler system under its "Riemann variables" form.
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Submitted on : Friday, October 22, 2021 - 4:57:18 PM
Last modification on : Thursday, November 24, 2022 - 3:30:32 AM
Long-term archiving on: : Monday, January 24, 2022 - 4:59:36 PM

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  • HAL Id : tel-03396661, version 1

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Guillaume Ferriere. Equation de schrödinger non-linéaire avec non-linéarité logarithmique. Equations aux dérivées partielles [math.AP]. Université Montpellier, 2021. Français. ⟨NNT : 2021MONTS015⟩. ⟨tel-03396661⟩

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