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Analyse et simulation d'équations de Schrödinger déterministes et stochastiques. Applications aux condensats de Bose-Einstein en rotation

Abstract : The aim of this Thesis is to study various mathematical and numerical aspects related to the Gross-Pitaevskii and nonlinear Schrödinger equations. We begin (chapter 1) by introducing a few models starting from the physics of Bose-Einstein condensates and optical fibers. This naturally leads to introducing a stochastic Gross-Pitaevskii equation and a nonlinear Schrödinger equation with random dispersion. Next, in the second chapter, we analyze the existence and uniqueness problem for these two equations. We prove that the Cauchy problem admits a solution for the stochastic Gross-Pitaevskii equation with a rotational term by constructing the solution associated with the linear. The third chapter is concerned with the computation of stationary states for the Gross-Pitaevskii equation. We develop a pseudo-spectral approximation scheme for the Continuous Normalized Gradient Flow formulation, combined with preconditioned Krylov subspace methods. This original approach leads to the robust and efficient computation of ground states for fast rotations and strong nonlinearities. In the fourth chapter, we consider some pseudo-spectral schemes for computing the dynamics of the Gross-Pitaevskii and nonlinear Schrödinger equations. These schemes (the Lie's and Strang's splitting schemes and the relaxation scheme) are numerically studied. Moreover, we proceed to a rigorous numerical analysis of the Lie scheme for the associated stochastic PDEs. Finally, we present in the fifth chapter a Matlab toolbox (called GPELab) that provides computational solutions based on the schemes previously introduced in the Thesis
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https://hal.univ-lorraine.fr/tel-01750566
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Submitted on : Thursday, March 29, 2018 - 12:49:21 PM
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Romain Duboscq. Analyse et simulation d'équations de Schrödinger déterministes et stochastiques. Applications aux condensats de Bose-Einstein en rotation. Mathématiques générales [math.GM]. Université de Lorraine, 2013. Français. ⟨NNT : 2013LORR0198⟩. ⟨tel-01750566⟩

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