PACHERIE , Eliot

In this thesis, we focus on the study of travelling waves in the Gross-Pitaevskii equation in dimension 2, with the condition a non-trivial condition at infinity. This equation has been studied extensively, both in physical and mathematical works. It is a model for Bose-Einstein condensates, and describes the behavior of superfluids.We are...

For the Nonlinear Schrödinger equation in dimension 2, the existence of a global minimizer of the energy at fixed momentum has been established by Bethuel-Gravejat-Saut [7] (see also [13]). This minimizer is a travelling wave for the Nonlinear Schrödinger equation. For large momentums, the propagation speed is small and the minimizer behaves...

In a previous paper, we constructed a smooth branch of travelling waves for the 2 dimensional Gross-Pitaevskii equation. Here, we continue the study of this branch. We show some coercivity results, and we deduce from them the kernel of the linearized operator, a spectral stability result, as well as a uniqueness result in the energy space. In...

We construct a smooth branch of travelling wave solutions for the 2 dimensional Gross-Pitaevskii equations for small speed. These travelling waves exhibit two vortices far away from each other. We also compute the leading order term of the derivatives with respect to the speed. We construct these solutions by an implicit function type argument....