For rather general nonlinearities, we prove that defocusing nonlinear Schrödinger equations in n (n ≤ 4), with non-vanishing initial data at infinity u 0 , are globally well-posed in u 0 + H 1 . The same result holds in an exterior domain in n , n = 2 3.
We give an upper bound on the growth rate of the Schrödinger group on Zhidkov spaces. In dimension 1, we prove that this bound is sharp. To cite this article: C. Gallo, C. R. Acad. Sci. Paris, Ser. I 342 (2006). 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.
RésuméTaux de croissance du groupe de Schrödinger sur les espaces de Zhidkov. On donne une borne supérieure au taux de croissance du groupe de Schrödinger sur les espaces de Zhidkov. En dimension 1, on montre que cette borne est optimale. Pour citer cet article : C. Gallo, C. R. Acad. Sci. Paris, Ser. I 342 (2006).Les équations de Schrödinger non linéaires défocalisantes, comme l'équation de Gross-Pitaevskii, admettent des ondes progressives non nulles à l'infini, appelées dark solitons [1][2][3]5,7,8]. Ces solutions interviennent dans de nombreux contextes physiques, en particulier en optique non linéaire (voir [6]) et dans l'étude de la superfluidité.Afin de mieux comprendre le comportement asymptotique en temps des solutions de ces équations, notamment à proximité des dark solitons, il est intéressant d'étudier l'effet du propagateur de Schrödinger sur un espace qui contient ces dark solitons. On s'intéresse ici à l'équation de Schrödinger linéaire sur R n avec données non nulles à l'infini. Dans [4], on montre que (1) est bien posée sur les espaces de Zhidkov X k R n := u ∈ L ∞ R n , ∇u ∈ H k−1 R n , sous la condition k > n/2. On améliore ici l'estimation montrée dans [4] sur le taux de croissance du groupe de Schrödinger sur X k (R n ) (noté S(t)). Plus précisément, on montre qu'il existe une constante C > 0 telle que
We study non-linear ground states of the Gross-Pitaevskii equation in the space of one, two and three dimensions with a radially symmetric harmonic potential. The Thomas-Fermi approximation of ground states on various spatial scales was recently justified using variational methods. We justify here the Thomas-Fermi approximation on an uniform spatial scale using the Painlevé-II equation. In the space of one dimension, these results allow us to characterize the distribution of eigenvalues in the point spectrum of the Schrödinger operator associated with the non-linear ground state.
In this paper, we prove a criterion to determine if a black soliton solution (which is an odd solution that does not vanish at infinity) to a one-dimensional nonlinear Schrödinger equation is linearly stable or not. This criterion handles the sign of the limit at 0 of the Vakhitov-Kolokolov function. For some nonlinearities, we numerically compute the black soliton and the Vakhitov-Kolokolov function in order to investigate linear stability of black solitons. We then show that linearly unstable black solitons are also orbitally unstable. In the Gross-Pitaevskii case, we rigorously prove the linear stability of the black soliton. Finally, we numerically study the dynamical stability of these solutions solving both linearized and fully nonlinear equations with a finite differences algorithm.
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