Quelques problemes lies a la dynamique des equatións de gross-pitaevskii et de landau-lifshitz

Abstract

This thesis is devoted to the study of the Gross-Pitaevskii equation and the LandauLifshitz equation, which have important applications in physics. The Gross- Pitaevskii equation models phenomena of nonlinear optics, superfl.uidity and Base-Einstein condensation, while the Landau-Lifshitz equation describes the dynamics of magnetization in ferromagnetic materials. When modeling matter at very low temperatures, it is usual to suppose that the interaction between particles is punctual. Then the classical Gross-Pitaevskii equation is derived by taking as interaction the Dirac delta function. However, different types of nonlocal potentials, probably more realistic, have also been proposed by physicists to model more general interactions. First, we will focus on provide sufficient conditions that cover a broad variety of nonlocal interactions and such that the associated Cauchy problem is globally well-posed with nonzero conditions at infinity. After that, we will study the traveling waves for this nonlocal model and we will provide conditions such that we can compute a range of speeds in which nonconstant finite energy solutions do not exist. Concerning the Landau- Lifshitz equation, we will also be interested in finite energy traveling waves. We will prove the nonexistence of nonconstant traveling waves with small energy in dimensions two, three and four, provided that the energy is less than the momentum in the two-dimensional case. In addition, we will also give, in the two-dimensional case, the description of a minimizing curve which could give a variational approach to build solutions of the LandauLifshitz equation. Finally, we describe the asymptotic behavior at infinity of the finite energy traveling waves.