Tesis Doctorado
Quelques problemes lies a la dynamique des equatións de gross-pitaevskii et de landau-lifshitz
Autor
De Laire-Peirano, André Jea Pierre
Institución
Resumen
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. PFCHA-Becas Doctor Matemáticas Aplicadas 180p. PFCHA-Becas TERMINADA