Critical Nonlinear Schrödinger Equations with and without Harmonic Potential
نویسندگان
چکیده
Abstract. We exhibit a change of variables that turns the critical nonlinear Schrödinger equation into the critical nonlinear Schrödinger equation with isotropic harmonic potential, in any space dimension. This change of variables is isometric on L, and bijective on some time intervals. Using the known results for the critical nonlinear Schrödinger equation, this provides information for the properties of Bose-Einstein condensate in space dimension one and two. We discuss in particular the wave collapse phenomenon.
منابع مشابه
Long time dynamics for the one dimensional non linear Schrödinger equation
— In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byp...
متن کاملChanging blow-up time in nonlinear Schrödinger equations
Abstract Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic poten...
متن کاملRemarks on nonlinear Schrödinger equations with harmonic potential
Bose-Einstein condensation is usually modeled by nonlinear Schrödinger equations with harmonic potential. We study the Cauchy problem for these equations. We show that the local problem can be treated as in the case with no potential. For the global problem, we establish an evolution law, which is the analogue of the pseudo-conformal conservation law for the nonlinear Schrödinger equation. With...
متن کاملSemi-classical Schrödinger Equations with Harmonic Potential and Nonlinear Perturbation
Solutions of semi-classical Schrödinger equation with isotropic harmonic potential focus periodically in time. We study the perturbation of this equation by a nonlinear term. If the scaling of this perturbation is critical, each focus crossing is described by a nonlinear scattering operator, which is therefore iterated as many times as the solution passes through a focus. The study of this nonl...
متن کاملRemarks on the nonlinear Schrödinger equation with harmonic potential
Bose-Einstein condensation is usually modeled by nonlinear Schrödinger equations with harmonic potential. We study the Cauchy problem for these equations. We show that the local problem can be treated as in the case with no potential. For the global problem, we establish an evolution law, which is the analogue of the pseudo-conformal conservation law for the nonlinear Schrödinger equation. With...
متن کامل