arXiv subjects

Remi Carles

Publications and source records attributed to Remi Carles.

26 records · Page 2Linked to original sources

arXiv2003

Semiclassical Nonlinear Schrodinger equations with potential and focusing initial data

We study the asymptotic behaviour of solutions to semi-classical nonlinear Schrodinger equations with a potential, for concentrating and oscillating initial data, when the nonlinearity is repulsive and the potential is a polynomial of degree at most two. We describe the separate roles of the nonlinearity and of the potential, with tools which seem to be specific to the class of potentials that we consider. We also discuss the case of more general subquadratic potentials.

math.AP↗

arXiv2003

Nonlinear Schrodinger equations with Stark potential

We study the nonlinear Schrodinger equations with a linear potential. A change of variables makes it possible to deduce results concerning finite time blow up and scattering theory from the case with no potential.

math.AP↗

arXiv2003

Nonlinear Schrodinger equations with repulsive harmonic potential and applications

We study the Cauchy problem for Schrodinger equations with repulsive quadratic potential and power-like nonlinearity. The local problem is well-posed in the same space as that used when a confining harmonic potential is involved. For a defocusing nonlinearity, it is globally well-posed, and a scattering theory is available, with no long range effect for any superlinear nonlinearity. When the nonlinearity is focusing, we prove that choosing the harmonic potential sufficiently strong prevents blow-up in finite time. Thanks to quadratic potentials, we provide a method to anticipate, delay, or prevent wave collapse; this mechanism is explicit for critical nonlinearity.

math.AP↗

arXiv2003

On the role of quadratic oscillations in nonlinear Schrodinger equations

We consider a nonlinear semi-classical Schrodinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrodinger equation with harmonic potential.

math.AP↗

arXiv2003

Focusing at a point with caustic crossing for a class of nonlinear equations

We consider the asymptotic behavior of solutions to nonlinear partial differential equations in the limit of short wavelength. For initial data which cause focusing at one point, we highlight critical indexes as far as the influence of the nonlinearity in the neighborhood of the caustic is concerned. Our results generalize some previous ones proved by the first author in the case of nonlinear Schrodinger equations. We apply them to Hartree, Klein-Gordon and wave equations.

math.AP↗

arXiv2002

Focusing of Spherical Nonlinear Pulses in ${\mathbb R}^{1+3}$, III. Sub and Supercritical cases

We study the validity of geometric optics in $L^\infty$ for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is subcritical, then no nonlinear effect occurs at leading order. If the amplitude of the initial data is sufficiently big, strong nonlinear effects occur; we study the cases where the equation is either dissipative or accretive. When the equation is dissipative, pulses are absorbed before reaching the focal point. When the equation is accretive, the family of pulses becomes unbounded.

math.AP↗

arXiv2002

Critical nonlinear Schroedinger equations with and without harmonic potential

We use a change of variables that turns the critical nonlinear Schroedinger equation into the critical nonlinear Schroedinger equation with isotropic harmonic potential, in any space dimension. This change of variables is isometric on $L^2$, and bijective on some time intervals. Using the known results for the critical nonlinear Schroedinger 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.

cond-mat↗

arXiv2001

Remarks on nonlinear Schroedinger equations with harmonic potential

Bose-Einstein condensation is usually modeled by nonlinear Schroedinger 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 Schroedinger equation. With this evolution law, we give wave collapse criteria, as well as an upper bound for the blow up time. Taking the physical scales into account, we finally give a lower bound for the blow up time.

cond-mat↗