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Pascal Bégout

Publications and source records attributed to Pascal Bégout.

23 records · Page 2Linked to original sources
arXiv2012

Maximum Decay Rate for Finite-Energy Solutions of Nonlinear Schrödinger Equations

We give explicit time lower bounds in the Lebesgue spaces for all nontrivial solutions of nonlinear Schrödinger equations bounded in the energy space. The result applies for these equations set in any domain of $\R^N,$ including the whole space. This also holds for a large class of nonlinearities, thereby extending the results obtained by Hayashi and Ozawa in \cite{MR91d:35035} and by the author in \cite{beg3}.

math.AP↗
arXiv2012

Maximum Decay Rate for the Nonlinear Schrödinger Equation

In this paper, we consider global solutions for the following nonlinear Schrödinger equation $iu_t+Δu+λ|u|^αu=0,$ in $\R^N,$ with $λ\in\R$ and $0\leα<\frac{4}{N-2}$ $(0\leα<\infty$ if $N=1).$ We show that no nontrivial solution can decay faster than the solutions of the free Schrödinger equation, provided that $u(0)$ lies in the weighted Sobolev space $H^1(\R^N)\cap L^2(|x|^2;dx),$ in the energy space, namely $H^1(\R^N),$ or in $L^2(\R^N),$ according to the different cases.

math.AP↗
arXiv2012

Convergence to Scattering States in the Nonlinear Schrödinger Equation

In this paper, we consider global solutions of the following nonlinear Schrödinger equation $iu_t+Δu+λ|u|^αu = 0,$ in $\R^N,$ with $λ\in\R,$ $α\in(0,\frac{4}{N-2})$ $(α\in(0,\infty)$ if $N=1)$ and \linebreak $u(0)\in X\equiv H^1(\R^N)\cap L^2(|x|^2;dx).$ We show that, under suitable conditions, if the solution $u$ satisfies $e^{-itΔ}u(t)-u_ \pm\to0$ in $X$ as $t\to\pm\infty$ then $u(t)-e^{itΔ}u_\pm\to0$ in $X$ as $t\to\pm\infty.$ We also study the converse. Finally, we estimate $|\:\|u(t)\|_X-\|e^{itΔ}u_\pm\|_X\:|$ under some less restrictive assumptions.

math.AP↗
arXiv2012

Localizing Estimates of the Support of Solutions of some Nonlinear Schrödinger Equations - The Stationary Case

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrödinger equations mainly the compactness of the support and its spatial localization. This question is very related with pure essence of the derivation of the Schrödinger equation since it is well-known that if the linear Schrödinger equation is perturbated with "regular" potentials then the corresponding solution never vanishes on a positive measured subset of the domain, which corresponds with the impossibility of localize the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution becomes a compact set and so any estimate on its spatial localization implies a very rich information on places which can not be. Our results are obtained by the application of some energy methods which connect the compactness of the support with the local vanishing of a suitable "energy function" which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.

math.AP↗