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Article Dans Une Revue Communications in Contemporary Mathematics Année : 2001

Convergence to Scattering States in the Nonlinear Schrödinger Equation

Résumé

In this paper, we consider global solutions of the following nonlinear Schrödinger equation $iu_t+\Delta u+\lambda|u|^\alpha u = 0,$ in $\R^N,$ with $\lambda\in\R,$ $\alpha\in(0,\frac{4}{N-2})$ $(\alpha\in(0,\infty)$ 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 $e^{-it\Delta}u(t)-u_ \pm\to0$ in $X$ as $t\to\pm\infty$ then $u(t)-e^{it\Delta}u_\pm\to0$ in $X$ as $t\to\pm\infty.$ We also study the converse. Finally, we estimate $|\:\|u(t)\|_X-\|e^{it\Delta}u_\pm\|_X\:|$ under some less restrictive assumptions.
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Dates et versions

hal-00715801 , version 1 (09-07-2012)

Identifiants

Citer

Pascal Bégout. Convergence to Scattering States in the Nonlinear Schrödinger Equation. Communications in Contemporary Mathematics, 2001, 3 (3), pp.403-418. ⟨10.1142/S0219199701000421⟩. ⟨hal-00715801⟩
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