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Communications in Mathematical Physics

, Volume 149, Issue 2, pp 377–414 | Cite as

Limit behavior of saturated approximations of nonlinear Schrödinger equation

  • F. Merle
Article

Abstract

We consider the solutionuɛ(t) of the saturated nonlinear Schrödinger equation
$$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} u + \varepsilon \left| u \right|^{q - 1} uandu(0,.) = \varphi (.)$$
(1)
where\(N \geqslant 2,\varepsilon > 0,1 + 4/N< q< (N + 2)/(N - 2),u:\mathbb{R} \times \mathbb{R}^N \to \mathbb{C},\varphi \), ϕ is a radially symmetric function inH1(R N ). We assume that the solution of the limit equation is not globally defined in time. There is aT>0 such that\(\mathop {\lim }\limits_{t \to T} \left\| {u(t)} \right\|_{H^1 } = + \infty \), whereu(t) is solution of
$$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} uandu(0,.) = \varphi (.)$$
(1)
For ɛ>0 fixed,uɛ(t) is defined for all time. We are interested in the limit behavior as ɛ→0 ofuɛ(t) fort≥T. In the case where there is no loss of mass inuɛ at infinity in a sense to be made precise, we describe the behavior ofuɛ as ɛ goes to zero and we derive an existence result for a solution of (1) after the blow-up timeT in a certain sense. Nonlinear Schrödinger equation with supercritical exponents are also considered.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • F. Merle
    • 1
  1. 1.Laboratoire d'analyse numériqueUniversités de Cergy-Pontoise et Pierre et Marie CurieParisFrance

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