### Time Dependent Nonlinear Schrödinger Equations.

Wolf von Wahl, Hartmut Pecher (1979)

Manuscripta mathematica

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Wolf von Wahl, Hartmut Pecher (1979)

Manuscripta mathematica

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Thierry Cazenave, Fred B. Weissler (1988)

Manuscripta mathematica

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Antonio Ambrosetti (2008)

Bollettino dell'Unione Matematica Italiana

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We survey some recent results dealing with some classes of systems of nonlinear Schrödinger equations.

Zhengping Wang, Huan-Song Zhou (2009)

Journal of the European Mathematical Society

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N. Hayashi, K. Nakamitsu, M. Tsutsumi (1986)

Mathematische Zeitschrift

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Nakao Hayashi, Masayoshi Tsutsumi (1981)

Mathematische Zeitschrift

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Jean Bourgain, W. Wang (1997)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Arne Jensen (1978)

Manuscripta mathematica

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J. Bourgain (1993)

Geometric and functional analysis

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Christophe Besse, Brigitte Bidégaray (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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In this article we implement different numerical schemes to simulate the Schrödinger-Debye equations that occur in nonlinear optics. Since the existence of blow-up solutions is an open problem, we try to compute such solutions. The convergence of the methods is proved and simulations seem indeed to show that for at least small delays self-focusing solutions may exist.

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina (2008)

Journal of the European Mathematical Society

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We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

Patrick Gérard, Vittoria Pierfelice (2010)

Bulletin de la Société Mathématique de France

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We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness...

Laurent Thomann (2008)

Bulletin de la Société Mathématique de France

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In this paper we are interested in constructing WKB approximations for the nonlinear cubic Schrödinger equation on a Riemannian surface which has a stable geodesic. These approximate solutions will lead to some instability properties of the equation.

Fabrice Planchon, Luis Vega (2009)

Annales scientifiques de l'École Normale Supérieure

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We prove bilinear virial identities for the nonlinear Schrödinger equation, which are extensions of the Morawetz interaction inequalities. We recover and extend known bilinear improvements to Strichartz inequalities and provide applications to various nonlinear problems, most notably on domains with boundaries.