Classical Solutions of Nonlinear Schrödinger Equations.

Nakao Hayashi

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

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Numerical study of self-focusing solutions to the Schrödinger-Debye system

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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.

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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.