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The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation

Published online by Cambridge University Press:  14 November 2011

Thierry Cazenave  and
Fred B. Weissler
Thierry Cazenave
Affiliation:
Analyse Numérique, Université Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex 05, France
Fred B. Weissler
Affiliation:
Centre de Mathématiques, Ecole Normale Supérieure de Cachan, 61, Avenue du Président Wilson, 94235 Cachan Cedex, France
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Synopsis

We study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 3-4 , 1991 , pp. 251 - 273
Copyright
Copyright © Royal Society of Edinburgh 1991

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