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THE FOCUSING NONLINEAR SCHRÖDINGER EQUATION ON THE QUARTER PLANE WITH TIME-PERIODIC BOUNDARY CONDITION: A RIEMANN–HILBERT APPROACH

Published online by Cambridge University Press:  12 February 2007

Anne Boutet de Monvel  and
Vladimir Kotlyarov
Anne Boutet de Monvel
Affiliation:
Institut de Mathématiques de Jussieu, Case 7012, Université Paris 7, 2 Place Jussieu, 75251 Paris, France (aboutet@math.jussieu.fr)
Vladimir Kotlyarov
Affiliation:
Mathematical Division, Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine (kotlyarov@ilt.kharkov.ua)
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Abstract

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic. The main goal of this paper is to introduce a Riemann–Hilbert problem whose solution gives the solution of our initial–boundary-value problem. This is a preliminary step to obtain uniform long-time asymptotics for the solution of this equation using both the stationary phase and the Deift–Zhou methods.

Keywords

nonlinear Schrödinger equationinverse scatteringRiemann–Hilbert probleminitial–boundary-value problemtime-periodic boundary conditionPrimary 35Q55Secondary 37K1535Q15
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 6 , Issue 4 , October 2007 , pp. 579 - 611
Copyright
2007 Cambridge University Press

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