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Journal of the Institute of Mathematics of Jussieu (1)

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THE FOCUSING NONLINEAR SCHRÖDINGER EQUATION ON THE QUARTER PLANE WITH TIME-PERIODIC BOUNDARY CONDITION: A RIEMANN–HILBERT APPROACH
Anne Boutet de Monvel, Vladimir Kotlyarov
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 6 / Issue 4 / October 2007

Published online by Cambridge University Press:

12 February 2007, pp. 579-611

Print publication:

October 2007
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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.