On the long‐time limit of semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation: Pure radiation case

In a previous paper [13] we calculated the leading-order term q0(x,t ,e )of the solution ofq(x,t ,e ), the focusing nonlinear (cubic) Schrodinger (NLS) equation in the semiclassical limit (e → 0) for a certain one-parameter family of initial conditions. This family contains both solitons and pure radiation. In the pure radiation case, our result is valid for all times t ≥ 0. The aim of the present paper is to calculate the long-term behavior of the semiclassical solution q(x,t ,e )in the pure radiation case. As before, our main tool is the Riemann-Hilbert problem (RHP) formulation of the inverse scattering problem and the corresponding system of “moment and integral conditions,” known also as a system of “modulation equations.” c � 2006 Wiley Periodicals, Inc.

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