On the exact solution for smooth pulses of the defocusing nonlinear Schrödinger modulation equations prior to breaking

Abstract

The modulation equations for the amplitude and the phase of the defocusing nonlinear Schrödinger (NLS) equation in the semi-classical limit are solved exactly for smooth pulse initial data using an implicit hodograph representation of Tsarev (1985 Sov. Math.—Dokl. 31 448) combined with an extension of Riemann's method on multi-sheeted characteristic planes developed by Ludford (1952 Proc. Camb. Phil. Soc. 48 499–510, 1954 J. Ration. Mech. 3 77–88). Our results extend previous exact solutions of the modulation equations for piecewise step function data (Biondini and Kodama 2006 J. Nonlinear Sci. 16 435–81, Kodama and Wabnitz 1995 Opt. Lett. 20 2291–3, Kodama 1999 SIAM J. Appl. Math. 59 2162–92) and for smooth monotone data (Wright et al 1999 Phys. Lett. A 257 170–4) to more physically relevant smooth pulse data (a finite number of pulses). Our results also provide an exact characterization of the estimates for smooth pulse data of first breaking time and location, previously based on analysis of the modulation equations as hyperbolic conservation laws (Forest and McLaughlin 1998 J. Nonlinear Sci. 7 43–62). Extensions to other integrable nonlinear equations of NLS-type are also discussed in the appendix.