Periodic boundary data for an integrable model of stimulated Raman scattering: long-time asymptotic behavior

Abstract

We study the long-time asymptotic behavior of the solution to the initial-boundary value (IBV) problem in the quarter plane (x > 0, t > 0) for nonlinear integrable equations of stimulated Raman scattering. We consider the case of zero initial condition and periodic boundary data (p eiωt). Using the steepest descent method for oscillatory matrix Riemann–Hilbert problems we show that the solution of the IBV problem has different asymptotic behavior in different regions. The solution takes the form ofa plane wave of finite amplitude, when 0 < x < ω20t,a modulated elliptic wave of finite amplitude, when ω20t < x < ω2t, anda self-similar vanishing (as t → ∞) wave, when x > ω2t.For the IBV problem with nonzero initial condition and the same periodic boundary data, the solution to this problem is qualitatively similar to that of this study with the only difference that the solitons (of finite amplitude) can appear in the region x > ω2t.