Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data - Moskovchenko, E. A.; Kotlyarov, V. P. (Журнал математичної фізики, аналізу, геометрії, 2009)
The long-time asymptotic behavior of the initial-boundary value (IBV) problem in the quarter plane (x>0, t>0) for nonlinear integrable equations of the stimulated Raman scattering is studied. Considered is the case of zero initial condition and single-phase boundary data (pe^iω t). By using the steepest descent method for oscillatory matrix Riemann-Hilbert problems it is shown that the solution of the IBV problem has different asymptotic behavior in different regions, namely: the selfsimilar vanishing (as t→ ∞) wave, when x>ω^2 t; the modulated elliptic wave of finite amplitude, when ω_0^2 t<x<ω^2 t; the plane wave of finite amplitude, when 0<x<ω_0^2 t. The similar results are true for the same IBV problem with nonzero initial condition vanishing as t→∞. Mathematics Subject Classification: 37K15, 35Q15, 35B40.