Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data

Анотація

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 $(pe^{i\omega t})$. 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 $t\to \infty$) wave, when $x>\omega^2 t$;
• $\;$ the modulated elliptic wave of finite amplitude, when $\omega_0^2 t<x<\omega^2 t$;
• $\;$ the plane wave of finite amplitude, when $0<x<\omega_0^2 t$.

The similar results are true for the same IBV problem with nonzero initial condition vanishing as $t\to\infty$.

Mathematics Subject Classification: 37K15, 35Q15, 35B40.