INTEGRABILITY AND SELF-SIMILARITY IN TRANSIENT STIMULATED RAMAN-SCATTERING

The phenomenon of stimulated Raman scattering (SRS) can be described b y three coupled PDEs which define the pump electric field, the Stokes electric field, and the material excitation as functions of distance a nd time. In the transient limit these equations are integrable, i.e., they admit a Lax pair formulation. Here we study this transient limit. The relevant physical problem can be formulated as an initial-boundar y value (IBV) problem where both independent variables are on a finite domain. A general method for solving IBV problems for integrable equa tions has been introduced recently. Using this method we show that the solution of the equations describing the transient SRS can be obtaine d by solving a certain linear integral equation. It is interesting tha t this equation is identical to the linear integral equation character izing the solution of an IBV problem of the sine-Gordon equation in li ght-cone coordinates. This integral equation can be solved uniquely in terms of the values of the pump and Stokes fields at the entry of the Raman cell. The asymptotic analysis of this solution reveals that the long-distance behavior of the system is dominated by the underlying s elf-similar solution which satisfies a particular case of the third Pa inleve transcendent. This result is consistent with both numerical sim ulations and experimental observations. We also discuss briefly the ef fect of frequency mismatch between the pump and the Stokes electric fi elds.