Integrability and Self-Similarity in Transient Stimulated Raman Scattering |
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Authors: | A S Fokas C R Menyuk |
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Institution: | (1) Department of Mathematics, Imperial College, University of London, London SW7 2BZ, UK, GB;(2) Institute of Nonlinear Studies, Clarkson University, Potsdam, NY 13699, USA, US;(3) Department of Computing Science and Electrical Engineering, University of Maryland, Baltimore County, Baltimore, MD 21250, USA, US |
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Abstract: | Summary. The phenomenon of stimulated Raman scattering (SRS) can be described by three coupled PDEs which define the pump electric
field, the Stokes electric field, and the material excitation as functions of distance and 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-boundary value (IBV) problem where both independent variables are on a finite domain. A general method for solving IBV problems for integrable equations has been introduced recently. Using this method
we show that the solution of the equations describing the transient SRS can be obtained by solving a certain linear integral
equation. It is interesting that this equation is identical to the linear integral equation characterizing the solution of
an IBV problem of the sine-Gordon equation in light-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 self-similar solution which satisfies a particular
case of the third Painlevé transcendent. This result is consistent with both numerical simulations and experimental observations.
We also discuss briefly the effect of frequency mismatch between the pump and the Stokes electric fields.
Received December 10, 1996; second revision received October 10, 1997; final revision received January 20, 1998 |
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Keywords: | , Raman cell, Painlevé equations, integrable equations, Riemann-Hilbert problems |
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