Abstract
We study solutions of the nonlinear Schrödinger equation (NLSE) with gain, describing optical pulse propagation in an amplifying medium. We construct a semiclassical self-similar solution with a parabolic temporal variation that corresponds to the energy-containing core of the asymptotically propagating pulse in the amplifying medium. We match the self-similar core through Painlevé functions to the solution of the linearized equation that corresponds to the low-amplitude tails of the pulse. The analytic solution accurately reproduces the numerically calculated solution of the NLSE.
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Boscolo, S., Turitsyn, S.K., Novokshenov, V.Y. et al. Self-Similar Parabolic Optical Solitary Waves. Theoretical and Mathematical Physics 133, 1647–1656 (2002). https://doi.org/10.1023/A:1021402024334
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DOI: https://doi.org/10.1023/A:1021402024334