Ported from Self-Similar Analytic Solutions of Ginzburg--Landau Equation with Varying Coefficients

Employing the technique of symmetry reduction of analytic method, we solve the Ginzburg--Landau equation with varying nonlinear, dispersion, gain coefficients, and gain dispersion which originates from the limiting effect of transition bandwidth in the realistic doped fibres. The parabolic asymptotic
self-similar analytical solutions in gain medium of the normal GVD is found for the first time to our best knowledge. The evolution of pulse amplitude, strict linear phase chirp and effective temporal width are given with self-similarity results in longitudinal nonlinearity distribution and longitudinal gain fibre. These analytical solutions are in good agreement with the numerical simulations. Furthermore, we theoretically prove that pulse evolution has the characteristics of parabolic asymptotic self-similarity in doped ions dipole gain fibres.