Moving fronts for complex Ginzburg-Landau equation with Raman term

Abstract

The nonlinear Schrodinger equation ~NSE! is a model equation describing a variety of short-pulse propagation phe- nomena in optics @1,2#. The NSE with nonconservative terms added is usually called the complex Ginzburg-Landau equa- tion ~CGLE!. Particular areas of application of the CGLE are all-optical fiber transmission lines and passively mode- locked fiber and solid-state lasers. The NSE can be modified to include the influence of various physical phenomena on short-pulse generation and propagation. The behavior of ultra-short pulses changes under the influence of these terms. For example, third-order dispersion results in pulse asymme- try and radiation phenomena @2#. Fourth-order dispersion may result in solitons with oscillating tails @2#. Self- steepening causes the leading edge of a pulse to become more sharp @1#. There are various works on ''optical wave breaking'' as well. The inclusion of the Raman term results in a continuous downward shift in pulse frequency @3-6# .I n the time domain, this represents the fact that the glass ~opti- cal fiber! response to the imposed field is not instantaneous. This delay of a few femtoseconds ~fs! can affect propagation of fs signals. Gagnon and Belanger @7# showed that the exact form of the soliton self-frequency shift follows from a sym- metry analysis of the equation. Kink-type solutions for the NSE with Raman term present were discovered in @8#. In nonlinear optics, a kink is a shock wave which propagates undistorted in a dispersive nonlinear medium. Interestingly enough, when gain and loss terms are added to the NSE with the Raman term, the kink solution can still exist. In contrast to the NSE case, the front moves with a certain velocity which depends on the parameters of the equation. In the present work we consider moving front solutions for the modified Ginzburg-Landau equation. We have devel- oped a special technique to find the solution in analytic form which is based on energy and momentum balance equations @9#. In order to do this, we use the ansatz which follows from the point symmetry of the equation and use this ansatz in the energy and balance equations. Allowing for nonzero velocity generalizes this ansatz to cover the case of moving fronts. This technique allows us to find the coefficients of the solu- tion in terms of the equation coefficients. We consider here two examples which allow us to obtain the solution this way: the CGLE with the Raman term and the CGLE with quintic terms. Other cases can also be considered and exact solutions can be found.