Ultrashort self-similar periodic waves and similaritons in an inhomogeneous optical medium with an external source and modulated coefficients

In this paper, we use the similarity transformation to reduce the generalized higher-order nonlinear Schrodinger equation with space- and time-modulated coefficients incorporating group velocity dispersion, self-phase modulation, self-steepening, self-frequency shift, gain or loss term, differential gain or loss term, group velocity of the modes, and inhomogeneous source to Hirota equation with a varying source. Afterward, we reduce this Hirota equation to a second-order nonhomogeneous nonlinear ordinary differential equation with constant source via a plane wave transformation and some constraint conditions. Finally, by using Mobius transformation, various types of exact self-similar femtosecond solutions are deduced such as the bright and dark-type solitons with W-shaped profiles, the bell-shaped bright solitons, and the periodic wave solutions. The evolutional dynamics of these self-similar structures are investigated in periodic distributed system. Our analysis shows that a suitable choice of the amplitude of the source term, the gain or loss parameter, and the differential gain or loss parameter allows us to control the similariton structures. In addition, the stability analysis of the solutions is discussed numerically. Our results may be useful to explain some nonlinear wave phenomena in nonlinear optics and related fields.

Automated summary

In this paper, the generalized higher-order nonlinear Schrodinger equation is reduced to Hirota equation using similarity transformation. Various types of exact self-similar femtosecond solutions are then obtained. The results show that the shape of similariton structures can be controlled by adjusting the amplitude of the source term and the gain or loss parameter.