Snakelike self-similar solutions in a graded-index grating waveguide amplifier with an external source

We present a series of snakelike self-similar solutions to the nonlinear Schrödinger equation with a gain in the presence of an external source through asymmetric dual-core waveguide amplifiers with a long-period grating. The asymmetric dual-core waveguide is composed of two adjoining, closely spaced, upper and lower waveguides, in which the upper one is an active graded-index waveguide with a long-period grating along the medium, while the lower one is a passive step-index waveguide. The results show that: (1) the self-similar optical structures can be generated and effectively controlled by modulating the amplitude of the source; (2) the long-period grating can change the motion of the solutions and preserve their shapes. These results may raise the possibility of some experiments and potential applications in the graded-index grating waveguide amplifier with an external source.