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.

中文翻译：

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具有外部源的渐变折射率光栅波导放大器中的蛇形自相似解

我们为非线性Schrödinger方程提供了一系列类似蛇形的自相似​​解，并且在存在外部源的情况下，通过具有长周期光栅的非对称双芯波导放大器来获得增益。非对称双芯波导由两个相邻的，间隔紧密的上下波导组成，其中上一个是有源渐变折射率波导，沿着介质具有长周期光栅，下一个是无源阶梯折射率波导。结果表明：（1）通过调制光源的振幅可以产生并有效地控制自相似的光学结构；（2）长周期光栅可以改变溶液的运动并保持其形状。