This work is reported the analytical solutions for describing the nonlinear directional couplers with metamaterials by including spatial–temporal fractional beta derivative evolution. The auxiliary ordinary differential equation method and the generalized Riccati method with the properties of beta derivative are implemented to secure such solutions. The solutions are obtained in the new forms by involving of some useful mathematical functions. The constraint conditions among the traveling wave parameters are evaluated. Some of the obtained solutions are presented graphically to illustrate the effectiveness of beta derivative parameter and mathematical techniques. It is investigated that the amplitudes of soliton are increased with the increase of fractional beta derivative parameter. The obtained results would be very useful to examine and understand the physical issues in nonlinear optics, especially in twin-core couplers with optical metamaterials.

这项工作被报道了通过包括时空分数β导数演化来描述具有超材料的非线性定向耦合器的解析解决方案。实现具有β导数性质的辅助常微分方程方法和广义Riccati方法以确保这种解。通过涉及一些有用的数学函数，可以以新形式获得解决方案。评估行波参数之间的约束条件。以图形方式介绍了一些获得的解决方案，以说明β导数参数和数学技术的有效性。研究表明，孤子的振幅随分数β导数参数的增加而增加。