In this paper, we mainly study the time-space fractional strain wave equation in microstructured solids. He’s variational method, combined with the two-scale transform are implemented to seek the solitary and periodic wave solutions of the time-space strain wave equation. The main advantage of the variational method is that it can reduce the order of the differential equation, thus simplifying the equation, making the solving process more intuitive and avoiding the tedious solving process. Finally, the numerical results are shown in the form of 3D and 2D graphs to prove the applicability and effectiveness of the method. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations in physics.

在本文中，我们主要研究微结构固体中的时空分数应变波方程。将他的变分方法与两尺度变换相结合，以寻求时空应变波动方程的孤立波和周期波解。变分法的主要优点是可以减少微分方程的阶数，从而简化了方程，使求解过程更加直观，避免了繁琐的求解过程。最后，数值结果以3D和2D图形的形式显示，证明了该方法的适用性和有效性。在这项工作中获得的结果有望为物理学中的分数阶非线性偏微分方程的研究提供一个亮点。