The nonlinear fractional differential equations (FDEs) are composed by mathematical modeling through nonlinear corporeal structures. The study of these kinds of models has an energetic position in different fields of applied sciences. In this study, we observe the dynamical behavior of nonlinear traveling waves for the $M$-fractional $(3+1)$-dimensional Wazwaz–Benjamin–Bona–Mohany (WBBM) equation. Novel exact traveling wave solutions in the form of trigonometric, hyperbolic and rational functions are derived using $(1/G^{\prime })$, modified $(G^{\prime }/G^2)$ and new extended direct algebraic methods with the help of symbolic soft computation. We guarantee that all the obtained results are new and verified the main equation. To promote the essential propagated features, some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the precise values to the parameters under the constrain conditions, and this provides useful information about the dynamical behavior. Further, bifurcation behavior of nonlinear traveling waves of the proposed equation is studied with the help of bifurcation theory of planar dynamical systems. It is also observed that the proposed equation support the nonlinear solitary wave, periodic wave, kink and antikink waves and most important supernonlinear periodic wave.

非线性分数阶微分方程 (FDE) 是通过非线性物质结构的数学建模组成的。对这些模型的研究在不同的应用科学领域都具有重要的地位。在这项研究中，我们观察了非线性行波的动力学行为$米$-分数$(\mathrm{3个}+\mathrm{1个})$维 Wazwaz–Benjamin–Bona–Mohany (WBBM) 方程。以三角函数、双曲函数和有理函数的形式导出了新的精确行波解，使用$(\mathrm{1个}/G^{‘})$， 修改的$(G^{‘}/G^{\mathrm{2个}})$以及借助符号软计算的新扩展直接代数方法。我们保证所有获得的结果都是新的，并验证了主方程。为了促进基本的传播特征，一些研究的解决方案通过在约束条件下将精确值传递给参数以 2D 和 3D 图形的形式展示，这提供了有关动态行为的有用信息。此外，借助平面动力系统的分岔理论，研究了所提出方程的非线性行波的分岔行为。还观察到所提出的方程支持非线性孤立波、周期波、扭结波和反扭结波以及最重要的超非线性周期波。