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In this article, a new (2 + 1)‐dimensional local fractional breaking soliton equation is derived with the local fractional derivative. Applying the traveling wave transform of the non‐differentiable type, the (2 + 1)‐dimensional local fractional breaking soliton equation is converted into a nonlinear local fractional ordinary differential equation. By defining a set of elementary functions on Cantor sets, a novel analytical technique namely the Mittag–Leffler function‐based method is employed for the first time ever to construct the exact solutions. The solutions on the Cantor sets are presented via the 3‐D contours. It reveals that the proposed method is effective and powerful and is expected to give some inspiration for the study of the local fractional PDEs. [ABSTRACT FROM AUTHOR]

In this current work, we main to find the abundant explicit solutions of the Benney–Luke equation by a novel approach, which is called the variational direct method (VDM). The VDM based on the variational theory and Ritz‐like method can reduce the order of the differential equation to make the equation simpler and obtain the optimal solution by the stationary conditions. The explicit solutions in terms of dark solitary, dark‐like solitary, kinky‐dark solitary, and periodic wave solutions are presented through the numerical results in the form of 3‐D plot, 2‐D curve, and 2‐D contour. The results show that the VDM is simple, straightforward, and powerful, which is expected to open new perspectives for the traveling wave theory. [ABSTRACT FROM AUTHOR]