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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]

Variational method is important since it can reduce the order of the differential equation to make the equation simpler and obtain the optimal solution by the stationary condition. In this paper, we mainly study the complex nonlinear Fokas–Lenells equation (CNFLE), which is used to describe the propagation of short pulses in optical fibers. The traveling wave transformation is used to convert the CNFLE into the ODE, and the variational principle of the solution is obtained by the semi‐inverse method. Based on the variational principle, the extended He′s variational method, which is based on the Ritz‐like method, is employed to investigate the bright soliton, bright‐like soliton, bright–dark soliton, and periodic wave solution. The absolute, real, and imaginary parts of the novel computational solutions are plotted through one example in the form of 3‐D and 2‐D contours. In addition, the physical explanation of the solutions is elaborated in detail. The results reveal that the variational method is straightforward, simple, and effective, which is expected to bring a light to the study of the traveling wave theory. [ABSTRACT FROM AUTHOR]

The foam drainage mainly depends on the influence of gravity and foam surface tension. In this work, we consider the foam drainage in a microgravity condition, and its fractal modification is established by using the fractal derivative. The variational principle of the novel fractal model is successfully obtained by the semi‐inverse method. Finally, an analytical solution is obtained by the variational iteration method; the effect of the fractal order on the solution property is illustrated. [ABSTRACT FROM AUTHOR]