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This research mainly concerned with some new solutions of the (3 + 1)‐dimensional nonlinear evolution equation (NEE). First, we extract the resonant multiple soliton solutions (RMSSs) by taking advantage of the linear superposition principle (LSP) and weight algorithm (WA). Then the non‐singular complexiton and singular complexiton solutions are developed by introducing pairs of the conjugate parameters. Besides, the complex multiple‐soliton solutions are also explored with the aid of the bilinear approach. The graph descriptions of the attained solutions are paraded to show the dynamical properties. The outcomes of this work are all new and are desirous to bring some new perspective to the investigation of the complexiton solutions to the other partial differential equations (PDEs). [ABSTRACT FROM AUTHOR]

This research focuses on the nonlinear coupled Davey–Stewartson Fokas system, which models pulse propagation in monomode optical fibers. In order to find the novel periodic and solitary wave solutions of the nonlinear coupled Davey–Stewartson Fokas system, we have employed two effective mathematical techniques named as the Sine‐Gordon expansion method and the simple equation method. We found after researching into previous literature that these novel solutions are unique and have never been reported. Some 3D and 2D graphs are also used to discuss the dynamical behavior of these new solutions. [ABSTRACT FROM AUTHOR]

The well‐known Drinfeld–Sokolov–Wilson system is widely used to describe the flow of the shallow water. This paper considers the time–space fractional Drinfeld–Sokolov–Wilson system, where He's variational method together with the two‐scale transform is applied to find its solitary solutions. The numerical results have been shown through graphs to demonstrate the applicability and efficiency of our approach. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

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]

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]

Yu‐Lan Ma, et al. (Mathematical Methods in the Applied Sciences, 2019,42(1)) make outstanding contributions for the soliton solutions of the generalized fourth‐order Boussinesq equation, which is used to describe the wave motion in fluid mechanics. But the periodic wave solution is not reported. So in this paper, He's variational methods are employed to find the solitary and periodic wave solutions of the generalized fourth‐order Boussinesq equation. The greatest advantage of the variational method is that it can reduce the order of the research equation, make the equation more simple, and obtain the optimal solution. Finally, the numerical results are shown in the form of a graph to prove the applicability and effectiveness of the method. The results reveal that variational method is simple and straightforward and can avoid the tedious calculation process, which is expected to shed a light to the new research frontiers of solitary wave theory. [ABSTRACT FROM AUTHOR]

In this work, a nonlinear oscillator model with a fractal derivative is successfully established for a fractal nano/microelectromechanical system (MEMS), and its variational principle is obtained by the semi‐inverse method. The established variational formulation suggests an energy conservation for the fractal nonlinear oscillator, and its approximate analytical solution is found by the two‐scale transform and homotopy perturbation method; Laplace transform is used in the solution process. This paper sheds a new light on fractal MEMS. [ABSTRACT FROM AUTHOR]

This paper suggests a fractal mathematical model for Fangzhu's water collection by taking into account its nanoscale surface morphology. A fractal variational principle is established by the semi‐inverse method, which elucidates the conservation laws in the fractal space. Additionally, the variational formulation can also reveal the structure of the solution. An approximate analytical solution is obtained via the two‐scale transform method and He's frequency formula. Some special cases of the model are discussed numerically, showing the method is remarkably simple and capable to solve nonlinear oscillators with fractal derivatives. [ABSTRACT FROM AUTHOR]