Generalized Variational Principle for the Fractal (2 + 1)-Dimensional Zakharov–Kuznetsov Equation in Quantum Magneto-Plasmas

The unsmooth boundary will greatly affect the motion morphology of ion-acoustic waves in plasma, so a modified equal width equation with fractal derivatives is proposed. The fractal variational formulation of the problem is established by using the semi-inverse method, which provides the conservation laws in an energy form in the fractal space and reveals the possible solution structures of the equation. Then He’s variational method based on the variational theory and Ritz-like method, combined with the two-scale transform is used to seek the periodic and solitary wave solutions of the fractal modified equal width equation. The obtained results show that the variational method is simple but powerful, which is expected to open some new perspectives toward the study of traveling wave theory in fractal space.

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A New Perspective on the Study of the Fractal Coupled Boussinesq–burger Equation in Shallow Water

Wang

Zhu

2021

Fractals

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The well-known coupled Boussinesq–Burger equation can be used to describe the flow of the shallow water of the harbor. But when the boundary is nonsmooth, it becomes powerless. So, the fractal calculus is needed to be applied to it and the fractal coupled Boussinesq–Burger equation (FCBBe) is presented for the first time in this paper. By using the semiinverse method, we have successfully established the fractal variational formulation of the FCBBe, which can not only provide the conservation laws in an energy form in the fractal space but also reveal the possible solution structures of the equation. Then a novel variational approach based on He’s variational method and the two-scale transform are used to seek its periodic wave solutions. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. Finally, the numerical results have been shown through graphs to discuss the effect of different fractal orders on the wave motion. The obtained results in this work are expected to shed a bright light on the study of fractal nonlinear partial differential equations in the fractal space.

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Generalized Variational Principle for the Fractal (2 + 1)-Dimensional Zakharov–Kuznetsov Equation in Quantum Magneto-Plasmas

Cited by 4 publications

References 52 publications

Conservation laws, exact solutions and stability analysis for time-fractional extended quantum Zakharov–Kuznetsov equation

Conservation laws, exact solutions and stability analysis for time-fractional extended quantum Zakharov–Kuznetsov equation

Variational Principle, Solitary and Periodic Wave Solutions of the Fractal Modified Equal Width Equation in Plasma Physics

A New Perspective on the Study of the Fractal Coupled Boussinesq–burger Equation in Shallow Water

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