Time–space fractional ( 2 + 1 ) $(2+1)$ dimensional nonlinear Schrödinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions

Fu, Cuicui; Lu, Chang Na; Yang, Hong

doi:10.1186/s13662-018-1512-3

Cited by 50 publications

(13 citation statements)

References 58 publications

(59 reference statements)

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“…The last step is to plug expressions for F 1 , F 2 given by Equation (24) and fractional potential functions D…”

Section: Definition 2 ([42]mentioning

confidence: 99%

“…Therefore, this paper derives the (2+1)-dimensional coupled generalized ZK(gZK) equations set from the classical quasi-geostrophic vorticity equations [22,23]. The gZK equation is a class of important high-dimensional nonlinear evolution equation in mathematical physics, the gZK equations set as the extension of a single equation can be used to describe the interaction of nonlinear Rossby waves in two-layer fluids [24][25][26].…”

Section: Introductionmentioning

confidence: 99%

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Time-Space Fractional Coupled Generalized Zakharov-Kuznetsov Equations Set for Rossby Solitary Waves in Two-Layer Fluids

Chen

Yang

2019

Mathematics

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In this paper, the theoretical model of Rossby waves in two-layer fluids is studied. A single quasi-geostrophic vortex equation is used to derive various models of Rossby waves in a one-layer fluid in previous research. In order to explore the propagation and interaction of Rossby waves in two-layer fluids, from the classical quasi-geodesic vortex equations, by employing the multi-scale analysis and turbulence method, we derived a new (2+1)-dimensional coupled equations set, namely the generalized Zakharov-Kuznetsov(gZK) equations set. The gZK equations set is an extension of a single ZK equation; they can describe two kinds of weakly nonlinear waves interaction by multiple coupling terms. Then, for the first time, based on the semi-inverse method and the variational method, a new fractional-order model which is the time-space fractional coupled gZK equations set is derived successfully, which is greatly different from the single fractional equation. Finally, group solutions of the time-space fractional coupled gZK equations set are obtained with the help of the improved ( G ′ / G ) -expansion method.

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“…The last step is to plug expressions for F 1 , F 2 given by Equation (24) and fractional potential functions D…”

Section: Definition 2 ([42]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Time-Space Fractional Coupled Generalized Zakharov-Kuznetsov Equations Set for Rossby Solitary Waves in Two-Layer Fluids

Chen

Yang

2019

Mathematics

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“…It has been an important work to study nonlinear PDE [1] due to their rich mathematical structures and features [2][3][4][5] as well as important applications in fluid dynamics and plasma physics [6][7][8][9][10][11][12]. Although many theories and methods were proposed to discuss the PDE [13][14][15][16][17][18][19][20], however, most nonlinear PDE have no analytic solutions; numerical methods are necessary to study hydrodynamic characteristics of PDEs [21][22][23][24]. For the discretization of time derivative in time-dependent PDEs for numerical methods, the well-known ways is the ODE solver, such as multi-step method and Runge-Kutta method.…”

Section: Formulation Of the Problem Of Interest For This Investigationmentioning

confidence: 99%

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

Dong

Yang

2018

Mathematics

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We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes.

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“…Many of the physical processes that have been explored to date are nonconservative. It is important to be able to apply the power of fractional differentiation [8][9][10]. However, because of its nonlocal character, fractional calculus has not been used in physics and engineering.…”

Section: Complexitymentioning

confidence: 99%

Study of Ion‐Acoustic Solitary Waves in a Magnetized Plasma Using the Three‐Dimensional Time‐Space Fractional Schamel‐KdV Equation

Guo

Zhang

et al. 2018

Complexity

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The study of ion-acoustic solitary waves in a magnetized plasma has long been considered to be an important research subject and plays an increasingly important role in scientific research. Previous studies have focused on the integer-order models of ionacoustic solitary waves. With the development of theory and advancement of scientific research, fractional calculus has begun to be considered as a method for the study of physical systems. The study of fractional calculus has opened a new window for understanding the features of ion-acoustic solitary waves and can be a potentially valuable approach for investigations of magnetized plasma. In this paper, based on the basic system of equations for ion-acoustic solitary waves and using multi-scale analysis and the perturbation method, we have obtained a new model called the three-dimensional(3D) Schamel-KdV equation. Then, the integerorder 3D Schamel-KdV equation is transformed into the time-space fractional Schamel-KdV (TSF-Schamel-KdV) equation by using the semi-inverse method and the fractional variational principle. To study the properties of ion-acoustic solitary waves, we discuss the conservation laws of the new time-space fractional equation by applying Lie symmetry analysis and the RiemannLiouville fractional derivative. Furthermore, the multi-soliton solutions of the 3D TSF-Schamel-KdV equation are derived using the Hirota bilinear method. Finally, with the help of the multi-soliton solutions, we explore the characteristics of motion of ionacoustic solitary waves.

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Time–space fractional ( 2 + 1 ) $(2+1)$ dimensional nonlinear Schrödinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions

Cited by 50 publications

References 58 publications

Time-Space Fractional Coupled Generalized Zakharov-Kuznetsov Equations Set for Rossby Solitary Waves in Two-Layer Fluids

Time-Space Fractional Coupled Generalized Zakharov-Kuznetsov Equations Set for Rossby Solitary Waves in Two-Layer Fluids

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

Study of Ion‐Acoustic Solitary Waves in a Magnetized Plasma Using the Three‐Dimensional Time‐Space Fractional Schamel‐KdV Equation

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