Lumps and interaction solutions to the (4 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

Fan, Lulu; Bao, Taogetusang

doi:10.1142/s0217979221502337

Cited by 11 publications

(3 citation statements)

References 39 publications

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“…Using Hirota bilinear method [9][10][11] as a tool, the solutions of nonlinear evolution equations can be obtained. Besides, there are many methods to construct solutions of nonlinear evolution equations, such as backscattering method [12][13][14], Darboux transform method [15][16][17], variable separation method [18][19][20], Bäcklund transform method [21], auxiliary equation method [22][23][24][25][26] and so on.…”

Section: Introductionmentioning

confidence: 99%

Weierstrass elliptic function solutions and degenerate solutions of a variable coefficient higher-order Schrödinger equation

Fan,

Bao

2023

Phys. Scr.

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In this paper, the auxiliary equation method is used to study the Weierstrass elliptic function solutions and degenerate solutions of the variable coefficient higher order Schrödinger equation, including Jacobian elliptic function solutions, trigonometric function solutions and hyperbolic function solutions. The types of solutions of the variable coefficient higher-order Schrödinger equation are enriched, and the method of seeking precise and accurate solutions is extended. It is concluded that the types of degenerate solutions are related to the coefficients of the equation itself when the degenerate solutions are obtained from the solutions of the Weierstrass elliptic functions. In addition, the solutions form of the equation is extended from the power series expansion form to the Laurent series expansion form, and the corresponding solutions are obtained. After the conversion formula between the Weierstrass elliptic function solutions and the Jacobian elliptic function solutions is constructed, the Jacobian elliptic function solutions of the higher order Schrödinger equation with variable coefficients are also obtained. These have not been previously studied.

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Section: Introductionmentioning

confidence: 99%

Weierstrass elliptic function solutions and degenerate solutions of a variable coefficient higher-order Schrödinger equation

Fan,

Bao

2023

Phys. Scr.

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“…The research of variable-coefficient equations is gradually becoming popular [24,25], because we can obtain more other equations by taking different values of coefficients, and then generalize the conclusions.…”

Section: Introductionmentioning

confidence: 99%

Superposition solutions to a (3+1)-dimensional variable-coefficient Sharma-Tasso-Olver-Like equation

Fan¹,

Bao²

2022

Phys. Scr.

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In this paper, the superposition solutions of (3+1)-dimensional variable-coefficient Sharma-Tasso-Olver-Like(vcSTOL) equation are studied. The equation can illustrate various difficult sciences areas. Due to the wide application, it is very important to find the exact solutions of it. By introducing transformation, the equation is transformed into bilinear form. We use variable separation method and trial function method to obtain the superposition solutions of the equation containing different functions and forms. The images are drawn with the help of symbolic computing system Mathematica, and the properties of the solutions are analyzed. The analysis shows that different functions will affect the overall shape of waves, including the interaction between waves, the size, the direction and the number of waves, which can get more new phenomena. To our knowledge, those types of superposition solutions of (3+1)-dimensional vcSTOL equation mentioned in our work by variable separation method have not been reported before. Furthermore, we add the square terms to the expansion function, so that the obtained solutions have the characteristics of Lump solution, which has not been done in the previous literatures.

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“…Due to the dimensions of equations increasing, which causes difficulties in the analysis and calculation of high-dimensional partial differential equations, there are more studies on low-dimensional problems than on high-dimensional problems. However, the real world is (3+1)-dimensional, thus high-dimensional equations have important applications in real-world problems [1][2][3]. For the past few years, more and more scholars have begun to pay attention to high-dimensional integrable equations in the integrable field.…”

Section: Introductionmentioning

confidence: 99%

Analytical and Numerical Solutions for a Kind of High-Dimensional Fractional Order Equation

Hou

Zhang

2022

Fractal Fract

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In this paper, a (4+1)-dimensional nonlinear integrable Fokas equation is studied. It is rarely studied because the order of the highest-order derivative term of this equation is higher than the common generalized (4+1)-dimensional Fokas equation. Firstly, the (4+1)-dimensional time-fractional Fokas equation with the Riemann–Liouville fractional derivative is derived by the semi-inverse method and variational method. Further, the symmetry of the time-fractional equation is obtained by the fractional Lie symmetry analysis method. Based on the symmetry, the conservation laws of the time fractional equation are constructed by the new conservation theorem. Then, the G′G-expansion method is used here to solve the equation and obtain the exact traveling wave solutions. Finally, the spectral method in the spatial direction and the Gru¨nwald–Letnikov method in the time direction are considered to obtain the numerical solutions of the time-fractional equation. The numerical solutions are compared with the exact solutions, and the error results confirm the effectiveness of the proposed numerical method.

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Lumps and interaction solutions to the (4 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

Cited by 11 publications

References 39 publications

Weierstrass elliptic function solutions and degenerate solutions of a variable coefficient higher-order Schrödinger equation

Weierstrass elliptic function solutions and degenerate solutions of a variable coefficient higher-order Schrödinger equation

Superposition solutions to a (3+1)-dimensional variable-coefficient Sharma-Tasso-Olver-Like equation

Analytical and Numerical Solutions for a Kind of High-Dimensional Fractional Order Equation

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