Dimensional Variable-coefficient Kadomtsev Petviashvili Equation Research Articles

Controllable transformed waves of a (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids or plasma

In this paper, we study the modulations of nonlinear transformed waves for a (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids or plasma. By virtue of the phase shift analysis, the shape-changed and unchanged transformed waves are investigated, which shows the inhomogeneity can restrain the time-varying property. The deformation of waves is determined by the phase difference between two wave components. In addition, the evolutions of parabolic transformed waves are illustrated via characteristic lines analysis. The interactions are further explored, which involve the long- and short-lived collisions. In particular, we discuss the dynamics of unidirectional and reciprocating molecular waves based on the velocity resonance condition, including the shape-changed and unchanged atoms. Different from previous results, certain new types of transformed molecular waves with shape-unchanged atoms are discovered. Our results indicate that the inhomogeneity can produce novel transformed waves and further facilitate the modulation of phase transition mechanism.

Hybrid-wave solutions for a (2 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics and plasma physics

In this paper, a (2 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics and plasma physics is studied. Gram-type solutions are derived via the bilinear Kadomtsev-Petviashvili hierarchy reduction method. Taking different parameter conditions in the Gram-type solutions, we construct the Y-shaped breather solutions and two types of the hybrid-wave solutions. Asymptotic forms for the aforementioned solutions are given. Based on the asymptotic forms, influences of the variable coefficients on the interactions of the breathers and solitons are studied. We obtain three types of the hybrid-wave solutions, which consist of several breathers and solitons. When those breathers and solitons interact, they form the evolving polyhedral arrangement. Changes of the entire arrangement of the breathers and solitons, and the processes of fission or fusion, are discussed and presented.

On the study of the higher-order and multiple lump/rogue waves to the (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation

In the text, rational solutions to a (3 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in determinant form are solved by means of the Kadomtsev-Petviashvili hierarchy reduction method and Hirota’s bilinear form. The first-order, higher-order, multi-lump waves and multi-rogue waves on the basis of the solution in Gramian are presented in graphs. Periodic and parabolic line rogue waves are obtained by choosing different dispersion coefficient functions. The second-order rogue waves are also depicted, which can be seen as the composition of two first-order rogue waves. For multiple rogue waves, taking the periodic and parabolic line rogue waves as examples, it can be observed the change tendency of amplitudes of the intersection region is different from that in branches of line rogue waves. Besides, propagation of the first-order lump wave, interaction between one-lump-soliton and one-rogue-wave, and two kinds of collisions of two lump waves, i.e., elastic and inelastic, are also shown in the plane.

Application of the polynomial function method to the variable-coefficient Kadomtsev–Petviashvili equation

In this paper, we research a (2+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics. The lump, lump-soliton and lump-periodic solutions are derived based on the variable-coefficient polynomial function method. The effect of variable coefficients on the amplitude and velocity of solitons is analyzed and shown by some 3D graphs, contour plots and density graphs.

Multiple rogue wave solutions of a generalized (3+1)-dimensional variable-coefficient Kadomtsev--Petviashvili equation

Abstract Kadomtsev–Petviashvili equation is used for describing the long water wave and small amplitude surface wave with weak nonlinearity, weak dispersion, and weak perturbation in fluid mechanics. Based on the modified symbolic computation approach, the multiple rogue wave solutions of a generalized (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation are investigated. When the variable coefficient selects different functions, the dynamic properties of the derived solutions are displayed and analyzed by different three-dimensional graphics and contour graphics.

Nonlinear dynamics for different nonautonomous wave structures solutions of a (4+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

Nonlinear dynamics for different nonautonomous wave structures solutions of a (4+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

Studies on the breather solutions for the $$\mathbf{(2+1)}$$-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids and plasmas

Studies on the breather solutions for the $$\mathbf{(2+1)}$$-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids and plasmas

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

In this paper, we introduce a new nonlinear evolution equation, which is ([Formula: see text])-dimensional variable-coefficient Kadomtsev–Petviashvili equation. First, according to the Hirota bilinear method, we get some exact solutions of the equation, including lump solution, lump-soliton solution, rogue-soliton solution and lump-kink solution. Then, we obtain some new exact solutions by generalizing the form of the lump solution on a further solution. Finally, based on the symbolic calculation method with Mathematica, the characteristics of the interaction solutions are shown in the graphs and we analyze the dynamic change of the solutions. Furthermore, we discuss the applications of these solutions in physics via the analysis.

Variable-coefficient symbolic computation approach for finding multiple rogue wave solutions of nonlinear system with variable coefficients

In this paper, a variable-coefficient symbolic computation approach is proposed to solve the multiple rogue wave solutions of nonlinear equation with variable coefficients. As an application, a (2+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation is investigated. The multiple rogue wave solutions are obtained and their dynamics features are shown in some 3D and contour plots.

One-, two- and three-soliton, periodic and cross-kink solutions to the (2 + 1)-D variable-coefficient KP equation

This paper deals with [Formula: see text]-soliton solution of the (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation by virtue of the Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, periodic and cross-kink solutions, which have been investigated by the approach of the bilinear method. Mainly, by choosing specific parameter constraints in the [Formula: see text]-soliton solutions, all cases of the periodic and cross-kink solutions can be captured from the one-, two- and three-soliton solutions. The obtained solutions are extended with numerical simulation to analyze graphically, which results into one-, two- and three-soliton solutions and also periodic and cross-kink solutions profiles. That will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics and so on.

Gramian solutions and soliton interactions for a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in a plasma or fluid

Plasmas and fluids are of current interest, supporting a variety of wave phenomena. Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation for the nonlinear phenomena in a plasma or fluid. Based on the existing bilinear form, N-soliton solutions in the Gramian are derived, where N = 1, 2, 3…. With N = 3, three-soliton solutions are constructed. Fission and fusion for the three solitons are presented. Effects of the variable coefficients, i.e. h(t), l(t), q(t), n(t) and m(t), on the soliton fission and fusion are revealed: soliton velocity is related to h(t), l(t), q(t), n(t) and m(t), while the soliton amplitude cannot be affected by them, where t is the scaled temporal coordinate, h(t), l(t) and q(t) give the perturbed effects, and m(t) and n(t), respectively, stand for the disturbed wave velocities along two transverse spatial coordinates. We show the three parallel solitons with the same direction.

Lumps and rouge waves for a $$(3+1)$$ ( 3 + 1 ) -dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

In this paper, a $$(3+1)$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation, which describes the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in fluid mechanics, is investigated. Lump, lump–soliton and rouge–soliton solutions are obtained with the aid of symbolic computation. For the lump and soliton, amplitudes are related to the nonlinearity coefficient and dispersion coefficient, while velocities are related to the perturbation coefficients. Fusion and fission phenomena between the lump and soliton are observed, respectively. Graphic analysis shows that: (i) soliton’s amplitude becomes larger after the fusion interaction, and becomes smaller after the fission interaction; (ii) after the interaction, the soliton propagates along the opposite direction to before when any one of the perturbation coefficients is a time-dependent function. For the interactions between the rogue wave and two solitons, the rogue wave splits from one soliton and merges into the other one, and the two solitons exchange the amplitudes through the energy transfer by the rogue wave.

Fusion and fission phenomena for the soliton interactions in a plasma

Investigation in this paper is given to a generalized (3 + 1) -dimensional variable-coefficient Kadomtsev-Petviashvili equation for a plasma. Via the bilinear form, the singular and double Wronskian soliton solutions are derived, respectively, under the different variable-coefficient constraints. Interactions between the two solitons are depicted, where the soliton fusion and fission phenomena are respectively pictured out, both for the velocity-unvarying and velocity-varying two solitons. Soliton velocity is related to the variable coefficients h(t), l (t), q(t), m(t) and n(t), while the soliton amplitude is not affected by them, where h(t), l(t) and q(t) are the perturbed effects, m(t) and n(t) stand for the disturbed wave velocities along the transverse spatial coordinates.

Solitons and rouge waves for a generalized ([formula omitted])-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

Evolution of the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in fluid mechanics in three spatial dimensions can be described by a generalized (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation, which is studied in this paper with symbolic computation. Via the truncated Painlevé expansion, an auto-Bäcklund transformation is derived, based on which, under certain variable-coefficient constraints, one-soliton, two-soliton, homoclinic breather-wave and rouge-wave solutions are respectively obtained via the Hirota method. Graphic analysis shows that the soliton propagates with the varying soliton direction. Change of the value of any one of g(t), m(t), n(t), h(t), q(t) and l(t) in the equation can cause the change of the soliton shape, while the soliton amplitude cannot be affected by that change, where g(t) represents the dispersion, m(t) and n(t) respectively stand for the disturbed wave velocities along the y and z directions, h(t), q(t) and l(t) are the perturbed effects, y and z are the scaled spatial coordinates, and t is the temporal coordinate. Soliton direction and type of the interaction between the two solitons can vary with the change of the value of g(t), while they cannot be affected by m(t), n(t), h(t), q(t) and l(t). Homoclinic breather wave and rouge wave are respectively displayed, where the rouge wave comes from the extreme behaviour of the homoclinic breather wave.

Shock wave solution of the variable coefficient nonlinear partial differential equations

In this paper, we consider a variable coefficient Calogero–Degasperis equation, a variable coefficient potential Kadomstev–Petviashvili equation and the generalized (3+1)‐dimensional variable coefficient Kadomtsev–Petviashvili equation with time‐dependent coefficients. Shock wave solutions for the considered models are obtained by using ansatz method in the form of tanhp function. The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Copyright © 2016 John Wiley & Sons, Ltd.

Direct Similarity Reduction and New Exact Solutions for the Variable-Coefficient Kadomtsev–Petviashvili Equation

Abstract In this paper, the generalized (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation (VCKPE), which can describe nonlinear phenomena in fluids or plasmas, is studied by using two different Clarkson and Kruskal (CK) direct methods, namely, the classical CK and the modified enlarged CK method. A similarity reduction to a (2+1)-dimensional nonlinear partial differential equation and a direct similarity reduction to a nonlinear ordinary differential equation are obtained, respectively. By solving the reduced ordinary differential equation, new solitary, periodic, and singular solutions for the VCKPE are obtained. Some figures for the soliton and periodic wave solutions are given to reflect the effect of the variable coefficients on the solution propagation. Finally, the comparison between the two different CK techniques indicates that the modified enlarged CK technique is clearly more powerful and simple than the classical CK technique.

Soliton Solutions, Bäcklund Transformations and Lax Pair for a (3 + 1)-Dimensional Variable-Coefficient Kadomtsev—Petviashvili Equation in Fluids

Under investigation in this paper is a (3 + 1)-dimensional variable-coefficient Kadomtsev—Petviashvili equation, which describes the propagation of surface and internal water waves. By virtue of the binary Bell polynomials, symbolic computation and auxiliary independent variable, the bilinear forms, soliton solutions, Bäcklund transformations and Lax pair are obtained. Variable coefficients of the equation can affect the solitonic structure, when they are specially chosen, while curved and linear solitons are illustrated. Elastic collisions between/among two and three solitons are discussed, through which the solitons keep their original shapes invariant except for some phase shifts.

Exact Solutions to a (3+1)-Dimensional Variable-Coefficient Kadomtsev–Petviashvilli Equation via the Bilinear Method and Wronskian Technique

By truncating the Painlevé expansion at the constant level term, the Hirota bilinear form is obtained for a (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation. Based on its bilinear form, solitary-wave solutions are constructed via the ∊-expansion method and the corresponding graphical analysis is given. Furthermore, the exact solution in the Wronskian form is presented and proved by direct substitution into the bilinear equation.