Computational soliton solutions to (3+1)(3+1)−dimensionalgeneralisedKadomtsev–Petviashviliand( 3 + 1 ) -dimensional generalised Kadomtsev–Petviashvili and(3+1)−dimensionalgeneralisedKadomtsev–Petviashviliand(2+1)$$ ( 2 + 1 ) -dimensional Gardner–Kadomtsev–Petviashvili models and their applications (original) (raw)
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Some New Exact Traveling Wave Solutions to the (3+1)-dimensional Kadomtsev-Petviashvili equation
2012
Mathematical modeling of numerous physical phenomena often leads to high-dimensional partial differential equations and thus the higher dimensional nonlinear evolution equations come into further attractive in many branches of physical sciences. In this article, we propose a new technique of the (G′/G)-expansion method combine with the Riccati equation for searching new exact traveling wave solutions of the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation. Consequently, some new solutions of the KP are successfully obtained in a unified way involving arbitrary parameters. When the parameters take special values, solitary waves are derived from the traveling waves. The obtained solutions are expressed by the hyperbolic, trigonometric and rational functions. The method can be applied to many other nonlinear partial differential equations.
A Soliton Solution for the Kadomtsev–Petviashvili Model Using Two Novel Schemes
Symmetry
Symmetries are crucial to the investigation of nonlinear physical processes, particularly the evaluation of a differential problem in the real world. This study focuses on the investigation of the Kadomtsev–Petviashvili (KP) model within a (3+1)-dimensional domain, governing the behavior of wave propagation in a medium characterized by both nonlinearity and dispersion. The inquiry employs two distinct analytical techniques to derive multiple soliton solutions and multiple solitary wave solutions. These methods include the modified Sardar sub-equation technique and the Darboux transformation (DT). The modified Sardar sub-equation technique is used to obtain multiple soliton solutions, while the DT is introduced to develop two bright and two dark soliton solutions. These solutions are presented alongside their corresponding constraint conditions and illustrated through 3-D, 2-D, and contour plots to physically portray the derived solutions. The results demonstrate that the employed an...
2013
In this article, the improved ()/G G ′-expansion method has been implemented to generate travelling wave solutions, where ()G ξ satisfies the second order linear ordinary differential equation. To show the advantages of the method, the (3+1)-dimensional Kadomstev-Petviashvili (KP) equation has been investigated. Higher-dimensional nonlinear partial differential equations have many potential applications in mathematical physics and engineering sciences. Some of our solutions are in good agreement with already published results for a special case and others are new. The solutions in this work may express a variety of new features of waves. Furthermore, these solutions can be valuable in the theoretical and numerical studies of the considered equation. Also, in order to understand the behaviour of solutions, the graphical representations of some obtained solutions have been presented.
Romanian Reports in Physics, 2013
This paper studies the (3+1)-dimensional extended Kadomtsev-Petviashvili equation with power law nonlinearity that apperas in the study of multi-component plasmas. The solutions are obtained by several methods such as modified F-expansion method, exp-function method, / G G ′ expansion method, ansatz method, traveling wave hypothesis, the improved Jacobi's elliptic function method and Lie symmetry analysis. These method lead to several closed form exact solutions. Some of these solutions are topological, non-topological and singular solitons, cnoidal, snoidal waves. It is also shown that in the limiting case, these doubly periodic functions lead to singular periodic functions, complexitons and linear waves. The domain restrictions are also identfified in order for the soliton solutions to exist.
Filomat
The Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) model equations as a water wave model, are governing equations, for fluid flows, describes bidirectional propagating water wave surface. The soliton solutions for (2+1) and (3+1)-Dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equations have been extracted. The solitary wave ansatz method are adopted to approximate the solutions. The corresponding integrability criteria, also known as constraint conditions, naturally emerge from the analysis of the problem.
Journal of Applied Mathematics, 2012
The two variables -expansion method is proposed in this paper to construct new exact traveling wave solutions with parameters of the nonlinear -dimensional Kadomtsev-Petviashvili equation. This method can be considered as an extension of the basic -expansion method obtained recently by Wang et al. When the parameters are replaced by special values, the well-known solitary wave solutions and the trigonometric periodic solutions of this equation were rediscovered from the traveling waves.
The Improved (G’/G)-Expansion Method to the (3+1)-Dimensional Kadomstev-Petviashvili Equation
American Journal of Applied Mathematics and Statistics, 2013
In this article, the improved ( ) / G G ′ -expansion method has been implemented to generate travelling wave solutions, where ( ) G ξ satisfies the second order linear ordinary differential equation. To show the advantages of the method, the (3+1)-dimensional Kadomstev-Petviashvili (KP) equation has been investigated. Higherdimensional nonlinear partial differential equations have many potential applications in mathematical physics and engineering sciences. Some of our solutions are in good agreement with already published results for a special case and others are new. The solutions in this work may express a variety of new features of waves. Furthermore, these solutions can be valuable in the theoretical and numerical studies of the considered equation. Also, in order to understand the behaviour of solutions, the graphical representations of some obtained solutions have been presented. Keywords: the improved ( ) '/ G G -expansion method, the Kadomstev-Petviashvili equation, traveling wave solutions, nonlinear evolution equations Cite This Article: Hasibun Naher, and Farah Aini Abdullah, "The Improved ( ) '/ G G -Expansion Method to the (3+1)-Dimensional Kadomstev-Petviashvili Equation.
GG -Expansion Method to the (3+1)- Dimensional Kadomstev-Petviashvili Equation
2013
In this article, the improved ( ) / GG ' -expansion method has been implemented to generate travelling wave solutions, where ( ) G ξ satisfies the second order linear ordinary differential equation. To show the advantages of the method, the (3+1)-dimensional Kadomstev-Petviashvili (KP) equation has been investigated. Higher- dimensional nonlinear partial differential equations have many potential applications in mathematical physics and engineering sciences. Some of our solutions are in good agreement with already published results for a special case and others are new. The solutions in this work may express a variety of new features of waves. Furthermore, these solutions can be valuable in the theoretical and numerical studies of the considered equation. Also, in order to understand the behaviour of solutions, the graphical representations of some obtained solutions have been presented.
Physica Scripta, 2012
We give an introduction to a new direct computational method for constructing multiple soliton solutions to nonlinear equations with variable coefficients in the Kadomtsev-Petviashvili (KP) hierarchy. We discuss in detail how this works for a generalized (3 + 1)-dimensional KP equation with variable coefficients. Explicit soliton, multiple soliton and singular multiple soliton solutions of the equation are obtained under certain constraints on the coefficient functions. Furthermore, the characteristic-line method is applied to discuss the solitonic propagation and collision under the effect of variable coefficients.