Journal ArticleDOI
Extended generalized \((Zakh\frac{G^{\prime }}{G})\)-expansion method for solving the nonlinear quantum Zakharov–Kuznetsov equation
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In this paper, the authors applied the extended generalized (G^{G^{\prime }}{G})-expansion method combined with the Jacobi elliptic equation to find new exact solutions of the nonlinear quantum Zakharov-Kuznetsov (QZK) equation with the aid of computer algebraic system Maple.Abstract:
In this article, we apply the extended generalized \((\frac{G^{\prime }}{G})\)-expansion method combined with the Jacobi elliptic equation to find new exact solutions of the nonlinear quantum Zakharov–Kuznetsov (QZK) equation with the aid of computer algebraic system Maple. Soliton solutions, periodic solutions, rational functions solutions and Jacobi elliptic functions solutions are obtained. Based on reductive perturbation technique and a series of transformation, the nonlinear QZK had been derived by many authors which can be reduced to a nonlinear ordinary differential equation (ODE) using the wave transformation. The extended generalized \((\frac{G^{\prime }}{G})\)-expansion method is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics.read more
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The (G′/G)-expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines
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TL;DR: In this article, the generalized Riccati equation (G′/G)-expansion method was applied based on three auxiliary equations, namely, the Jacobi Elliptic Equation (JE), Jacobi Eq.
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Solitons and Other Exact Solutions for Two Nonlinear PDEs in Mathematical Physics Using the Generalized Projective Riccati Equations Method
TL;DR: In this paper, the generalized projective Riccati equations method with the aid of Maple software is applied to construct many new soliton and periodic solutions with parameters for two higher-order nonlinear partial differential equations (PDEs), namely, the nonlinear Schrodinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity and the quantum Zakharov-Kuznetsov (QZK) equation.
References
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Book
Solitons, Nonlinear Evolution Equations and Inverse Scattering
M. A. Ablowitz,Peter A. Clarkson +1 more
TL;DR: In this article, the authors bring together several aspects of soliton theory currently only available in research papers, including inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multidimensional space, and the ∂ method.
Journal ArticleDOI
Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons
TL;DR: An exact solution for the Korteweg-de Vries equation for the case of multiple collisions of $N$ solitons with different amplitudes was obtained in this paper, which is the only known exact solution.
Journal ArticleDOI
The Painlevé property for partial differential equations
TL;DR: In this paper, the authors define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well-known partial differential equation (Burgers' equation, KdV equation, and modified KDV equation).
Journal ArticleDOI
Extended tanh-function method and its applications to nonlinear equations
TL;DR: In this article, an extended tanh-function method is proposed for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way, and the key idea of this method is to take full advantage of a Riccati equation involving a parameter and use its solutions to replace the tanh function.