Abstract
In this paper, we establish exact solutions for the non-linear coupled KdV equations. The exp-function method is used to construct the solitary travelling wave solutions for these equations. The numerical adaptive moving mesh PDEs (MMPDEs) method is also implemented in order to solve the proposed coupled KdV equations. The achieved results may be applicable to some plasma environments, such as ionosphere plasma. Some numerical simulations compared with the exact solutions are provided to illustrate the validity of the proposed methods. Furthermore, the modulational instability is analyzed based on the standard linear-stability analysis. The depiction of the techniques are straight, powerful, robust and can be applied to other nonlinear systems of partial differential equations.
Highlights
The nonlinear partial differential equations (NPDEs) are widely used to express a variety of physical circumstances in fields such as biology, economy, engineering, elastic media, meteorology, plasma physics, optics, fluid mechanics and chemical physics, see [1,2,3,4,5,6,7,8,9,10,11,12]
In order to reduce Equation (3) to an ordinary differential equation (ODE), we introduce the transformation ξ = x ± wt, φ( x, t) = φ0 (ξ ), (4)
We examine the stability of the achieved traveling wave solutions by using the Hamiltonian system form [36,37], which is ρ1 ( w ) =
Summary
The nonlinear partial differential equations (NPDEs) are widely used to express a variety of physical circumstances in fields such as biology, economy, engineering, elastic media, meteorology, plasma physics, optics, fluid mechanics and chemical physics, see [1,2,3,4,5,6,7,8,9,10,11,12]. Many authors mainly had paid attention to study solutions of coupled equations by using various methods. Among these are the trigonometric function transform method [30], F-expansion method [13], the homotopy perturbation [31] and differential transformation Method [32]. We achieve some new solutions for the coupled KdV Equation (1), utilizing the exp-function method [17,35]. We implement the adaptive moving mesh PDEs (MMPDEs) scheme to solve the coupled KdV Equation (1). We compare these numerical solutions with the exact solutions obtained by the exp-function method. For more details about the stability of the exact solutions, we refer to [37,38,39] and references therein
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