A numerical solution of the coupled sine-Gordon equation using the modified decomposition method
Introduction
In this paper, we shall consider a system of coupled sine-Gordon equations in the formwhich was introduced recently by Khusnutdinova and Pelinovsky [1]. The coupled sine-Gordon equations generalize the Frenkel–Kontorova dislocation model [2], [3]. The system (1.1) with c = 1 was also proposed to describe the open states in DNA [4].
In Ref. [1], the authors construct the linear and nonlinear wave process involving the exchange of energy between the two physical components of the system. They construct the linear solutions by considering ∣u − w∣ ≪ 1, the exact nonlinear solutions for the case c = 1 and the weakly nonlinear solutions, for general case, by means of asymptotic methods. We used to get the explicit series solutions of the coupled sine-Gordon equations without use of any linearization or transformation method by the Adomian’s decomposition method (in short ADM) [5], [6]. The method presented here is also very easy to use for obtain numerical solutions of the Eq. (1.1) without using any discretization techniques.
In this paper, we use the modified decomposition method (in short MDM) to obtain the numerical solutions of the coupled sine-Gordon Eq. (1.1). Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the Adomian decomposition method [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. A reliable modification of Adomian decomposition method has been done by Wazwaz [27]. The decomposition method provides an effective procedure for analytical solution of a wide and general class of dynamical systems representing real physical problems [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Recently, the implementations of Adomian decomposition method for the solutions of sine-Gordon equation, generalized Hirota–Satsuma coupled KdV equation, generalized regularized long-wave (RLW) and Korteweg–de Vries (KdV) equations have been well established by the notable researchers [13], [14], [15], [16], [17], [18]. This method efficiently works for initial-value or boundary-value problems and for linear or nonlinear, ordinary or partial differential equations and even for stochastic systems. Moreover, we have the advantage of a single global method for solving ordinary or partial differential equations as well as many types of other equations. Recently, the solution of fractional differential equation has been obtained through Adomian decomposition method by the researchers [19], [20], [21], [22], [23], [24], [25], [26]. The method has features in common with many other methods, but it is distinctly different on close examinations, and one should not be mislead by apparent simplicity into superficial conclusions [5], [6].
Section snippets
Analysis of the method
We consider the coupled sine-Gordon equations (1.1) in the operator formwhere , symbolize the linear differential operators and the notation N symbolize the nonlinear operator.
Applying the two-fold integration inverse operator to the system (2.1) and using the specified initial conditions yieldsThe Adomain decomposition method
Implementation of the method
We consider the coupled sine-Gordon equationswith initial conditions [1]We adopt modified decomposition method for solving Eq. (3.1). In the light of this method we can write
Conclusion
In this paper, the modified decomposition method was used for finding the numerical solutions for the coupled sine-Gordon equations with initial conditions. The approximate solutions to the equations have been calculated by using the MDM without any need to a transformation techniques and linearization of the equations. Additionally, it does not need any discretization method to get numerical solutions. This method thus eliminates the difficulties and massive computation work. The decomposition
References (32)
- et al.
On the exchange of energy in coupled Klein–Gordon equations
Wave Motion
(2003) - et al.
Nonlinear dynamics of the Frenkel–Kontorova model
Phys. Rep.
(1998) Solution of physical problems by decomposition
Comput. Math. Appl.
(1994)Solutions of nonlinear P.D.E.
Appl. Math. Lett.
(1998)- et al.
A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations
Math. Comput. Simul.
(2002) A numerical solution of the sine-Gordon equation using the modified decomposition method
Appl. Math. Comput.
(2003)- et al.
On a generalized fifth order KdV equations
Phys. Lett. A
(2003) - et al.
An application of the decomposition method for the generalized KdV and RLW equations
Chaos, Solitons & Fractals
(2003) An explicit and numerical solutions of some fifth-order KdV equation by decomposition method
Appl. Math. Comput.
(2003)Solitary wave solutions for a generalized Hirota–Satsuma coupled KdV equation
Appl. Math. Comput.
(2004)
A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation
Appl. Math. Comput.
The Adomian method applied to some extraordinary differential equations
Appl. Math. Lett.
Solutions of non-integer order differential equations via the Adomian decomposition method
Appl. Math. Lett.
Analytical approximate solutions for nonlinear fractional differential equations
Appl. Math. Comput.
An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method
Appl. Math. Comput.
A reliable modification of Adomian decomposition method
Appl. Math. Comput.
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2018, Computers and Mathematics with ApplicationsCitation Excerpt :Therefore, certain attentions have been paid to those models which possess rich localized structures in higher dimensions [17–21]. To obtain the analytic solutions for those integrable (2+1)-dimensional sG systems, several analytic methods including Painlevé analysis, bilinear method, inverse scattering transform, binary Darboux transformation, multi-linear variable separation approach, symmetry group analysis, Lamb’s method, and generalized tanh function expansion method have been employed [17–27]. Subsequently, different localized coherent structures of those systems, such as the multiple and vortex-like solitons, line and ring solitons, doubly periodic waves, kinks/antikinks, breathers, instantons, dromions, and peakons, have been reported [17–27].