Research Article  Open Access
Analytical and Numerical Methods for the CMKdVII Equation
Abstract
Hirota's bilinear form for the Complex Modified Kortewegde VriesII equation (CMKdVII) is derived. We obtain one and twosoliton solutions analytically for the CMKdVII. Onesoliton solution of the CMKdVII equation is obtained by using finite difference method by implementing an iterative method.
1. Introduction
It has been known that quasilinear parabolic equations or nonlinear reactiondiffusion systems arise in physics, chemistry, biology, and other applied sciences. The following three equations are the examples of this type of partial differential equations (PDEs). First one is called the Kortewegde Vries equation and first encountered in the study of waters, Korteweg [1], denoted by KdV. The other is called the Complex Modified Kortewegde VriesI equation (CMKdVI) which arises both in the asymptotic investigation of electrostatic waves in a magnetized plasma and in the asymptotic investigation of onedimensional planewave propagation in a micropolar medium, Erbay [2]. The last one is the Complex Modified Kortewegde VriesII equation (CMKdVII) which is another example for quasilinear parabolic equations or nonlinear reactiondiffusion systems, Ablowitz [3]. Equation (1.2) does not hold the PainlevΓ© property but the (1.1) and (1.3) do, Mohammad [4]. The equations which have PainlevΓ© property may be solved by the method of Inverse Scattering Transformations (IST) and hence they are completely integrable [5, 6]. Sometimes it is not easy to solve IST problems [3], such as for CMKdVII equation. Therefore the need for an easy and useful method which has to give soliton solutions for a given PDE is emerged. An important method is developed by Hirota for finding Nsoliton solutions of nonlinear PDE [5, 6].
In this paper, the Hirota's method is applied to the CMKdVII equation. The Hirota's method generally requires the transformation of PDE into homogeneous bilinear forms of degree two. Only specific PDEs can be transformed in this way. This means, when a bilinear (form) equation can be solved, then parameter solution can be obtained as a series which selftruncates at finite length. These expansions that selftruncate in this way give automatically exact solutions. Selftruncation, however, does not occur for all bilinear equations; if it does, then the equation in question possesses multiple soliton solutions. The reason for this situation has never been adequately explained. In other words, selftruncation which is equivalent to complete integrability would require a connection with the conserved quantities of the original equation.
In this study, it is proven that the CMKdVII equation has selftruncated Hirota expansions. It is shown that there is a direct equivalence between the soliton solutions of Hirota's bilinear form of CMKdVII and the Backlund transformations proposed by Weiss, Tabor, and Carnevale [7, 8].
Now, the question here is where the soliton comes from. Firstly, J. S. Russel in 1834 recorded his observations of great solitary wave as a mean of developing the mathematical properties of a large class of solvable nonlinear evolution equations. Solitary waves, solitons, Backlund transformations, conserved quantities and integrable evolutions which can be also named as completely integrable Hamiltonian systems are in the class of solvable nonlinear evolution equations. The description of John Scott Russel has aroused among mathematicians and physicists one hundred and forty years later, Zabusky [9]. In their paper, they were the first ones who defined the solution for the following KdV equation: For the CMKdVII Equation, the Hirota's bilinear form is given in Section 2 and the analytical one and twosoliton solutions are presented in Section 3. The numerical procedure and results for onesoliton solution are outlined in Section 4.
2. Hirota's Bilinear Form of the CMKdVII Equation
It is known that the equation is the complex modified Korteweg de Vries II equation (CMKdVII). Let and be the complex and real valued functions, respectively, satisfying By using the transformation above, CMKdVII becomes Let then (2.3) becomes which is a homogeneous bilinear form and called Hirota's form of CMKdVII equation. Let be the truncated solution of CMKdVII. Then Here and are both separated solutions of the CMKdVII equation. Hence it is an onto Backlund transformation of CMKdVII equation. The system of these equations are called the PainlevΓ© relations. There is a relation between the soliton of the bilinear (2.5) and the function of the PainlevΓ© relations (2.7). Consider the solitons having the properties It can be shown that
Theorem 2.1. If and satisfy (2.5) for all , with and if then the resulting equations in , and are satisfied by the PainlevΓ© relations (2.7). Furthermore with
Proof. When substituting (2.12) into (2.5) and using the PainlevΓ© relations when necessary yields the claim of the theorem. Using (2.12) succesively we obtain the relation (2.13) that completes the proof of the theorem [4].
3. Solitons for the CMKdVII Equation
By using the usual perturbation method, parameter exact solitary wave solutions of can be obtained, Nayfeh [10]. The power series of and which are given in Hirota [11] in a small parameter are: where and are the solutions of the (2.5). Then considering the increasing powers of from (2.5) it is clear that
and (2.2) yields where stands for the complex conjugate.
3.1. OneSoliton Solution for the CMKdVII Equation
To obtain onesoliton solution of the CMKdVII equation, let's take Hence and , for all where stands for the complex number . Therefore is a solution of Hirota bilinear form (2.5) and is the corresponding onesoliton solution of the CMKdVII equation.
3.2. TwoSoliton Solution for the CMKdVII Equation
To find twosoliton solution, let's take where with Hence from (3.8), it can immediately be found that Since the right hand side of (3.3) is not zero, a special solution for can be found by the method of undetermined coefficients. Considering (3.7), is a nonzero real function. The right hand side of (3.4) is not zero, and we can take . From (3.8), it is seen that is zero and that is taken for convenience. Further computations have shown that Hence is a solution of the Hirota's bilinear equation (2.5) and is the corresponding twosoliton solution of the CMKdVII (1.3). Similarly, it is possible to find solitary wave solutions by taking where but the computations are very tedious for .
4. Numerical results
4.1. Iterative Methods Using Finite Difference Schemes
Previously many researchers have used the finite difference methods to solve the KdV equation, Feng [12]. In the last decade, the CMKdVII type equations were solved numerically by using splitstep Fourier method [13β15]. Also parallel implementation of the splitstep Fourier method using Fast Fourier Transform (FFT) has been studied by Taha [16] (see references therein). Here, in this work, the onesoliton solution of the CMKdVII equation is considered. A finite interval for our numerical purposes is subjected, namely, . The constants and can be chosen sufficiently large so that the boundaries do not affect the propagation of solitons. For the CMKdVII (1.3), a numerical (finite difference) method of solution using iterative method is introduced. is approximated by using forward time difference scheme, and by the centralspace difference scheme using fourpoints. Equation (1.3) becomes where , , .
Multiplying both sides by and rearranging the terms we get for . For and , the backward difference scheme is chosen in and . Three more equations come from the boundary conditions, namely, , , and at . Thus, unknowns, namely, , and equations are obtained. Since the value of the nonlinear term is known here, a system of linear equations is obtained. The initial guess is taken as which represents a solitary wave initially at moving to the right with velocity and is the polarization angle. The main idea is to assume that the nonlinear term is zero first and then solve the problem for whole time domain. Afterwards, this solution is taken and substituted for the nonlinear term and solved again iteratively. The following two norms, namely, and are used to measure the accuracy of the approximate solutions for stopping criteria. These norms are defined as following: where and are the two consecutive new and old approximate solutions, respectively, at point for all , where is the final or terminating time. FORTRAN and MATLAB are used to obtain the results and figures, respectively. The graph of onesoliton numerical solution is shown in Figure 1.
5. Conclusions
In this study, Hirota's bilinear form for the complex modified Kortewegde VriesII equation is derived. One and twosoliton solutions of the CMKdVII equation are obtained analytically. Onesoliton solution of the CMKdVII equation is obtained by using finite difference method by implementing an iterative method. The computational cost is due to only finding the inverse of the matrix. The difference of two consecutive solution values according to the formula which is given in (4.3) is shown in Table 1 result. The convergence rate in the method presented above is quadratic as it can be seen in Table 1 result. It would be interesting to see what happens if this numerical scheme for the interaction of twosoliton waves for the CMKdVII equation is applied. The numerical scheme deserves further study according to its application to the CMKdVII equation.

Acknowledgments
The authors would like to thank the referees for the various suggestions to improve the paper, and would like also to thank Professor Allaberen Ashyralyev for pointing out useful numerical algorithms.
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Copyright © 2009 Ömer Akin and Ersin Özuğurlu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.