Analytical and Numerical Methods for the CMKdV-II Equation
Hirota's bilinear form for the Complex Modified Korteweg-de Vries-II equation (CMKdV-II) is derived. We obtain one- and two-soliton solutions analytically for the CMKdV-II. One-soliton solution of the CMKdV-II equation is obtained by using finite difference method by implementing an iterative method.
It has been known that quasilinear parabolic equations or non-linear reaction-diffusion 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 Korteweg-de Vries equation and first encountered in the study of waters, Korteweg , denoted by KdV. The other is called the Complex Modified Korteweg-de Vries-I equation (CMKdV-I) which arises both in the asymptotic investigation of electrostatic waves in a magnetized plasma and in the asymptotic investigation of one-dimensional plane-wave propagation in a micropolar medium, Erbay . The last one is the Complex Modified Korteweg-de Vries-II equation (CMKdV-II) which is another example for quasilinear parabolic equations or non-linear reaction-diffusion systems, Ablowitz . Equation (1.2) does not hold the Painlevé property but the (1.1) and (1.3) do, Mohammad . The equations which have Painlevé 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 , such as for CMKdV-II 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 N-soliton solutions of non-linear PDE [5, 6].
In this paper, the Hirota's method is applied to the CMKdV-II 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 self-truncates at finite length. These expansions that self-truncate in this way give automatically exact solutions. Self-truncation, 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, self-truncation 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 CMKdV-II equation has self-truncated Hirota expansions. It is shown that there is a direct equivalence between the -soliton solutions of Hirota's bilinear form of CMKdV-II 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 non-linear 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 non-linear evolution equations. The description of John Scott Russel has aroused among mathematicians and physicists one hundred and forty years later, Zabusky . In their paper, they were the first ones who defined the solution for the following KdV equation: For the CMKdV-II Equation, the Hirota's bilinear form is given in Section 2 and the analytical one- and two-soliton solutions are presented in Section 3. The numerical procedure and results for one-soliton solution are outlined in Section 4.
2. Hirota's Bilinear Form of the CMKdV-II Equation
It is known that the equation is the complex modified Korteweg de Vries II equation (CMKdV-II). Let and be the complex and real valued functions, respectively, satisfying By using the transformation above, CMKdV-II becomes Let then (2.3) becomes which is a homogeneous bilinear form and called Hirota's form of CMKdV-II equation. Let be the truncated solution of CMKdV-II. Then Here and are both separated solutions of the CMKdV-II equation. Hence it is an onto Backlund transformation of CMKdV-II equation. The system of these equations are called the Painlevé relations. There is a relation between the soliton of the bilinear (2.5) and the function of the Painlevé relations (2.7). Consider the solitons having the properties It can be shown that
Proof. When substituting (2.12) into (2.5) and using the Painlevé 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 .
3. Solitons for the CMKdV-II Equation
By using the usual perturbation method, -parameter exact solitary wave solutions of can be obtained, Nayfeh . The power series of and which are given in Hirota  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. One-Soliton Solution for the CMKdV-II Equation
To obtain one-soliton solution of the CMKdV-II 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 one-soliton solution of the CMKdV-II equation.
3.2. Two-Soliton Solution for the CMKdV-II Equation
To find two-soliton 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 two-soliton solution of the CMKdV-II (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 . In the last decade, the CMKdV-II type equations were solved numerically by using split-step Fourier method [13–15]. Also parallel implementation of the split-step Fourier method using Fast Fourier Transform (FFT) has been studied by Taha  (see references therein). Here, in this work, the one-soliton solution of the CMKdV-II 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 CMKdV-II (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 central-space difference scheme using four-points. 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 non-linear 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 non-linear term is zero first and then solve the problem for whole time domain. Afterwards, this solution is taken and substituted for the non-linear 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 one-soliton numerical solution is shown in Figure 1.
In this study, Hirota's bilinear form for the complex modified Korteweg-de Vries-II equation is derived. One- and two-soliton solutions of the CMKdV-II equation are obtained analytically. One-soliton solution of the CMKdV-II 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 two-soliton waves for the CMKdV-II equation is applied. The numerical scheme deserves further study according to its application to the CMKdV-II equation.
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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