Analytical and Numerical Methods for the CMKdV-II Equation

Hirota's bilinear form for the Complex Modified Korteweg-de Vries-II equation (CMKdV-II) 
 𝑈 𝑡 − 6 | 𝑈 | 2 𝑈 𝑥 + 𝑈 𝑥 𝑥 𝑥 = 0 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.


Introduction
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- Mathematical Problems in Engineering 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 2 . 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 3 . Equation 1.2 does not hold the Painlevé property but the 1.1 and 1.3 do, Mohammad 4 . 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 3 , 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 N-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 N-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 9 . 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.

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 g and f be the complex and real valued functions, respectively, satisfying By using the transformation above, CMKdV-II 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

2.9
It can be shown that

2.11
and if

2.12
then the resulting equations in Φ n−1 ,U 0 and U n−1 are satisfied by the Painlevé relations 2.7 . Furthermore 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 4 .

Solitons for the CMKdV-II Equation
By using the usual perturbation method, N-parameter exact solitary wave solutions of 7 can be obtained, Nayfeh 10 . The power series of g and f which are given in Hirota 11 in a small parameter ε are: 3.1 where g and f are the solutions of the 2.5 . Then considering the increasing powers of ε from 2.5 it is clear that where * stands for the complex conjugate.

One-Soliton Solution for the CMKdV-II Equation
To obtain one-soliton solution of the CMKdV-II equation, let's take Hence a 1 re a 1 I · im a 1 , 3.10 and f 2n 0, g 2n−1 0 for all n ≥ 2 where I stands for the complex number i. Therefore is a solution of Hirota bilinear form 2.5 and is the corresponding one-soliton solution of the CMKdV-II equation.

Two-Soliton Solution for the CMKdV-II Equation
To find two-soliton solution, let's take

3.15
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 g 3 can be found by the method of undetermined coefficients. Considering 3.7 , f 4 xx is a nonzero real function. The right hand side of 3.4 is not zero, and we can take g 5 0. From 3.8 , it is seen that f 6 xx is zero and that f 6 0 is taken for convenience. Further computations have shown that 3.17 2re a 2 im a 1 − I · re a 1 − im a 2 I · re a 2 2 e 2Φ 1 t,x Φ 2 t,x I·Ψ 2 t,x 3.18 8 Mathematical Problems in Engineering

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 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 16 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, a, b . The constants a and b 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. U t is approximated by using forward time difference scheme, U x and U xxx by the central-space difference scheme using four-points. 0, and ∂U/∂x 0 at x a. Thus, N unknowns, namely, z n 1 i , i 1, . . . , N and N 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 U x, 0 2c/α sech √ c x − x 0 e iθ 0 which represents a solitary wave initially at x 0 moving to the right with velocity c and θ 0 is the polarization angle. The main idea is to assume that the non-linear term |z n−1 m | 2 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, L ∞ and L 2 are used to measure the accuracy of the approximate solutions for stopping criteria. These norms are defined as following: where z n and z n are the two consecutive new and old approximate solutions, respectively, at point nΔx, T for all n, where T 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.

Conclusions
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.

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