Analytical and Numerical Methods for the CMKdV-II Equation

Abstract

Hirota's bilinear form for the Complex Modified Korteweg-de Vries-II equation (CMKdV-II) 𝑈_𝑡−6|𝑈|²𝑈_𝑥+𝑈_𝑥𝑥𝑥=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.

1. 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-de Vries equation 𝑈_𝑡+𝑐𝑈𝑈_𝑥+𝑈_𝑥𝑥𝑥=0, (1.1) and first encountered in the study of waters, Korteweg [1], denoted by KdV. The other is called the Complex Modified Korteweg-de Vries-I equation (CMKdV-I) 𝑈_𝑡+𝛼(|𝑈|²𝑈)_𝑥+𝛽𝑈_𝑥𝑥𝑥=0, (1.2) 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) 𝑈_𝑡−6|𝑈|²𝑈_𝑥+𝑈_𝑥𝑥𝑥=0, (1.3) 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: 𝑈_𝑡+6𝑈𝑈_𝑥+𝑈_𝑥𝑥𝑥=0. (1.4) 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 𝑈_𝑡−6|𝑈|²𝑈_𝑥+𝑈_𝑥𝑥𝑥=0 (2.1) is the complex modified Korteweg de Vries II equation (CMKdV-II). Let 𝑔 and 𝑓 be the complex and real valued functions, respectively, satisfying 𝑔𝑈=𝑓,|𝑔|²=−𝑓𝑓_𝑥𝑥+𝑓²_𝑥. (2.2) By using the transformation above, CMKdV-II becomes 𝑓𝑔_𝑡−𝑓_𝑡𝑔+3(𝑓_𝑥𝑥𝑔_𝑥−𝑓_𝑥𝑔_𝑥𝑥)+𝑓𝑔_𝑥𝑥𝑥−𝑔𝑓_𝑥𝑥𝑥=0. (2.3) Let 𝐷_𝑥=𝜕−𝜕𝜕𝑥𝜕𝑥^′,𝐷_𝑡=𝜕−𝜕𝜕𝑡𝜕𝑡^′ (2.4) then (2.3) becomes [𝐷³_𝑥+𝐷_𝑡]𝑔(𝑥,𝑡)𝑓(𝑥^′,𝑡^′𝑥)|_′=𝑥,𝑡^′=𝑡=0⟺(𝐷³_𝑥+𝐷_𝑡)𝑔𝑓=0, (2.5) which is a homogeneous bilinear form and called Hirota's form of CMKdV-II equation. Let 𝑈𝑈=⁰Φ+𝑈₁ (2.6) be the truncated solution of CMKdV-II. Then |𝑈₀|²=Φ²_𝑥,𝑈₀𝑈₁+𝑈₁𝑈₀=−Φ_𝑥𝑥,6𝑈_0𝑥|𝑈₁|²−6Φ_𝑥𝑥𝑈_1𝑥=𝑈_{0𝑥𝑥𝑥}+𝑈_0𝑡,6Φ_𝑥𝑈₀|𝑈₁|²−6Φ²_𝑥𝑈_1𝑥=3(Φ_𝑥𝑈_0𝑥𝑥−Φ_𝑥𝑥𝑈_0𝑥)+𝑈₀(Φ_𝑥𝑥𝑥𝑈+Φ𝑡),_1𝑡−6|𝑈₁|²𝑈_1𝑥+𝑈_{1𝑥𝑥𝑥}=0. (2.7) Here 𝑈 and 𝑈₁ 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 Φ of the Painlevé relations (2.7). Consider the solitons 𝑔𝑈=𝑓,𝑈₁=𝑔⁽¹⁾𝑓⁽¹⁾ (2.8) having the properties |𝑈|²=(log𝑓)_𝑥𝑥,|𝑈|²𝜕=−²𝜕𝑥²logΦ+|𝑈₁|². (2.9) It can be shown that 𝑓=Φ𝑓⁽¹⁾,𝑔=𝑈₀𝑓⁽¹⁾+Φ𝑔⁽¹⁾. (2.10)

Theorem 2.1. If 𝑓^(𝑛) and 𝑔^(𝑛) satisfy (2.5) for all 𝑛, with |𝑔^(𝑛)|=𝑓^(𝑛)𝑓^(𝑛)_𝑥𝑥−(𝑓_𝑥^(𝑛))², (2.11) and if 𝑓^(𝑛)=Φ_(𝑛−1)𝑓^(𝑛−1),𝑔^(𝑛)=𝑈₀𝑓^(𝑛−1)+Φ_(𝑛−1)𝑔^(𝑛−1), (2.12) then the resulting equations in Φ_𝑛−1,𝑈₀ and 𝑈^(𝑛−1) are satisfied by the Painlevé relations (2.7). Furthermore 𝑓^(𝑛)=^𝑛−1_𝑖=0Φ_𝑖, (2.13) with 𝑓⁽⁰⁾=1. (2.14)

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].

3. 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: 𝑔=𝜀𝑔⁽¹⁾+𝜀³𝑔⁽³⁾+𝜀⁵𝑔⁽⁵⁾+⋯𝑓=1+𝜀²𝑓⁽²⁾+𝜀⁴𝑓⁽⁴⁾+⋯, (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

𝑔⁽¹⁾_𝑥𝑥𝑥+𝑔_𝑡⁽¹⁾𝑔=0,_{(3)𝑥𝑥𝑥}+𝑔_𝑡⁽³⁾=−(𝐷³_𝑥+𝐷_𝑡)𝑓⁽²⁾𝑔⁽¹⁾𝑔_{(5)𝑥𝑥𝑥},+𝑔_𝑡⁽⁵⁾=−(𝐷³_𝑥+𝐷_𝑡)[𝑓⁽⁴⁾𝑔⁽¹⁾+𝑓⁽²⁾𝑔⁽³⁾𝑔],_{(5)𝑥𝑥𝑥}+𝑔_𝑡⁽⁵⁾=−(𝐷³_𝑥+𝐷_𝑡)[𝑓⁽⁶⁾𝑔⁽¹⁾+𝑓⁽⁴⁾𝑔⁽³⁾+𝑓⁽²⁾𝑔⁽⁵⁾] (3.2)(3.3)(3.4)(3.5) and (2.2) yields 𝑓⁽²⁾_𝑥𝑥=−𝑔⁽¹⁾𝑔^(1)∗𝑓_(4)𝑥𝑥,=−𝑔⁽¹⁾𝑔^(3)∗−𝑔^(1)∗𝑔⁽³⁾−𝑓_𝑥⁽²⁾𝑓_𝑥⁽²⁾−𝑓⁽²⁾𝑓⁽²⁾_𝑥𝑥𝑓_(6)𝑥𝑥,=−𝑔⁽³⁾𝑔^(3)∗−𝑔^(1)∗𝑔⁽⁵⁾−𝑔⁽¹⁾𝑔^(5)∗−2𝑓_𝑥⁽²⁾𝑓_𝑥⁽⁴⁾−𝑓⁽⁴⁾𝑓⁽²⁾_𝑥𝑥−𝑓⁽²⁾𝑓⁽⁴⁾_𝑥𝑥, (3.6)(3.7)(3.8) 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 𝑔⁽¹⁾=𝑒^Φ+𝐼Ψ;Φ+𝐼Ψ=𝑎₁𝑥+𝑏₁𝑡+𝑐₁. (3.9) Hence 𝑎₁=re(a₁)+𝐼⋅im(𝑎₁𝑏),₁=re(𝑏₁)+𝐼⋅im(𝑏₁𝑐),₁=re(𝑐₁)+𝐼⋅im(𝑐₁),re(𝑏₁)=−(re(𝑎₁))³+3∗re(𝑎₁)−(im(𝑎₁))²,im(𝑏₁)=(re(𝑎₁))³−3∗im(𝑎₁)−(re(𝑎₁))²,𝑓⁽²⁾=𝑒^2Φ4(re(𝑎₁))², (3.10) and 𝑓^(2𝑛)=0, 𝑔^(2𝑛−1)=0 for all 𝑛≥2 where 𝐼 stands for the complex number 𝑖. Therefore 𝜀𝑓(𝑡,𝑥)=1+²4(re(𝑎₁))²𝑒^2Φ,𝑔(𝑥,𝑡)=𝜀𝑒^Φ+𝐼Ψ (3.11) is a solution of Hirota bilinear form (2.5) and 𝑈=𝑔(𝑥,𝑡)𝑓(𝑥,𝑡) (3.12) 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 𝑔⁽¹⁾=𝑒^Φ1+𝐼Ψ1+𝑒^Φ2+𝐼Ψ2, (3.13) where Φ₁[𝑡,𝑥]=re(𝑎₁)∗𝑥+re(𝑏₁)∗𝑡+re(𝑐₁)Ψ₁[𝑡,𝑥]=im(𝑎₁)∗𝑥+im(𝑏₁)∗𝑡+im(𝑐₁)Φ₂[𝑡,𝑥]=re(𝑎₂)∗𝑥+re(𝑏₂)∗𝑡+re(𝑐₂)Ψ₂[𝑡,𝑥]=im(𝑎₂)∗𝑥+im(𝑏₂)∗𝑡+im(𝑐₂) (3.14) with re(𝑏₁)=−(re(𝑎₁))³+3∗re(𝑎₁)−(im(𝑎₁))²im(𝑏₁)=(re(𝑎₁))³−3∗im(𝑎₁)−(re(𝑎₁))²re(𝑏₂)=−(re(𝑎₂))³+3∗re(𝑎₂)−(im(𝑎₂))²im(𝑏₂)=(re(𝑎₂))³−3∗im(𝑎₂)−(re(𝑎₂))². (3.15) Hence from (3.8), it can immediately be found that 𝑓⁽²⁾=𝑒^2Φ14(re(𝑎₁))²+𝑒^2Φ24(re(𝑎₂))². (3.16) 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 𝑓^(2𝑛)=0,𝑔^(2𝑛−1)=0,𝑛≥3. (3.17) Hence 1𝑓[𝑡,𝑥]=1−4(re(𝑎₁))²𝑒^{2Φ1[𝑡,𝑥]}−14(re(𝑎₂))²𝑒^{2Φ2[𝑡,𝑥]}+((im(𝑎₁)−im(𝑎₂))²+(re(𝑎₁)−re(𝑎₂))²)²16(re(𝑎₁))²(re(𝑎₂))²[(im(𝑎₁)−im(𝑎₂))²+(re(𝑎₁)−re(𝑎₂))²]𝑒^{2(Φ1[𝑡,𝑥]+Φ2[𝑡,𝑥])}−1[re(𝑎₁)+𝐼⋅(im(𝑎₁)−im(𝑎₂))+re(𝑎₂)]²𝑒^{Φ1[𝑡,𝑥]+Φ2[𝑡,𝑥]+𝐼⋅Ψ1[𝑡,𝑥]−𝐼⋅Ψ2[𝑡,𝑥]}−1[re(𝑎₁)+𝐼⋅(im(𝑎₁)−im(𝑎₂))+re(𝑎₂)]²𝑒^{Φ1[𝑡,𝑥]+Φ2[𝑡,𝑥]−𝐼⋅Ψ1[𝑡,𝑥]+𝐼⋅Ψ2[𝑡,𝑥]},𝑔[𝑡,𝑥]=𝑒^{Φ1[𝑡,𝑥]+Φ2[𝑡,𝑥]+𝐼⋅Ψ1[𝑡,𝑥]+𝐼⋅Ψ2[𝑡,𝑥]}+(−𝐼⋅im(𝑎₁)−re(𝑎₁)+𝐼⋅im(𝑎₂)+re(𝑎₂))²[2re(𝑎₂)(im(𝑎₁)−𝐼⋅re(𝑎₁)−im(𝑎₂)+𝐼⋅re(𝑎₂))]²𝑒^{Φ1[𝑡,𝑥]+2Φ2[𝑡,𝑥]+𝐼⋅Ψ1[𝑡,𝑥]}+(−𝐼⋅im(𝑎₁)−re(𝑎₁)+𝐼⋅im(𝑎₂)+re(𝑎₂))²[2re(𝑎₂)(im(𝑎₁)−𝐼⋅re(𝑎₁)−im(𝑎₂)+𝐼⋅re(𝑎₂))]²𝑒^{2Φ1[𝑡,𝑥]+Φ2[𝑡,𝑥]+𝐼⋅Ψ2[𝑡,𝑥]} (3.18) is a solution of the Hirota's bilinear equation (2.5) and 𝑤=𝑔(𝑥,𝑡)𝑓(𝑥,𝑡) (3.19) is the corresponding two-soliton solution of the CMKdV-II (1.3). Similarly, it is possible to find 𝑁 solitary wave solutions by taking 𝑔^(𝑁)=^𝑛_𝑖=1𝑒^{𝑎𝑖𝑥+𝑏𝑖𝑡+𝑐𝑖}, (3.20) where 𝑏_𝑖=−𝑎_𝑖³ (3.21) but the computations are very tedious for 𝑖>3.

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 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. Equation (1.3) becomes (𝑧_𝑚^𝑛+1−𝑧^𝑛_𝑚Δ𝑡)−6|𝑧_𝑚^𝑛−1|²[−𝑧^𝑛+1_𝑚+2+8𝑧^𝑛+1_𝑚+1−𝑧^𝑛+1_𝑚+1+𝑧^𝑛+1_𝑚−2𝑧12Δ𝑥]+[^𝑛+1𝑚+2−2𝑧^𝑛+1_𝑚+1+2𝑧^𝑛+1_𝑚−1−𝑧^𝑛+1_𝑚−22(Δ𝑥)³]=0, (4.1) where 𝑧^𝑛_𝑚=𝑧(𝑡_𝑛,𝑥_𝑚)=𝑧(𝑛𝑘,𝑚ℎ), 𝑘=Δ𝑡, ℎ=Δ𝑥.

Multiplying both sides by 2(Δ𝑥)³ and rearranging the terms we get 𝑧^𝑛+1_𝑚+2[1+6(Δ𝑥)²(|𝑧_𝑚^𝑛−1|²)]+𝑧^𝑛+1_𝑚+1[−2−48(Δ𝑥)²|𝑧_𝑚^𝑛−1|²]+𝑧_𝑚^𝑛+1[2(Δ𝑥)²Δ𝑡]+𝑧^𝑛+1_𝑚−1[2+48(Δ𝑥)²(|𝑧_𝑚^𝑛−1|²)]+𝑧^𝑛+1_𝑚−2[−1−6(Δ𝑥)²(|𝑧_𝑚^𝑛−1|²)]=2(Δ𝑥)³𝑧Δ𝑡^𝑛𝑚 (4.2) for 3≤𝑚≤𝑁−2. For 𝑚=𝑁−1 and 𝑚=𝑁, the backward difference scheme is chosen in 𝑈_𝑥 and 𝑈_𝑥𝑥𝑥. Three more equations come from the boundary conditions, namely, 𝑈(𝑎)=0, 𝑈(𝑏)=0, and 𝜕𝑈/𝜕𝑥=0 at 𝑥=𝑎. Thus, 𝑁 unknowns, namely, 𝑧_𝑖^𝑛+1, 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/𝛼sech[√𝑐(𝑥−𝑥₀)]𝑒^(𝑖𝜃0) which represents a solitary wave initially at 𝑥₀ moving to the right with velocity 𝑐 and 𝜃₀ is the polarization angle. The main idea is to assume that the non-linear term |𝑧_𝑚^𝑛−1|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 𝐿₂ are used to measure the accuracy of the approximate solutions for stopping criteria. These norms are defined as following: 𝐿_∞=max_𝑛(||𝑧_𝑛|−|𝑧_𝑛||),𝐿₂=_𝑛(|𝑧_𝑛|−|𝑧_𝑛|)², (4.3) 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.

Figure 1 

The modulus of the one-soliton numerical solution for the CMKdV-II equation with 𝑁=256, −128≤𝑥≤128, 𝑀=100, 0≤𝑡≤500.

5. 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.

Table 1 

The difference of two consecutive solution values (𝐍=256, −128≤𝐱≤128, 𝐌=500, 0≤𝐭≤5).

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.