Periodic traveling wave, bright and dark soliton solutions of the (2 + 1)-dimensional complex modiﬁed Korteweg-de Vries system of equations by using three di ﬀ erent methods

: In this paper, the (2 + 1)-dimensional complex modiﬁed Korteweg-de Vries (cmKdV) equations are studied using the sine-cosine method, the tanh-coth method, and the Kudryashov method. As a result, analytical solutions in the form of dark solitons, bright solitons, and periodic wave solutions are obtained. Finally, the dynamic behavior of the solutions is illustrated by choosing the appropriate parameters using 2D and 3D plots. The obtained results show that the proposed methods are straightforward and powerful and can provide more forms of traveling wave solutions, which are expected to be useful for the study of the theory of traveling waves in physics.

The great interest in physics and mathematics is the study of the nonlinear excitations of the spin models [35][36][37][38][39]. In this motivation, Myrzakulov et.al had been presented various integrable spin systems in (2 + 1) dimensions by proposing the interaction of the spin field with vector potential or scalar potential in Ref. [39]. Researchers obtained Lax pairs and various interesting reductions in (1+1) and (2+1) dimensions.
In the current work, we mainly study the (2+1)-dimensional cmKdV system of equations that is given by [39] q t + q xxy + iqv + (qw) x = 0, v x + 2iδ(q * q xy − q * xy q) = 0, (1.1) The paper is organized as follows. Lax pair for the (2+1)-dimensional cmKdV equations are given in Sect. 2. Then the three methods are used to construct the exact solutions in Sect. 3-Sect. 5. The physical interpretation is presented in Sect. 6. Finally, we present concluding remarks in Sect. 7.

The sine-cosine method
We use the sine-cosine method to obtain sine and cosine solutions for the (2+1)-dimensional cmKdV system of equations (1.1). The description of the method used in the following subsection is given in [4].

Description of method
According to method the partial differential equation (PDE) can be transformed to ordinary differential equation (ODE) by applying a wave variable As long as all terms contain derivatives Eq (3.2) is integrated. The solutions of ODE (3.2) can be presented in the form or Q(x, y, t) = α sin β (µξ), (3.4) where ξ = x + y + ct and the parameters β, µ and α will be defined, c, µ are constants. The derivatives of Eq (3.3) are (3.6) and the derivatives of Eq (3.4) become (Q n ) = nβµα n sin nβ−1 (µξ) cos(µξ), (3.7) (Q n ) = −n 2 µ 2 β 2 α n sin nβ (µξ) + nµ 2 α n β(nβ − 1) sin nβ−2 (µξ), (3.8) and so on for the other derivatives. Applying (3.3)-(3.8) into the reduced ODE (3.2) yields a trigonometric equation of cos K (µξ) or sin K (µξ) terms. Thereafter, we define the parameters by first balancing exponents of each pair of sine or cosine to determine K. Further, all coefficients of the identical power in cos k (µξ) or sin k (µξ) are collected, where these coefficients have to vanish. Then, a system of algebraic equations with the unknown α, µ, β will be obtained and from that coefficients can be determined.

Implementation
For applying the sine-cosine method, we have to reduce Eqs. (1.1) to ODE. By taking transformation q(x, y, t) = e i(ax+by+dt) Q(x, y, t), (3.9) where a, b, d are real constants and Q(x, y, t) is the real valued function, Eqs (1.1) are reduced to the following system Substituting the wave transformation 14) w(x, y, t) = w(ξ) = w(x + y + ct), (3.15) into system of Eqs (3.10)-(3.12), we obtain that Integrating Eqs (3.17)-(3.18) once, with respect to ξ and taking constants of integration is zero, we obtain Substituting Eq (3.19) into Eq (3.16), we derive the following ODE where prime denotes the derivation with respect to ξ. By separating real and imaginary parts in Eq (3.20), we get the ordinary differential equations: Integrating Eq (3.21) once, with respect to ξ, gives where L is a constant of integration. As the same function Q(ξ) satisfies both Eqs (3.22) and (3.23), we have the next constraint condition: By using condition (3.24), we have We rewrite Eq (3.22) as In the next subsection, we solve Eq (3.26) by the sine-cosine method.

The sine solutions
According to method the solution of Eq (3.26) can be found by transformation To find the sine solution we use Eq (3.27) and its second order derivative Applying the balance method, by equating the exponents of sin(µξ), from (3.29) we determine β: We equate exponents and coefficients of each pair of the sin(µξ) functions and obtain a system of algebraic equations By solving the system (3.32)-(3.33) , we obtain: By substituting Eq (3.34) into Eq (3.27) and then obtained result in Eq (3.19) and Eq (3.9) we derive the exact solutions for the (2+1)-dimensional cmKdV equations (1.1)

The cosine solutions
The cosine solution of (3.26) can be found by transformation To find the cosine solution we use Eq (3.41) and its second order derivative Applying the balance method, by equating the exponents of cos(µξ), from Eq (3.43) we determine β: We equate exponents and coefficients of each pair of the cos(µξ) functions and obtain a system of algebraic equations Next, by solving the system (3.46)-(3.47) , we get: By substituting Eq (3.48) into Eq (3.41) and then obtained expression in Eq (3.19) and Eq (3.9) we derive the exact solutions for the (2+1)-dimensional cmKdV equations (1.1)

The tanh-coth method
We apply the tanh-coth method to derive traveling wave solutions for the (2+1)-dimensional cmKdV system of equations. The first, the tanh method was presented by Malfliet [26][27][28] and then was expanded by Wazwaz [4,29] In the next subsection, the description of the method is given by [4].

Description of method
Partial differential equation (PDE) can be transformed to an ordinary differential equation (ODE) by applying a wave variable where c is a constant. As long as all terms contain derivatives Eq (4.2) is integrated. Applying a new independent variable where µ is the wave number, we have the next change of derivatives: The tanh-coth method allows the application of the finite expansion in the next form: where a 0 , a 1 , a 2 , ..., a N , b 1 , b 2 , ..., b N are unknown coefficients. Parameter M is defined by balancing nonlinear terms and the highest order derivative term in Eq (4.2). By substituting the value of Q(ξ) from (4.4) in Eq (4.2), and comparing the coefficient of Y n we can derive the coefficients a 0 , a 1 , a 2 , ..., a N , b 1 , b 2 , ..., b N .

Description of method
Partial differential equation (PDE) can be transformed to an ordinary differential equation (ODE) by applying a wave variable where c is a constant. As long as all terms contain derivatives Eq (5.2) is integrated. To find dominant terms we substitute where R(ξ) is the following function We can calculate number of derivatives by Q ξξξ = N n=0 a n nR n (R − 1)[(n 2 + 3n + 2)R 2 − (2n 2 + 3n + 1)R + n 2 ]. (5.8)

Implementation
Let's study ODE (3.26) where prime denotes the derivation with respect to ξ. From Eq (3.26) we find N = 1 then we look for the solution of Eq (3.26) in the form The second derivative of Eq (5.9) is Substituting (5.9)-(5.10) into (3.26) we obtain the system of algebraic equations. By solving it we find coefficients as Substituting (5.11) in (5.9) and then obtained expressions in Eq (3.19) and Eq (3.9) we have solutions for Eqs (1.1) by the following form where ξ = x + y + ct, with c = 2ab + a 2 + d−a 2 b 2a+b .

Physical interpretation
In this section, we will give the physical explanation of the obtained exact solutions in Sect. 3-Sect. As we see, the solutions q 12 , v 12 (3.38)-(3.39) can be soliton solutions. It can be seen that the bright one-soliton q 12 and dark one-soliton v 12 keep their directions, widths, and amplitudes invariant during the propagation on the x−y plane. Figure 5 displays propagation of the bright soliton solutions q 22 , v 22 , w 22 in 2D plot at y = 0, t = −2, t = 0, t = 2. It is well known that bright soliton is a pulse on a zero-intensity background. However, the dark soliton is featured as a localized intensity dip below a continuous-wave background.  Figures 11-12. But for q 33 , v 33 the bright and dark soliton solutions can be derived that is Figures 13-14. In case q 33 , v 33 the values are taken as a = 1, b = 1, d = −2, δ = −1, c = 2, a 0 = 0, a 1 = − 1 2 , b 1 = 1 2 , µ = − 1 2 , (d−a 2 b) (2a+b) = −1 within the interval −5 ≤ x, y ≤ 5 for t = −3, t = 0, t = 3. The periodic solutions are obtained also for q 32 , v 32 , w 32 , q 34 , v 34 , w 34 .
Thus, the considered above cases show that the different choices of the parameters a, b, d, c yield a number of waveforms such as periodic solutions, bright soliton, and dark soliton. Moreover, the tanhcoth method yields more solutions compared to solutions by the sine-cosine method, the Kudryashov method.

Conclusions
In the paper, the (2+1)-dimensional cmKdV system of equations is studied using the sine-cosine method, the tanh-coth method, and the Kudryashov method. As a result, various types of exact solutions such as bright solitons, dark solitons, and periodic wave solutions are obtained. In addition, we have shown the graphical structures of some derived results in Figures 1-14 and then interpreted the nature of the profiles shown. The main advantage of the sine-cosine, tanh-coth and Kudryashov methods is that, unlike existing methods such as Hirota's bilinear method or the inverse scattering method, tedious algebra and guesswork can be avoided. By varying the choice of parameters, different waveforms can be generated, such as the bell shape, the anti-bell shape, and other forms of the solutions. The results obtained are new, as solutions for traveling waves have not been found before. Moreover, this work extends the work on the (2+1)-dimensional cmKdV equations [39][40][41][42][43][44] by deriving a variety of exact solutions. It is expected that the methods used in this work will open new horizons for the study of NPDEs arising in physics. Moreover, it will also be interesting to study the integrability properties such as the infinite number of conservation laws and geometry properties for Eqs (1.1). Related work is underway and results will be reported separately.