Identifying Combination of Dark–Bright Binary–Soliton and Binary–Periodic Waves for a New Two-Mode Model Derived from the (2 + 1)-Dimensional Nizhnik–Novikov–Veselov Equation

Abstract

In this paper, we construct a new two-mode model derived from the $(2+1)$-dimensional Nizhnik–Novikov–Veselov (TMNNV) equation. We generalize the concept of Korsunsky to accommodate the derivation of higher-dimensional two-mode equations. Since the TMNNV is presented here, for the first time, we identify some of its solutions by means of two recent and effective schemes. As a result, the Kudryashov-expansion method exports a combination of bright–dark binary solitons, which simulate many applications in optics, photons, and plasma. The modified rational sine and cosine functions export binary–periodic waves that arise in the field of surface water waves. Moreover, by using 2D and 3D graphs, some physical properties of the TMNNV were investigated by means of studying the effect of the following parameters of the model: nonlinearity, dispersion, and phase–velocity. Finally, we checked the validity of the obtained solutions by verifying the correctness of the original governing model.

1. Introduction

Two-mode equations are nonlinear partial differential equations of second order in the time coordinate and contain three main parameters, known as nonlinearity, dispersion, and phase-velocity. The propagation of these two-mode equations takes the pattern of moving symmetric or asymmetric binary waves, based on the values of the nonlinearity and dispersion parameters. Moreover, the overlapping of these binary waves is affected by the phase–velocity parameter. The idea of two-mode equations was first introduced by Korsunsky, when he noticed that wave profiles of the Hirota–Satsuma system moved in the form of binary long waves with different dispersion parameters. He then simulated this physical phenomenon by updating the KdV equation and presenting the first two-mode equation under the name of two-mode KdV (TMKdV) which reads, in scale form, as follows:

$${\theta }_{tt}-s^2{\theta }_{xx}+\left(\frac{\partial }{\partial t}-\alpha s\frac{\partial }{\partial x}\right)\theta {\theta }_x+\left(\frac{\partial }{\partial t}-\beta s\frac{\partial }{\partial x}\right){\theta }_{xxx}=0,$$

(1)

where $\theta =\theta (x,t)$ is the vector filed, $\alpha$ and $\beta$ refer to nonlinearity and dispersion parameters, and s represents the phase–velocity. According to Korsunsky’s observation, when no overlapping occurs “$s=0$”, the TMKdV reduces into the standard single-mode KdV, which is given by

$${\theta }_t+\theta {\theta }_x+{\theta }_{xxx}=0.$$

(2)

In this work we explored some physical insights regarding the field of two-mode equations through presenting a new two-mode version of the generalized $(2+1)$–dimensional Nizhnik–Novikov–Veselov (NNV) equation, which reads as:

$${\theta }_t+\mu {\left(\theta \int \theta dy\right)}_x+\lambda {\theta }_{xxx}=0,$$

(3)

where $\theta =\theta (x,y,t)$, $\mu$ is the nonlinearity coefficient, and $\lambda$ is the linearity coefficient. The NNV equation represents many physical phenomena arising in different media, such as modern string, optical fibers, biological membranes, magneto-electrodynamics, and incompressible fluid. Many studies were conducted to analyze the explicit solutions of (3). For example, Boiti et al. [1] derived the proposed form of the NNV equation, for the first time, and solved it by means of the inverse scattering method. The variable separation solutions were obtained by implementing the extended tanh method [2]. Multiple soliton solutions were extracted by using the simplified Hirota’s bilinear method [3]. Quasi-periodic solutions were derived by means of Riemann theta function and Hirota’s method [4]. Lump–stripe solutions were attained in [5]. Finally, both lump–solitons and breather–solitons were investigated for (3) in [6].

The contribution of this work is threefold. First, we generalize Korsunsky’s operators to formulate the two-mode version of NNV. Second, we construct explicit solutions to the new model by implementing the Kudryashov-expansion and modified rational sine–cosine methods. Finally, we investigate the symmetric and asymmetric propagations of the attained binary–soliton solutions to TMNNV.

2. Formulation of Two-Mode Nizhnik–Novikov–Veselov Model

Korsunsky and Wazwaz [7,8,9] presented the $(1+1)$-dimensional two-mode equation in the following form:

$${\theta }_{tt}-s^2{\theta }_{xx}+\left(\frac{\partial }{\partial t}-\alpha s\frac{\partial }{\partial x}\right)N(\theta ,{\theta }_x,\dots )+\left(\frac{\partial }{\partial t}-\beta s\frac{\partial }{\partial x}\right)L\left({\theta }_{nx}\right)=0,$$

(4)

where $\theta =\theta (x,t)$, $N(\theta ,{\theta }_x,\dots )$ is the nonlinear terms, $L\left({\theta }_{nx}\right):n\ge 2$ is the linear terms, $\left|\alpha \right|\le 1$ is the nonlinearity parameter, and $\left|\beta \right|\le 1$ is the dispersion parameter.

In this work, we suggested a new scale for the $(2+1)$-dimensional two-mode equation in the variables $x,y,t$. The new scale was:

$$\begin{array}{ccc}\hfill {\theta }_{tt}-s^2{\theta }_{xx}-s^2{\theta }_{yy}& +& \left(\frac{\partial }{\partial t}-{\alpha }_{1}s\frac{\partial }{\partial x}-{\alpha }_{2}s\frac{\partial }{\partial y}\right)N(\theta ,{\theta }_x,{\theta }_y,\dots )\hfill \\ & +& \left(\frac{\partial }{\partial t}-{\beta }_{1}s\frac{\partial }{\partial x}-{\beta }_{2}s\frac{\partial }{\partial y}\right)L({\theta }_{nx},{\theta }_{my})=0,\hfill \end{array}$$

(5)

where $\theta =\theta (x,y,t)$, $n\ge 2$, $m\ge 2$, $|{\alpha }_1|\le 1$ and $|{\alpha }_2|\le 1$ are the nonlinearity parameters, and $|{\beta }_1|\le 1$ and $|{\beta }_2|\le 1$ are the dispersions.

Now, applying (5) in (3) we reached the following $(2+1)$-dimensional two-mode NNV (TMNNV) equation:

$$\begin{array}{ccc}\hfill {\theta }_{tt}-s^2{\theta }_{xx}-s^2{\theta }_{yy}& +& \mu \left(\frac{\partial }{\partial t}-{\alpha }_{1}s\frac{\partial }{\partial x}-{\alpha }_{2}s\frac{\partial }{\partial y}\right){\left(\theta \int \theta dy\right)}_x\hfill \\ & +& \lambda \left(\frac{\partial }{\partial t}-{\beta }_{1}s\frac{\partial }{\partial x}-{\beta }_{2}s\frac{\partial }{\partial y}\right){\theta }_{xxx}=0.\hfill \end{array}$$

(6)

For more details about two-mode equations, types of its solutions, and some physical properties, we advise the reader to browse the following references [10,11,12,13].

3. Explicit Bidirectional Solutions to the TMNNV Model

In order to solve (6), we introduced a new dependent variable $\psi =\psi (x,y,t)$ that satisfies the relation:

$$\theta (x,y,t)={\psi }_y(x,y,t).$$

(7)

By considering (7), the TMNNV (6) was converted into:

$$\begin{array}{ccc}\hfill {\psi }_{tty}-s^2{\psi }_{xxy}-s^2{\psi }_{yyy}& +& \mu \left(\frac{\partial }{\partial t}-{\alpha }_{1}s\frac{\partial }{\partial x}-{\alpha }_{2}s\frac{\partial }{\partial y}\right){\left(\psi {\psi }_y\right)}_x\hfill \\ & +& \lambda \left(\frac{\partial }{\partial t}-{\beta }_{1}s\frac{\partial }{\partial x}-{\beta }_{2}s\frac{\partial }{\partial y}\right){\psi }_{xxxy}=0.\hfill \end{array}$$

(8)

Then, we converted (8) into an ordinary differential equation (ODE) with respect to the new independent variable $z=ax+by-ct$. We assumed that ${\alpha }_1={\alpha }_2=\alpha$ and ${\beta }_1={\beta }_2=\beta$. Accordingly, we obtained the following simplified ODE:

$$(c^2-(a^2+b^2)s^2)U\left(z\right)-\frac{1}{2}\mu a(c+(a+b)\alpha )U^2\left(z\right)-\lambda a^3(c+(a+b)\beta )U^{″}\left(z\right)=0,$$

(9)

where $U\left(z\right)=\psi (x,y,t)$. Next, we solved (9) by implementing the following two recent effective approaches the Kudryashov’s method and the modified rational sine–cosine form.

3.1. Approach I: Kudryashov-Expansion

The Kudryashov-expansion is one of the methods that imposes the solution in the form of a polynomial of degree n, the variable of which is a solution for a particular auxiliary differential equation of separable type [14,15]. The order of this polynomial is determined by the process of balancing the degree of highest nonlinear term versus the degree of highest linear derivative that appears in the equation. In particular, we compared the orders of $U^{″}\left(z\right)$ against $U^2\left(z\right)$, $n+2=2n$. Thus, the Kudryashov solution of (9) was:

$$U\left(z\right)=A+BY+D{Y}^2,$$

(10)

where $Y=Y\left(z\right)=\frac{1}{1+d{e}^z}$ satisfies $Y^{\prime }=Y^{\prime }\left(z\right)=Y(Y-1)$. By substituting (10) in (9) and collecting the coefficients of $Y^i$, we reached the following nonlinear algebraic system involving the unknowns $A,B,D,a,b,c$:

$$\begin{array}{ccc}\hfill 0& =& -A\left(s\left(a^2(\alpha A\mu +2s)+\alpha aAb\mu +2b^{2}s\right)+aAc\mu -2c^2\right),\hfill \\ \hfill 0& =& -2B\left(c\left(a^3\lambda +aA\mu \right)+s\left(a^4\beta \lambda +a^{3}b\beta \lambda +a^2(\alpha A\mu +s)+\alpha aAb\mu +b^{2}s\right)-c^2\right),\hfill \\ \hfill 0& =& c\left(a^3(6B\lambda -8D\lambda )-a\mu \left(2AD+B^2\right)\right)-s\left(\alpha ab\mu \left(2AD+B^2\right)+2b^{2}Ds\right)\hfill \\ & -& s\left(a^4(8\beta D\lambda -6\beta B\lambda )+2a^{3}b\beta \lambda (4D-3B)+a^2\left(2D(\alpha A\mu +s)+\alpha B^2\mu \right)\right)+2c^{2}D,\hfill \\ \hfill 0& =& -2a\left(2a^3\beta \lambda s(B-5D)+2a^2\lambda (B-5D)(b\beta s+c)+\alpha aBD\mu s+BD\mu (\alpha bs+c)\right),\hfill \\ \hfill 0& =& -aD\left(12a^3\beta \lambda s+12a^2\lambda (b\beta s+c)+\alpha aD\mu s+D\mu (\alpha bs+c)\right).\hfill \end{array}$$

(11)

By solving the above five equations, we obtained two sets of outputs:

Set 1.

$$\begin{array}{ccc}\hfill A& =& -\frac{2a^2(\beta \mp 1)\lambda }{(\alpha \mp 1)\mu },B=\frac{12a^2(\beta \mp 1)\lambda }{(\alpha \mp 1)\mu },D=-\frac{12a^2(\beta \mp 1)\lambda }{(\alpha \mp 1)\mu },\hfill \\ \hfill b& =& -\frac{a^3(\beta \mp 1)\lambda }{a^2(\beta \mp 1)\lambda +2s},c=\mp \frac{2a{s}^2}{a^2(\beta \mp 1)\lambda +2s}.\hfill \end{array}$$

(12)

Set 2.

$$\begin{array}{ccc}\hfill A& =& \frac{4(\beta -1)s}{(\alpha -1)(\beta +1)\mu },B=-\frac{24(\beta -1)s}{(\alpha -1)(\beta +1)\mu },D=\frac{24(\beta -1)s}{(\alpha -1)(\beta +1)\mu },\hfill \\ \hfill a& =& \mp \frac{\sqrt{2}\sqrt{s}}{\sqrt{-(\beta +1)\lambda }},b=\mp \frac{(\beta -1)\sqrt{s}}{\sqrt{2}\sqrt{-(\beta +1)\lambda }},c=\mp \frac{s^{3/2}\sqrt{-(\beta +1)\lambda }}{\sqrt{2}\lambda }.\hfill \end{array}$$

(13)

From Set 1, the first binary–soliton solution of TMNNV was:

$${\theta }_1(x,y,t)=\frac{12a^5{(\beta \mp 1)}^{2}d{\lambda }^2{e}^{\frac{a\left(a^2(\beta \mp 1)\lambda (x+y)\mp 2s^{2}t+2sx\right)}{a^2(\beta \mp 1)\lambda +2s}}\left(d{e}^{ax}-e^{\frac{a^3(\beta \mp 1)\lambda y\mp 2a{s}^{2}t}{a^2(\beta \mp 1)\lambda +2s}}\right)}{(\alpha \mp 1)\mu \left(a^2(\beta \mp 1)\lambda +2s\right){\left(d{e}^{ax}+e^{\frac{a^3(\beta \mp 1)\lambda y\mp 2a{s}^{2}t}{a^2(\beta \mp 1)\lambda +2s}}\right)}^3}.$$

(14)

From Set 2, the second binary–soliton solution of TMNNV was:

$${\theta }_2(x,y,t)=\mp \frac{12\sqrt{2}{(\beta -1)}^{2}d\lambda s^{3/2}{e}^{\pm (\frac{\sqrt{s}((\beta +1)st+2x+(\beta -1)y)}{\sqrt{2}\sqrt{-(\beta +1)\lambda }}}\left(e^{\pm \frac{\sqrt{s}((\beta +1)st+2x+(\beta -1)y)}{\sqrt{2}\sqrt{-(\beta +1)\lambda }}}-d\right)}{(\alpha -1)\mu {(-(\beta +1)\lambda )}^{3/2}{\left(d+e^{\pm (\frac{\sqrt{s}((\beta +1)st+2x+(\beta -1)y)}{\sqrt{2}\sqrt{-(\beta +1)\lambda }}}\right)}^3}.$$

(15)

3.2. Approach II: Modified Rational Sine–Cosine Functions

In this section, we explored periodic solutions to the TMNNV model by proposing a rational form in terms of both sine and cosine functions. We suggested the following construction [16,17]:

$$U\left(z\right)=\frac{1+Asin\left(z\right)}{B+Fcos\left(z\right)}.$$

(16)

Then, we inserted (16) in (9) and simplified the resulting substitutions to arrive at

$$\frac{\mathbf{Q}(1,sin\left(z\right),cos\left(z\right),sin\left(z\right)cos\left(z\right),{sin}^2\left(z\right),cos\left(z\right){sin}^2\left(z\right),{sin}^3\left(z\right))}{2{(B+Fcos\left(z\right))}^3}=0.$$

(17)

From (17), we set each of $1,sin\left(z\right),\dots ,{sin}^3\left(z\right)$ to zero to obtain an algebraic system which included the unknowns $A,B,F,a,b,c$. By doing so, we obtained the following system:

$$\begin{array}{ccc}\hfill 0& =& 2B^2\left(c^2-s^2\left(a^2+b^2\right)\right)-2F^2\left(a^{3}c\lambda +s\left(a^4\beta \lambda +a^{3}b\beta \lambda +a^{2}s+b^{2}s\right)-c^2\right)\hfill \\ & -& aB\mu \left(\alpha s\right(a+b)+c),\hfill \\ \hfill 0& =& F\left(-2B\left(a^{3}c\lambda +s\left(a^4\beta \lambda +a^{3}b\beta \lambda +2a^{2}s+2b^{2}s\right)-2c^2\right)-a\mu (\alpha s(a+b)+c)\right),\hfill \\ \hfill 0& =& 2A\left(B^2\left(a^{3}c\lambda +s\left(a^4\beta \lambda +a^{3}b\beta \lambda -a^{2}s-b^{2}s\right)+c^2\right)\right),\hfill \\ & +& 2A\left(F^2\left(-2a^{3}c\lambda -s\left(2a^4\beta \lambda +2a^{3}b\beta \lambda +a^{2}s+b^{2}s\right)+c^2\right)-aB\mu (\alpha s(a+b)+c)\right),\hfill \\ \hfill 0& =& AF\left(B\left(a^3(-c)\lambda -s\left(a^4\beta \lambda +a^{3}b\beta \lambda +2a^{2}s+2b^{2}s\right)+2c^2\right)-a\mu (\alpha s(a+b)+c)\right),\hfill \\ \hfill 0& =& s\left(2a^4\beta F^2\lambda +2a^{3}b\beta F^2\lambda +a^2\left(\alpha A^{2}B\mu -2F^{2}s\right)+\alpha a{A}^{2}bB\mu -2b^2{F}^{2}s\right)+2c^2{F}^2\hfill \\ & +& c\left(2a^3{F}^2\lambda +a{A}^{2}B\mu \right),\hfill \\ \hfill 0& =& -a{A}^{2}F\mu (\alpha s(a+b)+c),\hfill \\ \hfill 0& =& 2A{F}^2\left(s^2\left(a^2+b^2\right)-c^2\right).\hfill \end{array}$$

(18)

Now, by solving (18), we obtained two different sets of the unknown parameters:

Set I:

$$\begin{array}{ccc}\hfill A& =& 0,B=-\frac{(\alpha \mp 1)\mu }{6a^2(\beta \mp 1)\lambda },F=\frac{(\alpha \mp 1)\mu }{6a^2(\beta \mp 1)\lambda },b=-\frac{a^3(\beta \mp 1)\lambda }{a^2(\beta \mp 1)\lambda +2s},\hfill \\ \hfill c& =& \mp \frac{2a{s}^2}{a^2(\beta \mp 1)\lambda +2s}.\hfill \end{array}$$

(19)

Set II:

$$\begin{array}{ccc}\hfill A& =& 0,B=-\frac{(\alpha \mp 1)\mu }{6a^2(\beta \mp 1)\lambda },F=-\frac{(\alpha \mp 1)\mu }{6a^2(\beta \mp 1)\lambda },b=-\frac{a^3(\beta \mp 1)\lambda }{a^2(\beta \mp 1)\lambda +2s},\hfill \\ \hfill c& =& \mp \frac{2a{s}^2}{a^2(\beta \mp 1)\lambda +2s}.\hfill \end{array}$$

(20)

From Set I, the first binary–periodic solution of TMNNV was:

$${\theta }_3(x,y,t)=-\frac{3a^5{(\beta \mp 1)}^2{\lambda }^{2}cot\left(\frac{a\left(a^2(\beta \mp 1)\lambda (x-y)\pm 2s^{2}t+2sx\right)}{2a^2(\beta \mp 1)\lambda +4s}\right){csc}^2\left(\frac{a\left(a^2(\beta \mp 1)\lambda (x-y)\pm 2s^{2}t+2sx\right)}{2a^2(\beta \mp 1)\lambda +4s}\right)}{(\alpha \mp 1)\mu \left(a^2(\beta \mp 1)\lambda +2s\right)}.$$

(21)

From Set II, the second binary–periodic solution of TMNNV was:

$${\theta }_4(x,y,t)=-\frac{3a^5{(\beta \mp 1)}^2{\lambda }^{2}tan\left(\frac{a\left(a^2(\beta \mp 1)\lambda (x-y)\pm 2s^{2}t+2sx\right)}{2a^2(\beta \mp 1)\lambda +4s}\right){sec}^2\left(\frac{a\left(a^2(\beta \mp 1)\lambda (x-y)\pm 2s^{2}t+2sx\right)}{2a^2(\beta \mp 1)\lambda +4s}\right)}{(\alpha \mp 1)\mu \left(a^2(\beta \mp 1)\lambda +2s\right)}.$$

(22)

4. Physical Insights and Discussions

Different solutions were extracted in the new TMNNV model by imposing two approaches. Through investigation of these solutions, we always arrived at two instantaneous different values of the wave speed c, which, in turn, determined that the propagation of this model was due to moving binary waves with different wave heights. Sometimes, the height of a wave is known as the wave mode. The modes of these binary waves are determined by two parameters; the nonlinearity and the dispersion. In order to make it easier to name these binary waves, we refer to them as left-wave (L-wave) and right-wave (R-wave). The interaction of L-wave and R-wave is determined by one parameter which is the phase-velocity. As a result, the physical description becomes clear with regard to this type of two-mode equations, as it is characterized by containing three main factors that determine the dynamics of propagation of its binary waves.

Now, we identified the types of the obtained solutions and analyzed the impact of the TMNNV’s parameters. Both ${\theta }_1(x,y,t)$ and ${\theta }_2(x,y,t)$, obtained by Kudryashov’s solution, were of binary–soliton type, and each soliton was a combination of dark and bright waves. The Kudryashov method contains an index d that determines if the solution is singular ($d<0$) or nonsingular ($d>0$). Here we analyzed ${\theta }_1(x,y,t)$ for $d>0$. Figure 1 shows the wavefront of ${\theta }_1(x,y,t)$. Figure 2 shows that an increase in the phase–velocity s was followed by an increase in the distance between the L-wave and the R-wave. Finally, the modes of the obtained binary solitons varied according to the different assigned values of the nonlinearity $\alpha$ and the dispersion $\beta$. We observed optimal modes when $\alpha =0.53$ and $\beta =0.47$, (see Figure 3).

On the other hand, by applying the second approach, the modified rational sine–cosine form, we obtained two singular solutions ${\theta }_3(x,y,t)$ and ${\theta }_4(x,y,t)$, which were of the type binary–periodic waves, (see Figure 4) which represents the propagation of ${\theta }_3(x,y,t)$.

5. Conclusions

In this work, we presented a new model, TMNNV, that describes the motion of binary waves and has many applications in optics and water waves. Some solutions were obtained to the TMNNV by using two different schemes. We also investigated the role of the model’s parameters and its effect on the dynamics of propagation of the attained binary-wave solutions.

For future work, since the TMNNV is presented here for the first time, many solutions can be explored using other available techniques designed for constructing solitary wave solutions [18,19,20]. One may also investigate conservation laws, N-solitons, lumps, and breathers for the proposed model and other higher-dimensional problems [21,22,23,24].