Construction of Solitary Two-Wave Solutions for a New Two-Mode Version of the Zakharov-Kuznetsov Equation

Abstract

A new two-mode version of the generalized Zakharov-Kuznetsov equation is derived using Korsunsky’s method. This dynamical model describes the propagation of two-wave solitons moving simultaneously in the same direction with mutual interaction that depends on an embedded phase-velocity parameter. Three different methods are used to obtain exact bell-shaped soliton solutions and singular soliton solutions to the proposed model. Two-dimensional and three-dimensional plots are also provided to illustrate the interaction dynamics of the obtained two-wave exact solutions upon increasing the phase-velocity parameter.

1. Introduction

The work on nonlinear partial differential equations has been developed to get an insight through qualitative and quantitative features of many models arise in diverse fields, such as electro-magnetic waves, optics, nerve pulses, nonlinear dynamics, condensed matter physics and others. One of the essential properties of most of nonlinear equations is to capture the perfect balance between dispersion and nonlinearity effects which results in soliton pulse. The study of soliton solutions for nonlinear equations has been integrated by suggesting and developing ansatze methods that produce different types of solitons. Such compatible methods include Bernoulli sub-equation function method, $(G^{\prime }/G)$-expansion method, sine-cosine method, simplified-bilinear method, Kudryashov method, Unified methods, and many others listed in [1,2,3,4,5,6,7,8]. Motivated by exploring new physical insights for new models arise in physical sciences, we aim to propose new mathematical modification in the construction of one of the well-known physical models and recognize their dynamical soliton solutions. The suggested model to be addressed in this work is the Zakharov-Kuznetsov (ZK) equation.

The Zakharov-Kuznetsov (ZK) equation was first established to model the propagation of weakly nonlinear ion-acoustic waves in plasma, which includes cold ions and hot-isothermal electrons in a medium with a uniform magnetic field [9,10]. Moreover, it also describes the $(2+1)$-dimensional modulations of a KdV soliton equation in fluid mechanics [11]. The standard ZK equation reads

$$u_t+ku{u}_x+{(u_{xx}+u_{yy})}_x=0.$$

(1)

It has been shown that Equation (1) is not integrable by means of the inverse scattering transform test [12]. Thus, it is a difficult task to study it if compared with other integrable equations. In order to study the dynamics of ion-acoustic waves in cold-ion plasma when the behavior of electrons is not isothermal, Schamel [13] has derived a new $(1+1)$-dimensional ZK equation with a fractional power nonlinear term as follows:

$$u_t+u^{\frac{1}{2}}{u}_x+a{u}_{xxx}=0.$$

(2)

By means of the sine-cosine ansatz method [14], some special forms of exact solutions to the fractional ZK equation have been reported. Other related studies on the ZK equation can be found in [15,16]. A more general form of the $(2+1)$-dimensional ZK equation takes the form

$$u_t+{(u_{xx}+u_{yy}+k{u}^p)}_x=0.$$

(3)

In this paper, we aim to derive a two-mode version of the ZK equation given in (3). Two-mode equations are nonlinear partial differential equations of second-order in the time coordinate, and they describe the dynamics of the two-wave solitons propagating in the same direction, which overlap with one another without changing their shapes.

In [17], the overlapping of phase-locked waves and the over-taking waves have been observed, and their corresponding phase-speeds are found close to each other. These phenomena have been seen in the model of second-order in time of the Korteweg–de Vries equation, which reads as [18]

$$\begin{array}{ccc}{U}_{\tau \tau }+(c_1+c_2)U_{\chi \tau }+c_1{c}_2{U}_{\chi \chi }& +& (({\alpha }_1+{\alpha }_2)\frac{\partial }{\partial \tau }+({\alpha }_1{c}_2+{\alpha }_2{c}_1)\frac{\partial }{\partial \chi })U{U}_{\chi }\hfill \\ & +& (({\beta }_1+{\beta }_2)\frac{\partial }{\partial \tau }+({\beta }_1{c}_2+{\beta }_2{c}_1)\frac{\partial }{\partial \chi })U_{\chi \chi \chi }=0,\hfill \end{array}$$

(4)

where

• $\chi$ and $\tau$ are the scaled space and time coordinates.
• $U(\chi ,\tau )$ is the height of the water’s free surface above the flat bottom.
• $c_1$ and $c_2$ are the phase velocities.
• ${\alpha }_1$ and ${\alpha }_2$ are the linearity parameters.
• ${\beta }_1$ and ${\beta }_2$ are the dispersion parameters.

The model given in (4) describes the propagation of two-mode waves with the same dispersion relation and different phase-velocities, nonlinearity, and dispersion parameters [19]. In [18], Korsunsky reformulated (4) by using the following new variables defined as

$$\begin{array}{ccc}\hfill u& =& ({\alpha }_1+{\alpha }_2)U,\hfill \\ \hfill x& =& \frac{\chi -\frac{c_1+c_2}{2}\tau }{\sqrt{{\beta }_1+{\beta }_2}},\hfill \\ \hfill t& =& \frac{\tau }{\sqrt{{\beta }_1+{\beta }_2}},\hfill \end{array}$$

(5)

and proposing the constraints: $\left|\frac{{\alpha }_1-{\alpha }_2}{{\alpha }_1+{\alpha }_2}\right|\le 1$, $\left|\frac{{\beta }_1-{\beta }_2}{{\beta }_1+{\beta }_2}\right|\le 1$, and $c_2\le c_1$. Accordingly, (4) is converted into

$$u_{tt}-s^2{u}_{xx}+(\frac{\partial }{\partial t}-\alpha s\frac{\partial }{\partial x})u{u}_x+(\frac{\partial }{\partial t}-\beta s\frac{\partial }{\partial x})u_{xxx}=0.$$

(6)

Equation (6) is regarded as a two-mode KdV equation with $u=u(x,t)$ being the field function, $\alpha ,\beta$, respectively, are the nonlinearity and dispersion parameters that are less than 1, and s is the interaction phase velocity. Note here, that $\alpha ,\beta$ and s appear in (6) can be obtained if we let $c_1=-c_2=s$ and simplify algebraically (4). It is clear that, in case of $s=0$, no interaction occurs, and integrating once with respect to the time t, (6) is reduced to the standard KdV equation that describes the propagation of a single-moving wave.

Motivated by the Korsunsky’s technique, Wazwaz [20,21,22,23] has established the two-mode versions of Sharma–Tasso–Olver equation, fourth-order Burgers’ equation, fifth-order KdV equation, higher-order modified KdV and the KP equations, and has obtained multiple-kink solutions by adopting the simplified Hirota’s method. Furthermore, other two-mode models have been established by using Korsunsky’s scheme and their solutions have been obtained by means of simplified bilinear method, tanh-coth method, and the $(G^{\prime }/G)$-expansion method. Such types of two-mode equations have been derived for coupled Burgers’ equation, coupled KdV equation, coupled modified KdV equation, KdV–Burgers’ equation, third-order Fisher equation, Kuramoto-Sivashinsky equation, and higher-order Boussinesq-Burgers system [24,25,26,27,28,29,30]. Furthermore, in [31,32], the two-mode KdV equation and the two-mode Sharma-Tasso-Olver equation have been revisited and more new solitary wave solutions have been obtained. Moreover, the two-mode concept has been applied to the Schrödinger equations [33,34]. The dynamics of the two-mode phenomena have also been investigated in [35,36,37]. We should note here that the aforementioned works are devoted in presenting new two-mode equations based on Korsunsky’s method. We believe that other techniques will be further developed for the study of two-mode models.

The Korsunsky’s scheme to construct two-mode equations has the following scaled form

$$u_{tt}-s^2{u}_{xx}+(\frac{\partial }{\partial t}-\alpha s\frac{\partial }{\partial x})N(u,u_x,...)+(\frac{\partial }{\partial t}-\beta s\frac{\partial }{\partial x})L(u_{xx},u_{xxx},..)=0.$$

(7)

Here $N(u,u_x,...)$ and $L(u_{xx},u_{xxx},..)$ are the nonlinear and the linear terms of the model, respectively. In this work we extend (7) to construct $(2+1)$-dimensional two-mode equations that will take the form

$$\begin{array}{ccc}{u}_{tt}-s^2{u}_{xx}-s^2{u}_{yy}& +& (\frac{\partial }{\partial t}-{\alpha }_{1}s\frac{\partial }{\partial x}-{\alpha }_{2}s\frac{\partial }{\partial y})N(u,u_x,u_y,...)\hfill \\ & +& (\frac{\partial }{\partial t}-{\beta }_{1}s\frac{\partial }{\partial x}-{\beta }_{2}s\frac{\partial }{\partial y})L(u_{xx},u_{yy},u_{xy},u_{xxy},..)=0,\hfill \end{array}$$

(8)

where $u=u(x,y,t)$. Applying (8) on (3), we introduce the following $(2+1)$-dimensional two-mode Zakharov-Kuznetsov (TMZK) equation

$$\begin{array}{ccc}{u}_{tt}-s^2{u}_{xx}-s^2{u}_{yy}& +& (\frac{\partial }{\partial t}-{\alpha }_{1}s\frac{\partial }{\partial x}-{\alpha }_{2}s\frac{\partial }{\partial y})\left\{{(k{u}^p)}_x\right\}\hfill \\ & +& (\frac{\partial }{\partial t}-{\beta }_{1}s\frac{\partial }{\partial x}-{\beta }_{2}s\frac{\partial }{\partial y})\{u_{xxx}+u_{yyx}\}=0.\hfill \end{array}$$

(9)

We aim to study the solutions of the TMZK equation by implementing three different integration techniques: the sech-csch method, the Kudryashov’s expansion method, and the simplified bilinear method.

2. Bell-Shaped Soliton Solutions

To obtain a bell-shaped soliton solution for the TMZK equation, we consider the new variable $z=ax+by-ct$ to convert (9) into the following reduced-order differential equation

$$(c^2-(a^2+b^2)s^2)u-a(a^2+b^2)(c+{\beta }_{1}as+{\beta }_{2}bs)u^{″}-ak(c+{\alpha }_{1}as+{\alpha }_{2}bs)u^p=0,$$

(10)

where now $u=u\left(z\right)$. Then, we assume that the solution of (10) takes the form [38]

$$u\left(z\right)=Asec{h}^q\left(z\right).$$

(11)

We substitute (11) in (10), and we collect the coefficients of the same powers of $sech\left(z\right)$ to get the following outputs

$$\begin{array}{ccc}\hfill sec{h}^q\left(z\right)& :& A(c^2-(a^2+b^2)s^2)-aA{q}^2(a^2+b^2)(c+{\beta }_{1}as+{\beta }_{2}bs),\hfill \\ \hfill sec{h}^{pq}\left(z\right)& :& -ak{A}^p(c+{\alpha }_{1}as+{\alpha }_{2}bs),\hfill \\ \hfill sec{h}^{q+2}\left(z\right)& :& -aqA(1+q)(a^2+b^2)(c+{\beta }_{1}as+{\beta }_{2}bs).\hfill \end{array}$$

(12)

Equating the power indices of $sec{h}^{pq}$ against $sec{h}^{q+2}$, and setting the coefficient of the same power of $sech$ to zero, leads to the following system of equation

$$\begin{array}{ccc}\hfill 0& =& pq-(q+2),\hfill \\ \hfill 0& =& c^2-(a^2+b^2)s^2-a{q}^2(a^2+b^2)(c+{\beta }_{1}as+{\beta }_{2}bs),\hfill \\ \hfill 0& =& ak{A}^p(c+{\alpha }_{1}as+{\alpha }_{2}bs)+aqA(1+q)(a^2+b^2)(c+{\beta }_{1}as+{\beta }_{2}bs).\hfill \end{array}$$

(13)

Since the above system of equations involves many parameters, we require some reasonable restrictions. We may set

$$\begin{array}{ccc}\hfill {\alpha }_1& =& {\alpha }_2={\gamma }_1,\hfill \\ \hfill {\beta }_1& =& {\beta }_2={\gamma }_2.\hfill \end{array}$$

(14)

Solving (13) based on (14), we reach at the following findings

$$\begin{array}{ccc}\hfill q& =& \frac{2}{p-1}:p\ne 1,\hfill \\ \hfill c& =& \pm \sqrt{a^2+b^2}s,\hfill \end{array}$$

(15)

under the condition ${\gamma }_1=\frac{\pm \sqrt{a^2+b^2}-b{\gamma }_2}{a}$. Therefore, the two-mode bell-shaped soliton solution for (9) is

$$u(x,y,t)=Asec{h}^{\frac{2}{p-1}}(ax+by\pm \sqrt{a^2+b^2}st).$$

(16)

Figure 1, presents the dynamics of overlapping the obtained two-mode bell-shaped soliton solutions given by Equation (16) upon increasing the phase velocity s. Some remarks regarding the $(2+1)$-dimensional TMZK Equation (9) are as follows:

• If $u\left(z\right)=Acsc{h}^q\left(z\right)$ is considered instead of (11), a two-mode singular soliton solution for (9) will be obtained as $u(x,y,t)=Acsc{h}^{\frac{2}{p-1}}(ax+by\pm \sqrt{a^2+b^2}st)$, provided that ${\gamma }_1=\frac{\pm \sqrt{a^2+b^2}-b{\gamma }_2}{a}$.
• The obtained solution given in (16) preserve its bell-type shape when p varies.
• The overlapping of the obtained two-wave solutions does not change the shapes of these waves.

3. Kudryashov Expansion Method

In this section we solve the TMZK equation for the particular case $p=2$ by means of the Kudryashov expansion method. In particular, we study the following equation

$$u_{tt}-s^2{u}_{xx}-s^2{u}_{yy}+(\frac{\partial }{\partial t}-{\alpha }_{1}s\frac{\partial }{\partial x}-{\alpha }_{2}s\frac{\partial }{\partial y})\left\{ku{u}_x\right\}+(\frac{\partial }{\partial t}-{\beta }_{1}s\frac{\partial }{\partial x}-{\beta }_{2}s\frac{\partial }{\partial y})\left\{{(u_{xx}+u_{yy})}_x\right\}=0.$$

(17)

By using the new variable $\zeta =ax+by-ct$, (17) is converted into

$$(c^2-(a^2+b^2)s^2)u-a(a^2+b^2)(c+{\beta }_{1}as+{\beta }_{2}bs)u^{″}-a\frac{k}{2}(c+{\alpha }_{1}as+{\alpha }_{2}bs)u^2=0,$$

(18)

with $u=u\left(\zeta \right)$. The Kudryashov’s method [39,40] assumes the solution of (18) as a finite series in the variable Z:

$$\begin{array}{c}\hfill u\left(\zeta \right)=\sum _{i=0}^n{b}_i{Z}^i.\end{array}$$

(19)

The variable Z is the solution of the nonlinear differential equation

$$Z^{\prime }=\mu Z(Z-1).$$

(20)

Applying the separable method on (20) gives

$$Z\left(\zeta \right)=\frac{1}{1+d{e}^{\mu \zeta }}.$$

(21)

Performing the balance procedure on the terms $u^{″}$ and $u^2$, gives $n=2$ and accordingly we write (19) as

$$\begin{array}{ccc}\hfill u\left(\zeta \right)& =& b_0+b_{1}Z+b_2{Z}^2.\hfill \end{array}$$

(22)

Differentiating both (20) and (22), leads to

$$\begin{array}{ccc}\hfill Z^{″}& =& {\mu }^{2}Z(Z-1)(2Z-1),\hfill \end{array}$$

(23)

and

$$\begin{array}{ccc}\hfill u^{\prime }\left(z\right)& =& b_1{Z}^{\prime }+2b_{2}Z{Z}^{\prime },\hfill \\ \hfill u^{″}\left(z\right)& =& b_1{Z}^{″}+2b_2(Z{Z}^{″}+{\left(Z^{\prime }\right)}^2).\hfill \end{array}$$

(24)

Now, we substitute (20) through (24) in (18) to get a finite series in Z whose coefficients are identical to zero. To be able to solve the resulting system, we require the following two reasonable constraints:

$$\begin{array}{ccc}\hfill {\alpha }_1& =& {\alpha }_2={\gamma }_1,\hfill \\ \hfill {\beta }_1& =& {\beta }_2={\gamma }_2.\hfill \end{array}$$

(25)

Now, by solving the resulting system along with (25), we reach at the following outputs.

$$\begin{array}{ccc}\hfill b_0& =& 0,\hfill \\ \hfill b_1& =& \frac{12(a^2+b^2){\mu }^2}{k},\hfill \\ \hfill b_2& =& -b_1,\hfill \\ \hfill c& =& \frac{1}{2}(a(a^2+b^2){\mu }^2\mp \sqrt{(a^2+b^2)h}),\hfill \\ \hfill h& =& 4s^2+4as(a{\gamma }_1+b{\gamma }_2){\mu }^2+a^2(a^2+b^2){\mu }^4.\hfill \end{array}$$

(26)

Therefore, the two-wave solution of the TMZK model (17) is given by

$$u(x,y,t)=\frac{\frac{12{\mu }^2}{k}(a^2+b^2)d{e}^{\frac{\mu }{2}(-2(ax+by)+t(\mp \sqrt{(a^2+b^2)h}+a(a^2+b^2){\mu }^2))}}{{(d+e^{\frac{\mu }{2}(-2(ax+by)+t(\mp \sqrt{(a^2+b^2)h}+a(a^2+b^2){\mu }^2))})}^2}$$

(27)

Figure 2 is an overview of the dynamics of the profiles of solutions for the interaction of the two-wave solitons given by (27) upon increasing the phase velocity s.

4. Simplified Bilinear Method

In this section, we use the simplified bilinear method [41,42] to find the one-soliton solution for (17). This method requires the following functions

$$\begin{array}{ccc}\hfill h(x,y,t)& =& ax+by-ct,\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill v(x,y,t)& =& e^{h(x,y,t)},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill g(x,y,t)& =& 1+Bv(x,y,t):B=\pm 1,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill u(x,y,t)& =& A{(lng(x,y,t))}_{xx}.\hfill \end{array}$$

(31)

Substituting (29) in the linear terms of (17) and solving for c, we get

$$c=\frac{1}{2}(a^3+a{b}^2\mp \sqrt{a^6+2a^4{b}^2+a^2{b}^4+4a^2{s}^2+4b^2{s}^2+4a{b}^2(a+b)s{\gamma }_2}).$$

(32)

Considering the result obtained in (32), we substitute (31) in (17) and we solve it for A under the constraints ${\alpha }_1={\alpha }_2={\gamma }_1$ and ${\beta }_1={\beta }_2={\gamma }_2$, to get

$$\begin{array}{ccc}\hfill A& =& \frac{12a^5+24a^3{b}^2+12a(b^4+2b^{2}s{\gamma }_2)-12a^2▵-12b^2(-2bs{\gamma }_2+▵)}{a^{5}k+a^3{b}^{2}k+2a^{3}ks{\gamma }_1-a^{2}k(-2bs{\gamma }_1+▵)},\hfill \end{array}$$

(33)

where $▵=\sqrt{(a^2+b^2)(a^4+a^2{b}^2+4s^2)+4a{b}^2(a+b)s{\gamma }_2}$. Therefore, the one-soliton solution of Equation (17) is

$$\begin{array}{ccc}u(x,y,t)& =& -\frac{a^2{B}^{2}A{e}^{2ax+2by-t(a^3+a{b}^2-\sqrt{a^6+2a^4{b}^2+a^2{b}^4+4a^2{s}^2+4b^2{s}^2+4a^2{b}^{2}s{\gamma }_2+4a{b}^{3}s{\gamma }_2})}}{{(1+B{e}^{ax+by-\frac{1}{2}t(a^3+a{b}^2-\sqrt{a^6+2a^4{b}^2+a^2{b}^4+4a^2{s}^2+4b^2{s}^2+4a^2{b}^{2}s{\gamma }_2+4a{b}^{3}s{\gamma }_2})})}^2}\hfill \\ & +& \frac{a^{2}B{e}^{ax+by-\frac{1}{2}t(a^3+a{b}^2-\sqrt{a^6+2a^4{b}^2+a^2{b}^4+4a^2{s}^2+4b^2{s}^2+4a^2{b}^{2}s{\gamma }_2+4a{b}^{3}s{\gamma }_2})}}{1+B{e}^{ax+by-\frac{1}{2}t(a^3+a{b}^2-\sqrt{a^6+2a^4{b}^2+a^2{b}^4+4a^2{s}^2+4b^2{s}^2+4a^2{b}^{2}s{\gamma }_2+4a{b}^{3}s{\gamma }_2})}}.\hfill \end{array}$$

(34)

It is worth to mention that for $B=1$, (34) gives the bell-shaped soliton solution, while for $B=-1$, it gives the singular soliton-solution.

Furthermore, since the wave speed c has two different values as given by Equation (32), the soliton solution given by (34) describes the propagation of two-wave solitons moving simultaneously in the same direction with mutual interaction that depends on an embedded phase-velocity parameter and with no change in their shapes.

5. Conclusions

In this work, we have introduced a new nonlinear partial differential equation called the two-mode Zakharov-Kuznetsov (TMZK) equation. This model represents the overlapping of moving two-wave solitons that are propagating simultaneously, in the same direction. Three different methods are used to obtain exact soliton solutions of the TMZK equation. Both two-dimensional and three-dimensional plots are provided to show the profiles of the obtained soliton solutions upon increasing the phase-velocity parameter.

We should point here that the suggested model is proposed for the first time in this work. This study is adhered only with the mathematical modification of ZK equation and obtaining different forms of solutions but with the same soliton type which is found to be of bell-shaped type. In a future work, we aim to find a connection between these new equations with possible applications arise in physical sciences. Furthermore, one future goal, is to find possible soliton solutions of other two-mode equations that are relevant from the physics and engineering point of view by using either the Korsunsky’s method or similar integration techniques.