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Reliable methods to study some nonlinear conformable systems in shallow water

Abstract

In this paper, consider the eminent coupled Boussinesq–Burger (BB) equations and the coupled Whitham–Broer–Kaup (WBK) equations with time fractional derivative arising in the investigation of shallow water waves. The derivative is described in the sense of conformable derivative. We introduce the fundamental \({(G'} / {G} )\)-expansion method and its extension, namely the two-variable \({(G'} / {G}, {1} / {G} )\)-expansion method, to establish general solutions, some typical wave solutions existing in the literature, and some new and compatible soliton solutions comprised with certain parameters. For the definite values of these parameters, we derive and show in figures the well-known kink, singular kink, bell-shape soliton, periodic soliton, cuspon, and so on. The obtained solutions affirm that the introduced methods are reliable and efficient techniques to examine a wide variety of nonlinear fractional systems in the sense conformable derivative.

Introduction

Although the concept of fractional derivative is as old as that of the classical one, its advancement is not so old. Though relatively new, its use is increasing day by day due to its advantages in modeling and widespread applications to real-world problems, and thus it has generated much interest among researchers. There are many physical phenomena and processes, such as anomalous diffusion processes in physics, chemistry, and biology, complex diffusion process, diffusion in heterogeneous medium, diffusion processes in porous medium, viscoelasticity, viscoelastic deformation, viscous fluid, groundwater investigations [13], and so on, which can be analyzed more accurately through fractional differential equations than through integer-order differential equations. Fractional differential and integral operators have eliminated the drawback of classical integer-order difficulties considering their nonlocal characteristics [411]. There are different definitions of fractional derivatives and integrals, such as variable-order fractional derivative, Riemann–Liouville fractional derivative, Jumarie fractional derivative, Caputo fractional derivative, Weyl fractional derivative, and so on. To describe anomalous diffusion phenomena, constant-order fractional diffusion equations are introduced and have had great success. On the contrary, to characterize some complex diffusion processes, for instance, diffusion processes in heterogeneous medium [12] and diffusion processes in porous medium, if the medium structure or external field changes with time, then the variable-order fractional diffusion equations properly model the incidents [13]. The Riemann–Liouville and Jumarie derivatives are recognized as a powerful modeling approach in the fields of viscoelasticity, viscoelastic deformation, viscous fluid [1416], anomalous diffusion [17], and so on. The problem in groundwater investigation can be better analyzed by the Weyl fractional-order derivative [18]. The Caputo-type fractional derivative [19] is useful in investigating numerical solutions of a model equation. Therefore, in the recent years, fractional calculus has become an emerging and interesting branch of applied mathematics and analysis.

Referring to mathematical models of the problems, the estimation of traveling wave solutions of fractional nonlinear differential equations (FNDEs) helps us to better understand these phenomena. Therefore various mathematical approaches have been proposed over the past few decades to extract soliton solutions to FNDEs, such as the auxiliary equation method [20], the exp-function method [21, 22], the simplest equation method [23], the sine–cosine method [24], the first integral method [25, 26], the \(( {G'} / {G)}\)-expansion method [2729], the \({(G'} / {G}, {1} / {G} )\)-expansion method [3032], the Kudryashov method [3335], the subequation method [36], the Jacobi elliptic equation method [37], the Ricatti–Bernoulli sub-ODE method [3840], and so on.

The exact traveling solutions to the coupled BB equation have been established in [41]. By using the Lie symmetry analysis Mhlanga and Khalique [42] described the traveling wave solutions to the generalized coupled BB equation. The envelope soliton and periodic wave solutions have been studied by Ebadi et al. [43]. The coupled WBK equations are examined by other researchers using different analytical and numerical methods, such as the exp-function method [44], the Adomian decomposition method [45], the \(( {G '} / {G}^{2} )\)-expansion method [46], the hyperbolic function method [47], the Lie symmetry analysis [48], the differential transformation method [49], the homotopy analysis method [50], and so on. Recently, Amjad et al. [51] used the result of a standard order coupled fractional-order Whitham—Broer–Kaup equation by the Laplace decomposition method. To the best of our knowledge, the coupled Boussinesq–Burger and Whitham–Broer–Kaup nonlinear fractional differential equations have not been examined through the \(( {G'} / {G)}\)-expansion method [52] and the \({(G'} / {G}, {1} / {G} )\)-expansion method [53]. Therefore, motivated by the studies mentioned, the objective of our study is to extract general solutions, some classical wave solutions, and some compatible soliton solutions entangled with parameters to the equations mentioned. When we set definite values of the parameters, bell-shape soliton, kink, periodic, and other solitary wave solutions are originated from the broad-ranging general solution.

The rest of the paper is arranged as follows: In Sect. 2, we present the properties of the conformable derivative. In Sect. 3, we present the basic idea of the \(( {G'} / {G)}\)-expansion method. In Sect. 4, we give the algorithm of the \({(G'} / {G}, {1} / {G} )\)-expansion method. In Sect. 5, we implement the methods to extract soliton solutions to the coupled BB and coupled WBK systems of time-fractional order. In Sect. 6, we provide a physical explanation and graphs of the solutions. In Sect. 7, we present conclusions.

The conformable derivative and some its properties

The conformable derivative of a function g of order α is defined as [5457]

$$ T_{\alpha } (g) (t)= \lim_{x\rightarrow 0} \frac{g(t+x t^{1-\alpha } )-g(t)}{x},\quad t> 0, \alpha \in ( 0, 1), $$
(2.1)

where \(g: [ 0, \infty ) \rightarrow R\), \(t>0\), and \(\alpha \in ( 0, 1 )\).

Let \(\alpha \in ( 0, 1 )\), and let g, f be α-differentiable at the point t. Then \(T_{\alpha }\) satisfies the following important properties:

$$\begin{aligned}& T_{\alpha } ( a g+b f ) =a T_{\alpha } ( g ) +b T_{\alpha } ( f ),\quad a, b\in R; \end{aligned}$$
(2.2)
$$\begin{aligned}& T_{\alpha } \bigl( t^{\mu } \bigr) =\mu t^{\mu -\alpha },\quad \mu \in R; \end{aligned}$$
(2.3)
$$\begin{aligned}& T_{\alpha } ( gof ) ( t ) = t^{1-\alpha } f ' ( t ) g ' \bigl( f ( t ) \bigr); \end{aligned}$$
(2.4)
$$\begin{aligned}& T_{\alpha } ( g f ) ( t ) =g T_{\alpha } ( f ) +f T_{\alpha } ( g ); \end{aligned}$$
(2.5)
$$\begin{aligned}& T_{\alpha } \biggl( \frac{g}{f} \biggr) ( t ) = \frac{f T_{\alpha } ( g ) - gT_{\alpha } ( f )}{f^{2}}. \end{aligned}$$
(2.6)

If g is a differentiable function, then \(T_{\alpha } ( g ) ( t ) = t^{1 - \alpha } \frac{dg}{dt} ( t )\).

The algorithm of the \(( {G'} / {G)}\)-expansion method for FNDEs

Consider an FNDE of the form

$$ F \bigl( v, D_{t}^{\alpha } v, D_{t}^{\alpha } D_{t}^{\alpha } v,\ldots \bigr) = 0,\quad 0 < \alpha < 1, $$
(3.1)

where F is a polynomial of v and fractional partial derivatives of s, and \(v=v(x, t)\) is an unidentified function to be computed.

By using the wave transformation

$$ v ( x, t ) = V ( \zeta ),\qquad \zeta = k x - w\frac{t^{\alpha }}{\alpha }, $$
(3.2)

where w is the wave velocity, and k is the wave number, both nonzero constants, the FNDE (3.1) can be rewritten as the following ordinary differential equation (ODE):

$$ P \biggl( V, \frac{dV}{d\zeta }, \frac{d^{2}V}{d\zeta ^{2}}, \ldots \biggr) = 0. $$
(3.3)

Assume that Eq. (3.3) has the formal solution

$$ V ( \zeta ) = \sum_{i = 0}^{N} \alpha _{i} \biggl( \frac{G' ( \zeta )}{G ( \zeta )} \biggr)^{i},\quad \alpha _{N} \ne 0, $$
(3.4)

where \(\alpha _{i} \) (\(i= 0, 1, \ldots, N\)) are constants, and the function \(G ( \zeta )\) satisfies the auxiliary equation

$$ G'' ( \zeta ) + \lambda G' ( \zeta ) + \mu G ( \zeta ) = 0. $$
(3.5)

The constant N in Eq. (3.4) can be determined by balancing the highest-order derivatives and nonlinear terms turn up in Eq. (3.3). Substituting solution (3.4) into (3.3), using (3.5), summing up all terms of the same order of (\({G '} / {G} \)), and setting the coefficients to zero yield a system of algebraic equations for \(\alpha _{i} \) (\(i=0, 1, \dots , N\)), k, w, λ, and μ. We obtain the values of constants \(\alpha _{i} \) (\(i=0, 1, \dots , N\)), k, w, λ, and μ by unraveling this system, and substituting these constants and the general solutions of (3.5) into solution (3.4), we attain adequate travelling wave solutions to the FNDE (3.1).

Description of the \(( {G'} / {G, {1} / {G)}}\)-expansion method for FNDEs

In this section, we interpret the \(( {G'} / {G, {1} / {G)}}\)-expansion method as follows [53]. For the auxiliary differential equation

$$ G'' ( \zeta ) + \lambda G ( \zeta ) = \mu , $$
(4.1)

we set

$$ \phi = \frac{G'}{G},\qquad \psi = \frac{1}{G}, $$
(4.2)

Thus from (4.1) and (4.2) we can derive

$$ \phi ' = - \phi ^{2} + \mu \psi - \lambda ,\qquad \psi ' = - \phi \psi . $$
(4.3)

The solutions of Eq. (4.1) are subject to the following three cases.

Case 1: If \(\lambda < 0\), then the formal solution of Eq. (4.1) is

$$ G ( \zeta ) = A_{1}\sinh ( \sqrt{ - \lambda } \zeta ) + A_{2}\cosh ( \sqrt{ - \lambda } \zeta ) + \mu / \lambda , $$
(4.4)

and the corresponding relation is

$$ \psi ^{2} = - \lambda \biggl( \frac{\phi ^{2} - 2\mu \psi + \lambda }{\lambda ^{2}\sigma + \mu ^{2}} \biggr), $$
(4.5)

where \(\sigma = A_{1}^{2} - A_{2}^{2}\).

Case 2: If \(\lambda > 0\), then the standard solution of Eq. (4.1) is

$$ G ( \zeta ) = A_{1} \sin ( \sqrt{\lambda } \zeta ) + A_{2} \cos ( \sqrt{\lambda } \zeta ) + {\mu } / {\lambda }. $$
(4.6)

Thus the relation between ϕ and ψ is

$$ \psi ^{2} = \lambda \biggl( \frac{\phi ^{2} - 2\mu \psi + \lambda }{\lambda ^{2}\sigma - \mu ^{2}} \biggr), $$
(4.7)

where \(\sigma = A_{1}^{2} + A_{2}^{2}\).

Case 3: If \(\lambda = 0\), then the typical solution of Eq. (4.1) is

$$ G ( \zeta ) = \frac{\mu }{2}\zeta ^{2} + A_{1}\zeta + A_{2}, $$
(4.8)

and the relations between ϕ and ψ is

$$ \psi ^{2} = \frac{\phi ^{2} -2\mu \psi }{A_{1}^{2} -2\mu A_{2}}, $$
(4.9)

where \(A_{1}\), \(A_{2}\) are integral constants.

The major steps of the \(( {G'} / {G, {1} / {G)}}\)-expansion method are described as follows.

Step 1: We assume that Eq. (3.3) has the general solution

$$ V ( \zeta ) = \sum_{i = 0}^{S} \alpha _{i}\phi ^{i} + \sum_{i = 1}^{S} \beta _{i}\phi ^{i - 1}\psi , $$
(4.10)

where G is the solution of the auxiliary Eq. (4.1), \(\alpha _{i}\) (\(i= 0, 1, \ldots, S \)), \(\beta _{i}\) (\(i= 1, 2, \ldots, S\)), μ, and λ are constants, and S in solution (4.10) can be determined by the balancing theory from Eq. (3.3).

Step 2: Substituting (4.10) into (3.3) with (4.3) and (4.5), we obtain a polynomial in ψ and ϕ, where the degree of ψ is not greater than one. Setting all coefficients of the polynomial to zero yields a set of algebraic equations, which can be solved with the help of Maple software package, and substituting the values of k, w, μ, λ, \(\alpha _{i}\), \(\beta _{i}\) into (4.10), we get analytical exact solutions to Eq. (3.3) expressed by the hyperbolic function.

Step 3: Substituting (4.10) into (3.3) with (4.3) and (4.7) (or (4.3) and (4.9)), we get exact solutions to Eq. (3.1) expressed by trigonometric or rational functions, respectively.

Extraction of soliton solutions

In this section, we extract the traveling wave, including periodic, kink, bell-shape soliton, and so on wave solutions to the following fractional systems utilizing the methods described in Sects. 3 and 4.

First, we consider the model based on the nonlinear time fractional coupled BB equations [58, 59]:

$$ \begin{gathered} D_{t}^{\alpha } u- \frac{1}{2} v_{x} +2u u_{x} =0, \\ t> 0, \qquad 0< \alpha < 1, \\ D_{t}^{\alpha } v- \frac{1}{2} u_{xxx} +2 ( u v )_{x} =0, \end{gathered} $$
(5.1)

where \(u ( x, t )\) represents the horizontal velocity field, and \(v ( x, t )\) indicates the water surface height above a horizontal level from the bottom.

Second, we consider a model based on the nonlinear time fractional coupled WBK equations [49]:

$$ \begin{gathered} D_{t}^{\alpha } u+u u_{x} + v_{x} +c u_{xx} =0, \\ t> 0, \qquad 0< \alpha < 1, \\ D_{t}^{\alpha } v+ (uv)_{x} +b u_{xxx} -c v_{xx} =0, \end{gathered} $$
(5.2)

where the constants b and c represent the coefficients of diffusion and dissipation, respectively, \(u ( x, t )\) represents the horizontal field of horizontal velocity, and \(v ( x, t )\) indicates the height of deviation from the liquid equilibrium position.

The time-fractional coupled BB equations

For the time-fractional coupled BB equations, we put the wave transformation

$$ \begin{gathered} u ( x, t ) = U ( \zeta ),\qquad v ( x, t ) = V ( \zeta ), \\ \zeta = x - w \frac{t^{\alpha }}{\alpha }. \end{gathered} $$
(5.3)

We will take the advantage of fractional wave transformation to turn system (5.1) into the ODEs:

$$ \begin{gathered} wU' + \frac{1}{2}V' - 2UU' = 0, \\ wV' + \frac{1}{2}U''' - 2 ( UV )^{\prime } = 0, \end{gathered} $$
(5.4)

By integrating we obtain

$$ \begin{gathered} w U+ \frac{1}{2} V- U^{2} + c_{1} =0, \\ w V+ \frac{1}{2} U^{\prime \prime } -2 U V+ c_{2} =0, \end{gathered} $$
(5.5)

where \(c_{1}\) and \(c_{2}\) are integrating constants.

Balancing between V and \(U^{2}\), \(U''\), and UV in (5.5), we find \(N_{1} =1\) and \(N_{2} =2\).

Therefore the formal solutions of (5.5) can be presented by a polynomial in \(( G' / G )\):

$$ \begin{gathered} U ( \zeta ) = \alpha _{0} + \alpha _{1} \bigl( G' / G \bigr), \quad \alpha _{1} \ne 0, \\ V ( \zeta ) = \beta _{0} + \beta _{1} \bigl( G' / G \bigr) + \beta _{2} \bigl( G' / G \bigr)^{2},\quad \beta _{2} \ne 0. \end{gathered} $$
(5.6)

Embedding (5.6) into (5.5) and applying the procedure stated in Sect. 3, we get the following result:

$$ \begin{gathered} \alpha _{0} =\mp {\lambda } / {4} + {w} / {2},\qquad \alpha _{1} = \mp {1} / {2},\qquad \beta _{0} = {\mu } / {2}, \qquad \beta _{1} = {\lambda } / {2},\qquad \beta _{2} = {1} / {2},\\ c_{1} = {\lambda ^{2}} / {16} - {w^{2}} / {4} - {\mu } / {4},\qquad c_{2} =0, \end{gathered} $$
(5.7)

where w is an arbitrary constant.

By means of the values assembled in (5.7) and the general solutions of (3.5), from solution (5.6) we accomplish three types of solitary wave solutions to the coupled BB Eq. (5.1) as follows.

Type I: When \(( \lambda ^{2} - 4\mu ) > 0\), we attain the hyperbolic function solutions of (5.1):

$$ \begin{gathered} u_{1} ( x, t ) = \frac{w}{2} \mp \frac{\sqrt{\lambda ^{2} - 4\mu }}{4} \times \biggl( \frac{B_{1}\cosh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} + B_{2}\sinh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2}}{B_{1}\sinh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} + B_{2}\cosh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2}} \biggr), \\ v_{1} ( x, t ) = \frac{ ( \lambda ^{2} - 4\mu )}{8} \times \biggl[ \biggl( \frac{B_{1}\cosh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} + B_{2}\sinh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2}}{B_{1}\sinh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} + B_{2}\cosh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2}} \biggr)^{2} - 1 \biggr], \end{gathered} $$
(5.8)

where \(\zeta = x - w \frac{t^{\alpha }}{\alpha } \). Solution (5.8) is the general hyperbolic type of the coupled BB equation, from which different compact-form solutions can be extracted for definite values of the integral constants.

For \(B_{1} = 0\), \(B_{2} \ne 0\), we get the squeezed bell-shape and kink-type solitary wave solutions to the coupled BB Eqs. (5.1):

$$ \begin{gathered} u_{1_{1}} (x, t)= \frac{w}{2} \mp \frac{\sqrt{\lambda ^{2} -4\mu }}{4} \tanh \biggl( \sqrt{\lambda ^{2} -4\mu } \frac{\zeta }{2} \biggr), \\ v_{1_{1}} ( x, t ) =- \frac{ ( \lambda ^{2} -4\mu )}{8} \operatorname{sech}^{2} \biggl( \sqrt{\lambda ^{2} -4\mu } \frac{\zeta }{2} \biggr). \end{gathered} $$
(5.9)

On the other hand, if we put \(B_{2} = 0\), \(B_{1} \ne 0\), we gain singular solitary wave solutions of (5.1):

$$ \begin{gathered} u_{1_{2}} (x, t)= \frac{w}{2} \mp \frac{\sqrt{\lambda ^{2} -4\mu }}{4} \coth \biggl( \sqrt{\lambda ^{2} -4\mu } \frac{\zeta }{2} \biggr), \\ v_{1_{2}} ( x, t ) = \frac{ ( \lambda ^{2} -4\mu )}{8} \operatorname{csch}^{2} \biggl( \sqrt{\lambda ^{2} -4\mu } \frac{\zeta }{2} \biggr). \end{gathered} $$
(5.10)

Again, if we set \(B_{2} \ne 0\), \(B_{2}^{2} > B_{1}^{2}\), we accomplish the solitary wave solutions of (5.1):

$$ \begin{gathered} u_{1_{3}} ( x, t ) = \frac{w}{2} \mp \frac{\sqrt{\lambda ^{2} - 4\mu }}{4}\tanh \biggl( \zeta _{0} + \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} \biggr), \\ v_{1_{3}} ( x, t ) = - \frac{ ( \lambda ^{2} - 4\mu )}{8}\operatorname{sech}^{2} \biggl( \zeta _{0} + \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} \biggr), \end{gathered} $$
(5.11)

where \(\zeta _{0} = \tanh ^{ - 1}\frac{B_{1}}{B_{2}}\).

However, if we set \(B_{1} \ne 0\), \(B_{1}^{2} > B_{2}^{2}\), we attain the singular solitary wave solutions of (5.1):

$$ \begin{gathered} u_{1_{4}} ( x, t ) = \frac{w}{2} \mp \frac{\sqrt{\lambda ^{2} - 4\mu }}{4}\coth \biggl( \zeta _{0} + \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} \biggr), \\ v_{1_{4}} ( x, t ) = \frac{ ( \lambda ^{2} - 4\mu )}{8}\operatorname{csch}^{2} \biggl( \zeta _{0} + \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} \biggr), \end{gathered} $$
(5.12)

where \(\zeta _{0} = \tanh ^{ - 1}\frac{B_{2}}{B_{1}}\).

Type II: When \(( \lambda ^{2} - 4\mu ) < 0\), we derive the trigonometric function solutions of (5.1):

$$ \begin{gathered} u_{2} ( x, t ) = \frac{w}{2} \mp \frac{\sqrt{4\mu - \lambda ^{2}}}{4} \biggl( \frac{ - B_{1}\sin \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} + B_{2}\cos \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2}}{B_{1}\cos \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} + B_{2}\sin \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2}} \biggr), \\ v_{2} ( x, t ) = \frac{ ( 4\mu - \lambda ^{2} )}{8} \times \biggl[ 1 + \biggl( \frac{ - B_{1}\sin \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} + B_{2}\cos \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2}}{B_{1}\cos \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} + B_{2}\sin \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2}} \biggr)^{2} \biggr]. \end{gathered} $$
(5.13)

Since \(B_{1}\) and \(B_{2}\) are integral constants; someone is able to accept their values spontaneously. Therefore, if we accept \(B_{1} = 0\), \(B_{2} \ne 0\), we attain the subsequent singular periodic wave solutions to the nonlinear coupled Boussinesq–Burger equations:

$$ \begin{gathered} u_{2_{1}} (x, t)= \frac{w}{2} \mp \frac{\sqrt{4\mu - \lambda ^{2}}}{4} \cot \biggl( \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \\ v_{2_{1}} ( x, t ) = \frac{ ( 4\mu - \lambda ^{2} )}{8} \operatorname{csc}^{2} \biggl( \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr). \end{gathered} $$
(5.14)

Furthermore, if we accept \(B_{2} = 0\), \(B_{1} \ne 0\), then we determine the following singular periodic wave solutions to the nonlinear coupled BB equations:

$$ \begin{gathered} u_{2_{2}} (x, t)= \frac{w}{2} \pm \frac{\sqrt{4\mu - \lambda ^{2}}}{4} \tan \biggl( \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \\ v_{2_{2}} ( x, t ) = \frac{ ( 4\mu - \lambda ^{2} )}{8} \operatorname{sec}^{2} \biggl( \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr). \end{gathered} $$
(5.15)

However, if \(B_{2} \ne 0\), \(B_{2}^{2} > B_{1}^{2}\), then we obtain the singular periodic wave solutions of (5.1) as follows:

$$ \begin{gathered} u_{2_{3}} ( x, t ) = \frac{w}{2} \mp \frac{\sqrt{4\mu - \lambda ^{2}}}{4}\cot \biggl( \zeta _{0} + \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \\ v_{2_{3}} ( x, t ) = \frac{ ( 4\mu - \lambda ^{2} )}{8}\csc ^{2} \biggl( \zeta _{0} + \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \end{gathered} $$
(5.16)

where \(\zeta _{0} = \tan ^{ - 1}\frac{B_{1}}{B_{2}}\).

On the other hand, if \(B_{1} \ne 0\), \(B_{1}^{2} > B_{2}^{2}\), then we attain the periodic wave solutions of (5.1) as follows:

$$ \begin{gathered} u_{2_{4}} ( x, t ) = \frac{w}{2} \pm \frac{\sqrt{4\mu - \lambda ^{2}}}{4}\tan \biggl( \zeta _{0} - \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \\ v_{2_{4}} ( x, t ) = \frac{ ( 4\mu - \lambda ^{2} )}{8}\sec ^{2} \biggl( \zeta _{0} - \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \end{gathered} $$
(5.17)

where \(\zeta _{0} = \tan ^{ - 1}\frac{B_{2}}{B_{1}}\).

Type III: When \(( \lambda ^{2} - 4\mu ) = 0\), we ensure the subsequent rational function solutions of (5.1):

$$ \begin{gathered} u_{3} (x, t)= \frac{w}{2} \mp \frac{1}{2} \frac{B_{2}}{B_{1} + B_{2} \zeta }, \\ v_{3} ( x, t ) = \frac{1}{2} \biggl( \frac{ B_{2}}{B_{1} + B_{2} \zeta } \biggr)^{2}, \end{gathered} $$
(5.18)

wherein \(B_{1}\) could be zero, but \(B_{2}\) cannot be zero; otherwise, solution (5.18) would turn into steady solution, which has no physical significance.

Now we use the two-variable \(( G' / G, 1 / G )\)-expansion method to analyze the wave solutions to the coupled BB equation. Accordingly, we look for the solutions in the form

$$ \begin{gathered} U ( \zeta ) = a_{0} + a_{1} \phi + b_{1} \psi , \\ V ( \zeta ) = \alpha _{0} + \alpha _{1} \phi + \alpha _{2} \phi ^{2} + \beta _{1} \psi + \beta _{2} \phi \psi , \end{gathered} $$
(5.19)

where \(a_{0}\), \(a_{1}\), \(b_{1}\), \(\alpha _{0}\), \(\alpha _{1}\), \(\alpha _{2}\), \(\beta _{1}\), and \(\beta _{2}\) are constants to be determined. As we mentioned in Sect. 4, we have three cases.

Case 1: When \(\lambda < 0\), inserting (5.19) into (5.5), by (4.3), (4.4), and (4.5) system (5.5) turns into a polynomial in ψ and ϕ. The coefficients of this equation yield a system of algebraic equations in \(a_{0}\), \(a_{1}\), \(b_{1}\), \(\alpha _{0}\), \(\alpha _{1}\), \(\alpha _{2}\), \(\beta _{1}\), \(\beta _{2}\), w, λ, μ, σ, \(\zeta _{1}\), and \(\zeta _{2}\). Solving the algebraic equations via Maple software package, we obtain three different sets of results.

Result 1

$$ \begin{gathered} a_{0} = \pm \frac{1}{2}\sqrt{ - 4 \zeta _{1} + \frac{2\lambda ^{3}\sigma - \lambda \mu ^{2}}{4\lambda ^{2}\sigma + 4\mu ^{2}}} \pm \frac{\mu }{4}\sqrt{ - \frac{\lambda }{\lambda ^{2}\sigma + \mu ^{2}}},\\ a_{1} = 0,\qquad b_{1} = \pm \sqrt{ - \frac{\lambda ^{2}\sigma + \mu ^{2}}{4\lambda }}, \\ \alpha _{0} = \frac{1}{4}\frac{\lambda ^{3}\sigma + 2\lambda \mu ^{2}}{\lambda ^{2}\sigma + \mu ^{2}},\qquad \alpha _{1} = 0,\qquad \alpha _{2} = \frac{1}{2},\qquad \beta _{1} = - \frac{1}{2}\mu ,\qquad \beta _{2} = 0, \\ w = \pm \sqrt{ - 4\zeta _{1} + \frac{2\lambda ^{3}\sigma - \lambda \mu ^{2}}{4\lambda ^{2}\sigma + 4\mu ^{2}}},\qquad \zeta _{1} = \zeta _{1}, \qquad \zeta _{2} = \mp \frac{\sigma \lambda ^{3}\mu \sqrt{ - \lambda }}{8 ( \lambda ^{2}\sigma + \mu ^{2} )\sqrt{\lambda ^{2}\sigma + \mu ^{2}}}. \end{gathered} $$
(5.20)

From (4.4), (5.19), and (5.20) with (5.3) we derive the following hyperbolic function solutions of (5.1):

$$ \begin{gathered} u_{1} ( x, t ) = \pm \frac{1}{2} \sqrt{ - 4\zeta _{1} + \frac{2\lambda ^{3}\sigma - \lambda \mu ^{2}}{4\lambda ^{2}\sigma + 4\mu ^{2}}} \pm \frac{1}{4} \frac{\mu \sqrt{ - \lambda }}{\sqrt{\lambda ^{2}\sigma + \mu ^{2}}} \\ \hphantom{ u_{1} ( x, t ) =}{}\pm \frac{1}{2\sqrt{ - \lambda }} \frac{\sqrt{\lambda ^{2}\sigma + \mu ^{2}}}{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda }, \\ v_{1} ( x, t ) = \frac{\lambda ^{3}\sigma + 2\lambda \mu ^{2}}{4\lambda ^{2}\sigma + 4\mu ^{2}} - \frac{\lambda }{2} \biggl( \frac{A_{1}\cosh \sqrt{ - \lambda } \zeta + A_{2}\sinh \sqrt{ - \lambda } \zeta }{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda } \biggr)^{2} \\ \hphantom{v_{1} ( x, t ) =}{}- \frac{1}{2}\frac{\mu }{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda }, \end{gathered} $$
(5.21)

where \(\zeta = x \mp \sqrt{ - 4\zeta _{1} + \frac{2\lambda ^{3}\sigma - \lambda \mu ^{2}}{4\lambda ^{2}\sigma + 4\mu ^{2}}} \frac{t^{\alpha }}{\alpha } \) and \(\sigma = A_{1}^{2} - A_{2}^{2}\).

Inasmuch as \(A_{1}\), \(A_{2}\), and μ are free parameters, choosing \(A_{1} = 0\), \(\mu = 0\), and \(A_{2} > 0\), from (5.21) we attain the following solitary wave solutions:

$$ \begin{gathered} u_{1_{1}} (x, t)=\pm \frac{1}{2} \biggl( \sqrt{-4 \zeta _{1} + \frac{\lambda }{2}} +i \sqrt{-\lambda } \operatorname{sech} \sqrt{-\lambda } \zeta \biggr), \\ v_{1_{1}} ( x, t ) = \frac{\lambda }{4} - \frac{\lambda }{2} \tanh ^{2} \sqrt{-\lambda } \zeta . \end{gathered} $$
(5.22)

Alternatively, choosing \(A_{2} = 0\), \(\mu = 0\), and \(A_{1} > 0\), we attain the solitary wave solutions

$$ \begin{gathered} u_{1_{2}} ( x, t ) = \pm \frac{1}{2} \biggl( \sqrt{ - 4\zeta _{1} + \frac{\lambda }{2}} + \sqrt{ - \lambda } \operatorname{csch} \sqrt{ - \lambda } \zeta \biggr), \\ v_{1_{2}} ( x, t ) = \frac{\lambda }{4} - \frac{\lambda }{2}\coth ^{2}\sqrt{ - \lambda } \zeta . \end{gathered} $$
(5.23)

Result 2

$$ \begin{gathered} a_{0} = \pm \frac{1}{4}\sqrt{ - \lambda - 16\zeta _{1}},\qquad a_{1} = \frac{1}{4},\qquad b_{1} = \pm \frac{1}{4}\sqrt{ - \frac{\lambda ^{2}\sigma + \mu ^{2}}{\lambda }}, \\ \alpha _{0} = \frac{1}{4}\lambda ,\qquad \alpha _{1} = 0,\qquad \alpha _{2} = \frac{1}{4},\qquad \beta _{1} = - \frac{1}{4}\mu ,\qquad \beta _{2} = \pm \frac{1}{4}\sqrt{ - \frac{\lambda ^{2}\sigma + \mu ^{2}}{\lambda }}, \\ w = \pm \frac{1}{2}\sqrt{ - \lambda - 16\zeta _{1}},\qquad \zeta _{1} = \zeta _{1},\qquad \zeta _{2} = 0. \end{gathered} $$
(5.24)

From (4.4), (5.19), and (5.24) with (5.3) we find the following hyperbolic function solutions of (5.1):

$$ \begin{gathered} u_{2} ( x, t ) = \pm \frac{1}{4} \sqrt{ - \lambda - 16\zeta _{1}} \\ \hphantom{u_{2} ( x, t ) =}{}+ \frac{\sqrt{ - \lambda }}{4}\frac{A_{1}\cosh \sqrt{ - \lambda } \zeta + A_{2}\sinh \sqrt{ - \lambda } \zeta }{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda } \\ \hphantom{u_{2} ( x, t ) =}{}\pm \frac{1}{4\sqrt{ - \lambda }} \frac{\sqrt{\lambda ^{2}\sigma + \mu ^{2}}}{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda }, \\ v_{2} ( x, t ) = \frac{\lambda }{4} - \frac{1}{4} \frac{\mu }{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda } \\ \hphantom{v_{2} ( x, t ) =}{}- \frac{1}{4}\frac{A_{1}\cosh \sqrt{ - \lambda } \zeta + A_{2}\sinh \sqrt{ - \lambda } \zeta }{ ( A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda )^{2}} \\ \hphantom{v_{2} ( x, t ) =}{}\times \bigl[ A_{1}\lambda \cosh \sqrt{ - \lambda } \zeta + A_{2}\lambda \sinh \sqrt{ - \lambda } \zeta \mp \sqrt{\lambda ^{2}\sigma + \mu ^{2}} \bigr], \end{gathered} $$
(5.25)

where \(\zeta = x \mp \frac{1}{2}\sqrt{ - \lambda - 16\zeta _{1}} \frac{t^{\alpha }}{\alpha } \).

In particular, if we set \(A_{1} = 0\), \(\mu = 0\), and \(A_{2} > 0\) into (5.25), we attain the following solitary wave solutions:

$$ \begin{gathered} u_{2_{1}} ( x, t ) = \pm \frac{1}{4} \sqrt{ - \lambda - 16\zeta _{1}} + \frac{\sqrt{ - \lambda }}{4} [ \tanh \sqrt{ - \lambda } \zeta \pm i\operatorname{sech} \sqrt{ - \lambda } \zeta ], \\ v_{2_{1}} ( x, t ) = \frac{\lambda }{4} - \frac{\lambda }{4}\tanh \sqrt{ - \lambda } \zeta [ \tanh \sqrt{ - \lambda } \zeta \pm i\operatorname{sech} \sqrt{ - \lambda } \zeta ]. \end{gathered} $$
(5.26)

On the contrary, if we set \(A_{2} = 0\), \(\mu = 0\), and \(A_{1} > 0\), we attain the solitary wave solutions

$$ \begin{gathered} u_{2_{2}} ( x, t ) = \pm \frac{1}{4} \sqrt{ - \lambda - 16\zeta _{1}} + \frac{\sqrt{ - \lambda }}{4} [ \coth \sqrt{ - \lambda } \zeta \pm \operatorname{csch} \sqrt{ - \lambda } \zeta ], \\ v_{2_{2}} ( x, t ) = \frac{\lambda }{4} - \frac{\lambda }{4}\coth \sqrt{ - \lambda } \zeta [ \coth \sqrt{ - \lambda } \zeta \mp \operatorname{csch} \sqrt{ - \lambda } \zeta ]. \end{gathered} $$
(5.27)

Result 3

$$ \begin{gathered} a_{0} = \pm \frac{1}{4}\sqrt{ - \lambda - 16\zeta _{1}},\qquad a_{1} = - \frac{1}{4},\qquad b_{1} = \pm \frac{1}{4}\sqrt{ - \frac{\lambda ^{2}\sigma + \mu ^{2}}{\lambda }}, \\ \alpha _{0} = \frac{1}{4}\lambda ,\qquad \alpha _{1} = 0,\qquad \alpha _{2} = \frac{1}{4},\\ \beta _{1} = - \frac{1}{4}\mu ,\qquad \beta _{2} = \mp \frac{1}{4}\sqrt{ - \frac{\lambda ^{2}\sigma + \mu ^{2}}{\lambda }}, \\ w = \pm \frac{1}{2}\sqrt{ - \lambda - 16\zeta _{1}},\qquad \zeta _{1} = \zeta _{1},\qquad \zeta _{2} = 0. \end{gathered} $$
(5.28)

From result 3 we gain other hyperbolic function solutions of (5.1):

$$ \begin{gathered} u_{3} ( x, t ) = \pm \frac{1}{4} \sqrt{ - \lambda - 16\zeta _{1}} - \frac{\sqrt{ - \lambda }}{4}\frac{A_{1}\cosh \sqrt{ - \lambda } \zeta + A_{2}\sinh \sqrt{ - \lambda } \zeta }{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda } \\ \hphantom{u_{3} ( x, t ) =}{}\pm \frac{1}{4\sqrt{ - \lambda }} \frac{\sqrt{\lambda ^{2}\sigma + \mu ^{2}}}{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda }, \\ v_{3} ( x, t ) = \frac{\lambda }{4} - \frac{1}{4} \frac{\mu }{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda }\\ \hphantom{v_{3} ( x, t ) =}{} - \frac{1}{4}\frac{A_{1}\cosh \sqrt{ - \lambda } \zeta + A_{2}\sinh \sqrt{ - \lambda } \zeta }{ ( A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda )^{2}} \\ \hphantom{v_{3} ( x, t ) =}{}\times \bigl[ A_{1}\lambda \cosh \sqrt{ - \lambda } \zeta + A_{2}\lambda \sinh \sqrt{ - \lambda } \zeta \pm \sqrt{\lambda ^{2}\sigma + \mu ^{2}} \bigr], \end{gathered} $$
(5.29)

where \(\zeta = x \mp \frac{1}{2}\sqrt{ - \lambda - 16\zeta _{1}} \frac{t^{\alpha }}{\alpha } \).

Similarly, if the parameters take distinct values, then we deduce many other solitary wave solutions, but for conciseness, here we do not document the other solutions.

Case 2 When \(\lambda > 0\), substituting (5.19) into (5.5) and using (4.3), (4.6), and (4.7), system (5.5) can be expressed as a polynomial in ψ and ϕ. Vanishing all coefficients from this polynomial, we obtain a system of algebraic equations, which can be solved by utilizing Maple software package to get different results.

Result 1

$$ \begin{gathered} a_{0} = \pm \frac{1}{2}\sqrt{ - 4 \zeta _{1} + \frac{2\lambda ^{3}\sigma + \lambda \mu ^{2}}{4\lambda ^{2}\sigma - 4\mu ^{2}}} \pm \frac{\mu }{4}\sqrt{ \frac{\lambda }{\lambda ^{2}\sigma - \mu ^{2}}},\\ a_{1} = 0,\qquad b_{1} = \pm \frac{1}{2}\sqrt{\frac{\lambda ^{2}\sigma - \mu ^{2}}{\lambda }}, \\ \alpha _{0} = \frac{1}{4}\frac{\lambda ^{3}\sigma - 2\lambda \mu ^{2}}{\lambda ^{2}\sigma - \mu ^{2}},\qquad \alpha _{1} = 0,\qquad \alpha _{2} = \frac{1}{2},\qquad \beta _{1} = - \frac{1}{2}\mu ,\qquad \beta _{2} = 0, \\ w = \pm \sqrt{ - 4\zeta _{1} + \frac{2\lambda ^{3}\sigma + \lambda \mu ^{2}}{4\lambda ^{2}\sigma - 4\mu ^{2}}},\qquad \zeta _{1} = \zeta _{1},\qquad \zeta _{2} = \mp \frac{\sigma \lambda ^{3}\mu \sqrt{\lambda }}{8 ( \lambda ^{2}\sigma - \mu ^{2} )\sqrt{\lambda ^{2}\sigma - \mu ^{2}}}. \end{gathered} $$
(5.30)

From (4.6), (5.19), and (5.30) with (5.3), we deduce the following trigonometric function solutions of (5.1):

$$ \begin{gathered} u_{1} ( x, t ) = \pm \frac{1}{2} \sqrt{ - 4\zeta _{1} + \frac{2\lambda ^{3}\sigma + \lambda \mu ^{2}}{4\lambda ^{2}\sigma - 4\mu ^{2}}} \pm \frac{\mu }{4} \frac{\sqrt{\lambda }}{\sqrt{\lambda ^{2}\sigma - \mu ^{2}}} \\ \hphantom{u_{1} ( x, t ) =}{}\pm \frac{1}{2 \sqrt{\lambda }} \frac{\sqrt{\lambda ^{2}\sigma - \mu ^{2}}}{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda }, \\ v_{1} ( x, t ) = \frac{\lambda ^{3}\sigma - 2\lambda \mu ^{2}}{4\lambda ^{2}\sigma - 4\mu ^{2}} + \frac{\lambda }{2} \biggl( \frac{A_{1}\cos \sqrt{\lambda } \zeta - A_{2}\sin \sqrt{\lambda } \zeta }{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda } \biggr)^{2} \\ \hphantom{v_{1} ( x, t ) =}{}- \frac{1}{2}\frac{\mu }{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda }, \end{gathered} $$
(5.31)

where \(\zeta = x \mp \sqrt{ - 4\zeta _{1} + \frac{2\lambda ^{3}\sigma + \lambda \mu ^{2}}{4\lambda ^{2}\sigma - 4\mu ^{2}}} \frac{t^{\alpha }}{\alpha } \) and \(\sigma = A_{1}^{2} + A_{2}^{2}\).

In particular, by taking \(A_{1} = 0\), \(\mu = 0\), and \(A_{2} > 0\) in (5.31), we achieve the following periodic wave solutions:

$$ \begin{gathered} u_{1_{1}} ( x, t ) = \pm \frac{1}{2} \sqrt{ - 4\zeta _{1} + \frac{\lambda }{2}} \pm \frac{\sqrt{\lambda }}{2}\sec \sqrt{\lambda } \zeta , \\ v_{1_{1}} ( x, t ) = \frac{\lambda }{4} + \frac{\lambda }{2}\tan ^{2}\sqrt{\lambda } \zeta , \end{gathered} $$
(5.32)

whereas for \(A_{2} = 0\), \(\mu = 0\), and \(A_{1} > 0\), we deduce the periodic wave solutions

$$ \begin{gathered} u_{1_{2}} ( x, t ) = \pm \frac{1}{2} \sqrt{ - 4\zeta _{1} + \frac{\lambda }{2}} \pm \frac{\sqrt{\lambda }}{2}\csc \sqrt{\lambda } \zeta , \\ v_{1_{2}} ( x, t ) = \frac{\lambda }{4} + \frac{\lambda }{2}\cot ^{2}\sqrt{\lambda } \zeta . \end{gathered} $$
(5.33)

Result 2

$$ \begin{gathered} a_{0} = \pm \frac{1}{4}\sqrt{ - \lambda - 16\zeta _{1}},\qquad a_{1} = \frac{1}{4},\qquad b_{1} = \pm \frac{1}{4}\sqrt{\frac{\lambda ^{2}\sigma - \mu ^{2}}{\lambda }}, \\ \alpha _{0} = \frac{1}{4}\lambda ,\qquad \alpha _{1} = 0,\qquad \alpha _{2} = \frac{1}{4},\qquad \beta _{1} = - \frac{1}{4}\mu ,\qquad \beta _{2} = \pm \frac{1}{4}\sqrt{ \frac{\lambda ^{2}\sigma - \mu ^{2}}{\lambda }}, \\ w = \pm \frac{1}{2}\sqrt{ - \lambda - 16\zeta _{1}},\qquad \zeta _{1} = \zeta _{1},\qquad \zeta _{2} = 0. \end{gathered} $$
(5.34)

From (4.6), (5.19), and (5.34) with (5.3), we get the following trigonometric function solutions of (5.1):

$$ \begin{gathered} u_{2} ( x, t ) = \pm \frac{1}{4} \sqrt{ - \lambda - 16\zeta _{1}} + \frac{\sqrt{\lambda }}{4}\frac{A_{1}\cos \sqrt{\lambda } \zeta - A_{2}\sin \sqrt{\lambda } \zeta }{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda } \\ \hphantom{u_{2} ( x, t ) =}{}\pm \frac{1}{4 \sqrt{\lambda }} \frac{\sqrt{\lambda ^{2}\sigma - \mu ^{2}}}{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda }, \\ v_{2} ( x, t ) = \frac{\lambda }{4} - \frac{1}{4} \frac{\mu }{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda } \\ \hphantom{v_{2} ( x, t ) =}{}+ \frac{1}{4}\frac{A_{1}\cos \sqrt{\lambda } \zeta - A_{2}\sin \sqrt{\lambda } \zeta }{ ( A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda )^{2}} \\ \hphantom{v_{2} ( x, t ) =}{}\times \bigl[ A_{1}\lambda \cos \sqrt{\lambda } \zeta - A_{2}\lambda \sin \sqrt{\lambda } \zeta \pm \sqrt{\lambda ^{2}\sigma - \mu ^{2}} \bigr], \end{gathered} $$
(5.35)

where \(\zeta = x \mp \frac{1}{2}\sqrt{ - \lambda - 16\zeta _{1}} \frac{t^{\alpha }}{\alpha } \).

In particular, if we set \(A_{1} = 0\), \(\mu = 0\), and \(A_{2} > 0\) into (5.35), then we get the following periodic wave solutions:

$$ \begin{gathered} u_{2_{1}} ( x, t ) = \pm \frac{1}{4} \sqrt{ - \lambda - 16\zeta _{1}} - \frac{\sqrt{\lambda }}{4} [ \tan \sqrt{ \lambda } \zeta \mp \sec \sqrt{\lambda } \zeta ], \\ v_{2_{1}} ( x, t ) = \frac{\lambda }{4} + \frac{\lambda }{4}\tan \sqrt{ \lambda } \zeta [ \tan \sqrt{\lambda } \zeta \mp \sec \sqrt{\lambda } \zeta ]. \end{gathered} $$
(5.36)

Moreover, if we set \(A_{2} = 0\), \(\mu = 0\), and \(A_{1} > 0\), then we find the periodic wave solutions

$$ \begin{gathered} u_{2_{2}} ( x, t ) = \pm \frac{1}{4} \sqrt{ - \lambda - 16\zeta _{1}} + \frac{\sqrt{\lambda }}{4} [ \cot \sqrt{ \lambda } \zeta \pm \csc \sqrt{\lambda } \zeta ], \\ v_{2_{2}} ( x, t ) = \frac{\lambda }{4} + \frac{\lambda }{4}\cot \sqrt{ \lambda } \zeta [ \cot \sqrt{\lambda } \zeta \pm \csc \sqrt{\lambda } \zeta ]. \end{gathered} $$
(5.37)

Result 3

$$ \begin{gathered} a_{0} = \pm \frac{1}{4}\sqrt{ - \lambda - 16\zeta _{1}},\qquad a_{1} = - \frac{1}{4},\qquad b_{1} = \pm \frac{1}{4}\sqrt{\frac{\lambda ^{2}\sigma - \mu ^{2}}{\lambda }}, \\ \alpha _{0} = \frac{1}{4}\lambda ,\qquad \alpha _{1} = 0,\qquad \alpha _{2} = \frac{1}{4},\qquad \beta _{1} = - \frac{1}{4}\mu , \qquad \beta _{2} = \mp \frac{1}{4}\sqrt{ \frac{\lambda ^{2}\sigma - \mu ^{2}}{\lambda }}, \\ w = \pm \frac{1}{2}\sqrt{ - \lambda - 16\zeta _{1}},\qquad \zeta _{1} = \zeta _{1},\qquad \zeta _{2} = 0. \end{gathered} $$
(5.38)

From result, we have other trigonometric function solutions of (5.1):

$$ \begin{gathered} u_{3} ( x, t ) = \pm \frac{1}{4} \sqrt{ - \lambda - 16\zeta _{1}} - \frac{\sqrt{\lambda }}{4}\frac{A_{1}\cos \sqrt{\lambda } \zeta - A_{2}\sin \sqrt{\lambda } \zeta }{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda } \\ \hphantom{u_{3} ( x, t ) =}{}\pm \frac{1}{4 \sqrt{\lambda }} \frac{\sqrt{\lambda ^{2}\sigma - \mu ^{2}}}{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda }, \\ v_{3} ( x, t ) = \frac{\lambda }{4} - \frac{1}{4} \frac{\mu }{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda }\\ \hphantom{v_{3} ( x, t ) =}{} + \frac{1}{4}\frac{A_{1}\cos \sqrt{\lambda } \zeta - A_{2}\sin \sqrt{\lambda } \zeta }{ ( A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda )^{2}} \\ \hphantom{v_{3} ( x, t ) =}{}\times \bigl[ A_{1}\lambda \cos \sqrt{\lambda } \zeta - A_{2}\lambda \sin \sqrt{\lambda } \zeta \mp \sqrt{\lambda ^{2}\sigma - \mu ^{2}} \bigr], \end{gathered} $$
(5.39)

where \(\zeta = x \mp \frac{1}{2}\sqrt{ - \lambda - 16\zeta _{1}} \frac{t^{\alpha }}{\alpha } \).

Similarly, By taking special values of the parameters we deduce many other periodic wave solutions.

Case 3 When \(\lambda = 0\), substituting (5.19) into (5.5), by (4.3), (4.8), and (4.9) system (5.5) can be exposed as a polynomial in ψ and ϕ. Equating each coefficient of this polynomial to zero, we obtain a system of algebraic equations, which is analyzed by applying Maple software package, and get the following results:

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} a_{0} = a_{0},\qquad a_{1} = 0,\qquad b_{1} = \pm \frac{1}{2}\sqrt{A_{1}^{2} - 2A_{2}\mu },\qquad w = 2a_{0} \mp \frac{\mu }{2\sqrt{A_{1}^{2} - 2A_{2}\mu }}, \\ \alpha _{0} = - \frac{1}{4}\frac{\mu ^{2}}{A_{1}^{2} - 2A_{2}\mu },\qquad \alpha _{1} = 0,\qquad \alpha _{2} = \frac{1}{2},\qquad \beta _{1} = - \frac{1}{2}\mu ,\qquad \beta _{2} = 0, \\ \zeta _{1} = - a_{0}^{2} + \frac{\mu ^{2} \pm 4a_{0} \mu \sqrt{A_{1}^{2} - 2A_{2}\mu }}{8 ( A_{1}^{2} - 2A_{2}\mu )},\qquad \zeta _{2} = \mp \frac{\mu ^{3}}{8 ( A_{1}^{2} - 2A_{2}\mu )\sqrt{A_{1}^{2} - 2A_{2}\mu }} \end{array}\displaystyle \right \} , \end{aligned}$$
(5.40)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} a_{0} = a_{0},\qquad a_{1} = \frac{1}{4},\qquad b_{1} = \pm \frac{1}{4}\sqrt{A_{1}^{2} - 2A_{2}\mu }, \\ \alpha _{0} = 0,\qquad \alpha _{1} = 0,\qquad \alpha _{2} = \frac{1}{4},\qquad \beta _{1} = - \frac{1}{4}\mu ,\qquad \beta _{2} = \pm \frac{1}{4}\sqrt{A_{1}^{2} - 2A_{2}\mu }, \\ w = 2a_{0},\qquad \zeta _{1} = - a_{0}^{2},\qquad \zeta _{2} = 0 \end{array}\displaystyle \right \} , \end{aligned}$$
(5.41)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} a_{0} = a_{0},\qquad a_{1} = - \frac{1}{4},\qquad b_{1} = \pm \frac{1}{4}\sqrt{A_{1}^{2} - 2A_{2}\mu }, \\ \alpha _{0} = 0,\qquad \alpha _{1} = 0,\qquad \alpha _{2} = \frac{1}{4},\qquad \beta _{1} = - \frac{1}{4}\mu ,\qquad \beta _{2} = \mp \frac{1}{4}\sqrt{A_{1}^{2} - 2A_{2}\mu }, \\ w = 2a_{0},\qquad \zeta _{1} = - a_{0}^{2},\qquad \zeta _{2} = 0 \end{array}\displaystyle \right \} . \end{aligned}$$
(5.42)

In this case the rational function solutions to the coupled BB (5.1) are:

$$ \begin{gathered} u_{1} ( x, t ) = a_{0} \pm \frac{\sqrt{A_{1}^{2} - 2A_{2}\mu }}{\mu \zeta ^{2} + 2A_{1}\zeta + 2A_{2}}, \\ v_{1} ( x, t ) = - \frac{1}{4}\frac{\mu ^{2}}{A_{1}^{2} - 2A_{2}\mu } + \frac{1}{2} \biggl( \frac{\mu \zeta + A_{1}}{\frac{\mu }{2}\zeta ^{2} + A_{1}\zeta + A_{2}} \biggr)^{2} - \frac{\mu }{\mu \zeta ^{2} + 2A_{1}\zeta + 2A_{2}}, \end{gathered} $$
(5.43)

where \(\zeta =x- ( 2 a_{0} \mp \frac{\mu }{2 \sqrt{A_{1}^{2} -2 A_{2} \mu }} ) \frac{t^{\alpha }}{\alpha }\);

$$ \begin{gathered} u_{2} ( x, t ) = a_{0} + \frac{\mu \zeta + A_{1}}{2\mu \zeta ^{2} + 4A_{1}\zeta + 4A_{2}} \pm \frac{\sqrt{A_{1}^{2} - 2A_{2}\mu }}{2\mu \zeta ^{2} + 4A_{1}\zeta + 4A_{2}}, \\ v_{2} ( x, t ) = - \frac{\mu }{2\mu \zeta ^{2} + 4A_{1}\zeta + 4A_{2}} \\ \hphantom{v_{2} ( x, t ) = }{}+ \frac{1}{4} \frac{\mu \zeta + A_{1}}{ ( \frac{\mu }{2}\zeta ^{2} + A_{1}\zeta + A_{2} )^{2}} \times \Bigl[ \mu \zeta + A_{1} \pm \sqrt {A_{1}^{2} - 2A_{2}\mu } \Bigr], \end{gathered} $$
(5.44)

where \(\zeta =x-2 a_{0} \frac{t^{\alpha }}{\alpha }\); and

$$ \begin{gathered} u_{3} ( x, t ) = a_{0} - \frac{\mu \zeta + A_{1}}{2\mu \zeta ^{2} + 4A_{1}\zeta + 4A_{2}} \pm \frac{\sqrt{A_{1}^{2} - 2A_{2}\mu }}{2\mu \zeta ^{2} + 4A_{1}\zeta + 4A_{2}}, \\ v_{3} ( x, t ) = - \frac{\mu }{2\mu \zeta ^{2} + 4A_{1}\zeta + 4A_{2}}\\ \hphantom{v_{3} ( x, t ) =}{} + \frac{1}{4} \frac{\mu \zeta + A_{1}}{ ( \frac{\mu }{2}\zeta ^{2} + A_{1}\zeta + A_{2} )^{2}} \times \Bigl[ \mu \zeta + A_{1} \mp \sqrt {A_{1}^{2} - 2A_{2}\mu } \Bigr], \end{gathered} $$
(5.45)

where \(\zeta = x - 2a_{0}\frac{t^{\alpha }}{\alpha } \).

These solutions are generalized further and also contain extra free parameters. The definite values of these parameters yield some solutions available in the literature as particular cases. This modification validates the achieved results.

The nonlinear time-fractional coupled WBK equations

In this section, we use transformation (5.3) to reduce system (5.2) into the following ODEs:

$$ \begin{gathered} w U ' -U U ' - V ' -c U^{\prime \prime } = 0, \\ w V ' - ( UV ) ' -b U^{\prime \prime \prime } +c V^{\prime \prime } = 0. \end{gathered} $$
(5.46)

Integrating Eq. (5.46), we get

$$ \begin{gathered} wU- \frac{1}{2} U^{2} -V-c U ' + \zeta _{1} = 0, \\ wV-UV-b U^{\prime \prime } +c V ' + \zeta _{2} = 0. \end{gathered} $$
(5.47)

By balancing theory, from V and \(U^{2}\) and from \(U''\) and UV appearing in (5.47) we get \(N = 1\) and \(S = 2\). Therefore the formal solutions to Eq. (5.47) are of the following form:

$$ \begin{gathered} U ( \zeta ) = \alpha _{0} + \alpha _{1} \bigl( G' / G \bigr),\quad \alpha _{1} \ne 0, \\ V ( \zeta ) = \beta _{0} + \beta _{1} \bigl( G' / G \bigr) + \beta _{2} \bigl( G' / G \bigr)^{2},\quad \beta _{2} \ne 0. \end{gathered} $$
(5.48)

Substituting (5.48) into (5.47) and applying the same procedure discussed in Sect. 3, we get the following results:

$$ \begin{gathered} \alpha _{0} = \mp \lambda \sqrt{c^{2} + b} + w,\qquad \alpha _{1} = \mp 2 \sqrt{c^{2} + b},\\ \beta _{0} = 2\mu \bigl( \mp c \sqrt{c^{2} + b} - c^{2} - b \bigr),\qquad \beta _{1} = 2 \lambda \bigl( \mp c\sqrt{c^{2} + b} - c^{2} - b \bigr), \\ \beta _{2} = 2 \bigl( \mp c\sqrt{c^{2} + b} - c^{2} - b \bigr),\qquad w = w, \\ \xi _{1} = \frac{1}{2} \bigl( \lambda ^{2} - 4\mu \bigr) \bigl( c^{2} + b \bigr) - \frac{1}{2}w^{2},\qquad \xi _{2} = 0. \end{gathered} $$
(5.49)

By substituting (5.49) and general solutions of Eq. (3.5) into (5.48) with (5.3) we deduce three types of solitary wave solutions of the coupled WBK Eq. (5.2).

Type I: When \(( \lambda ^{2} - 4\mu ) > 0\), we acquire the hyperbolic function solutions of (5.2):

$$ \begin{gathered} u_{1} ( x, t ) = w \mp \sqrt{ \bigl( \lambda ^{2} - 4\mu \bigr) \bigl( c^{2} + b \bigr)} \biggl( \frac{B_{1}\cosh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} + B_{2}\sinh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2}}{B_{1}\sinh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} + B_{2}\cosh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2}} \biggr), \\ v_{1} ( x, t ) = \frac{\lambda ^{2} - 4\mu }{2} \bigl( \mp c\sqrt{c^{2} + b} - c^{2} - b \bigr) \\ \hphantom{v_{1} ( x, t ) =}{}\times \biggl[ \biggl( \frac{B_{1}\cosh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} + B_{2}\sinh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2}}{B_{1}\sinh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} + B_{2}\cosh \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2}} \biggr)^{2} - 1 \biggr], \end{gathered} $$
(5.50)

where \(\zeta = x - w \frac{t^{\alpha }}{\alpha } \).

Here \(B_{1}\) and \(B_{2}\) are integral constants. Therefore we can randomly select their values. Thus, if we select \(B_{1} = 0\), \(B_{2} \ne 0\), then we accomplish the kink and bell-shape solitary wave solutions of (5.2) of the form

$$ \begin{gathered} u_{1_{1}} (x, t)=w\mp \sqrt{\bigl( \lambda ^{2} -4\mu \bigr) \bigl( c^{2} +b\bigr)} \tanh \biggl( \sqrt{ \lambda ^{2} -4\mu } \frac{\zeta }{2} \biggr), \\ v_{1_{1}} ( x, t ) = \frac{\lambda ^{2} -4\mu }{2} \bigl( \pm c \sqrt{c^{2} +b} + c^{2} +b \bigr) \operatorname{sech}^{2} \biggl( \sqrt{\lambda ^{2} -4\mu } \frac{\zeta }{2} \biggr). \end{gathered} $$
(5.51)

Moreover, if we select \(B_{2} = 0\), \(B_{1} \ne 0\), then we obtain the singular solitary wave solutions of (5.2):

$$ \begin{gathered} u_{1_{2}} ( x, t ) = w \mp \sqrt{ \bigl( \lambda ^{2} - 4\mu \bigr) \bigl( c^{2} + b \bigr)} \coth \biggl( \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} \biggr), \\ v_{1_{2}} ( x, t ) = \frac{\lambda ^{2} - 4\mu }{2} \bigl( \mp c\sqrt{c^{2} + b} - c^{2} - b \bigr)\operatorname{csch}^{2} \biggl( \sqrt{ \lambda ^{2} - 4\mu } \frac{\zeta }{2} \biggr). \end{gathered} $$
(5.52)

In addition, if we select \(B_{2} \ne 0\), \(B_{2}^{2} > B_{1}^{2}\), then we get the solitary wave solutions of (5.2):

$$ \begin{gathered} u_{1_{3}} ( x, t ) = w \mp \sqrt{ \bigl( \lambda ^{2} - 4\mu \bigr) \bigl( c^{2} + b \bigr)} \tanh \biggl( \zeta _{0} + \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} \biggr), \\ v_{1_{3}} ( x, t ) = \frac{\lambda ^{2} - 4\mu }{2} \bigl( \pm c\sqrt{c^{2} + b} + c^{2} + b \bigr)\operatorname{sech}^{2} \biggl( \zeta _{0} + \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} \biggr), \end{gathered} $$
(5.53)

where \(\zeta _{0} = \tanh ^{ - 1}\frac{B_{1}}{B_{2}}\).

However, if we select \(B_{1} \ne 0\), \(B_{1}^{2} > B_{2}^{2}\), then we carry out the solitary wave solutions of (5.2):

$$ \begin{gathered} u_{1_{4}} ( x, t ) = w \mp \sqrt{ \bigl( \lambda ^{2} - 4\mu \bigr) \bigl( c^{2} + b \bigr)} \coth \biggl( \zeta _{0} + \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} \biggr), \\ v_{1_{4}} ( x, t ) = \frac{\lambda ^{2} - 4\mu }{2} \bigl( \mp c\sqrt{c^{2} + b} - c^{2} - b \bigr)\operatorname{csch}^{2} \biggl( \zeta _{0} + \sqrt{\lambda ^{2} - 4\mu } \frac{\zeta }{2} \biggr), \end{gathered} $$
(5.54)

where \(\zeta _{0} = \tanh ^{ - 1}\frac{B_{2}}{B_{1}}\).

Type II. When \(( \lambda ^{2} - 4\mu ) < 0\), we get trigonometric function solutions of (5.2) of the form

$$ \begin{gathered} u_{2} ( x, t ) = w \mp \sqrt{ \bigl( 4\mu - \lambda ^{2} \bigr) \bigl( c^{2} + b \bigr)} \biggl( \frac{ - B_{1}\sin \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} + B_{2}\cos \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2}}{B_{1}\cos \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} + B_{2}\sin \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2}} \biggr), \\ v_{2} ( x, t ) = \frac{4\mu - \lambda ^{2}}{2} \bigl( \mp c\sqrt{c^{2} + b} - c^{2} - b \bigr)\\ \hphantom{v_{2} ( x, t ) =}{}\times \biggl[ \biggl( \frac{ - B_{1}\sin \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} + B_{2}\cos \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2}}{B_{1}\cos \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} + B_{2}\sin \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2}} \biggr)^{2} + 1 \biggr]. \end{gathered} $$
(5.55)

We might accept \(B_{1} = 0\), \(B_{2} \ne 0\), Since \(B_{1}\) and \(B_{2}\) are integral constants, we find the following periodic wave solutions of (5.2):

$$ \begin{gathered} u_{2_{1}} (x, t)=w\mp \sqrt{\bigl(4\mu - \lambda ^{2} \bigr) \bigl( c^{2} +b\bigr)} \cot \biggl( \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \\ v_{2_{1}} ( x, t ) = \frac{4\mu - \lambda ^{2}}{2} \bigl( \mp c \sqrt{c^{2} +b} - c^{2} -b \bigr) \operatorname{csc}^{2} \biggl( \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr). \end{gathered} $$
(5.56)

In addition, if we put \(B_{2} = 0\), \(B_{1} \ne 0\), then we find the following the periodic wave solutions of (5.2):

$$ \begin{gathered} u_{2_{2}} (x, t)=w\pm \sqrt{\bigl(4\mu - \lambda ^{2} \bigr) \bigl( c^{2} +b\bigr)} \tan \biggl( \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \\ v_{2_{2}} ( x, t ) = \frac{4\mu - \lambda ^{2}}{2} \bigl( \mp c \sqrt{c^{2} +b} - c^{2} -b \bigr) \operatorname{sec}^{2} \biggl( \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr). \end{gathered} $$
(5.57)

Besides, if we set \(B_{2} \ne 0\), \(B_{2}^{2} > B_{1}^{2}\), then we achieve the following periodic wave solutions of (5.2):

$$ \begin{gathered} u_{2_{3}} ( x, t ) = w \mp \sqrt{ \bigl( 4\mu - \lambda ^{2} \bigr) \bigl( c^{2} + b \bigr)} \cot \biggl( \zeta _{0} + \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \\ v_{2_{3}} ( x, t ) = \frac{4\mu - \lambda ^{2}}{2} \bigl( \mp c\sqrt{c^{2} + b} - c^{2} - b \bigr)\csc ^{2} \biggl( \zeta _{0} + \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \end{gathered} $$
(5.58)

where \(\zeta _{0} = \tan ^{ - 1}\frac{B_{1}}{B_{2}}\).

Additionally, if we set \(B_{1} \ne 0\), \(B_{1}^{2} > B_{2}^{2}\), then we achieve the periodic wave solutions of (5.2):

$$ \begin{gathered} u_{2_{4}} ( x, t ) = w \mp \sqrt{ \bigl( 4\mu - \lambda ^{2} \bigr) \bigl( c^{2} + b \bigr)} \tan \biggl( \zeta _{0} - \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \\ v_{2_{4}} ( x, t ) = \frac{4\mu - \lambda ^{2}}{2} \bigl( \mp c\sqrt{c^{2} + b} - c^{2} - b \bigr)\sec ^{2} \biggl( \zeta _{0} - \sqrt{4\mu - \lambda ^{2}} \frac{\zeta }{2} \biggr), \end{gathered} $$
(5.59)

where \(\zeta _{0} = \tan ^{ - 1}\frac{B_{2}}{B_{1}}\).

Type III: When \(( \lambda ^{2} - 4\mu ) = 0\), we find the following rational function solutions of (5.2):

$$ \begin{gathered} u_{3} ( x, t ) = w \mp \frac{2B_{2}\sqrt{c^{2} + b}}{B_{1} + B_{2}\zeta }, \\ v_{3} ( x, t ) = 2 \bigl( \mp c\sqrt{c^{2} + b} - c^{2} - b \bigr) \biggl( \frac{ B_{2}}{B_{1} + B_{2}\zeta } \biggr)^{2}. \end{gathered} $$
(5.60)

Now we examine the coupled WBK equations by means of the \(( G' / G, 1 / G )\)-expansion method. For the balance number attained in the earlier section for this equation, the shape of the solution is of the form

$$ \begin{gathered} U ( \zeta ) = a_{0} + a_{1} \phi + b_{1} \psi , \\ V ( \zeta ) = \alpha _{0} + \alpha _{1} \phi + \alpha _{2} \phi ^{2} + \beta _{1} \psi + \beta _{2} \phi \psi , \end{gathered} $$
(5.61)

where \(a_{0}\), \(a_{1}\), \(b_{1}\), \(\alpha _{0}\), \(\alpha _{1}\), \(\alpha _{2}\), \(\beta _{1}\), and \(\beta _{2}\) are constants to be determined. Now we take into account the following three cases.

Case 1: When \(\lambda < 0\), inserting (5.61) into (5.47), by (4.3)–(4.5) system (5.47) will be transfigured to a polynomial in ψ and ϕ. A system of algebraic equations can be obtained by equalizing the coefficients of this polynomial for the unknown \(a_{0}\), \(a_{1}\), \(b_{1}\), \(\alpha _{0}\), \(\alpha _{1}\), \(\alpha _{2}\), \(\beta _{1}\), \(\beta _{2}\), w, λ, μ, σ, \(\zeta _{1}\), and \(\zeta _{2}\). Solving the algebraic equations via Maple software package, we get three different sets of solution.

Result 1

$$ \begin{gathered} a_{0} = w,\qquad a_{1} = 0,\qquad b_{1} = b_{1},\qquad \alpha _{0} = \frac{b_{1}^{2}}{4\sigma },\qquad \alpha _{1} = 0, \\ \alpha _{2} = \frac{b_{1}^{2}}{2\lambda \sigma },\qquad \beta _{1} = 0,\qquad \beta _{2} = b_{1}c,\qquad b = - \frac{4c^{2}\lambda \sigma + b_{1}^{2}}{4\lambda \sigma }, \\ w = w,\qquad \mu = 0,\qquad \zeta _{1} = - \frac{2\sigma w^{2} + b_{1}^{2}}{4\sigma },\qquad \zeta _{2} = 0. \end{gathered} $$
(5.62)

From (4.4), (5.61), and (5.62), with the help of (5.3), we deduce the following hyperbolic function solutions of (5.2):

$$ \begin{gathered} u_{1} ( x, t ) = w + \frac{b_{1}}{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta }, \\ v_{1} ( x, t ) = \frac{b_{1}^{2}}{4\sigma } - \frac{A_{1}b_{1}\cosh \sqrt{ - \lambda } \zeta + A_{2}b_{1}\sinh \sqrt{ - \lambda } \zeta }{ ( A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta )^{2}} \\ \hphantom{v_{1} ( x, t ) =}{}\times \biggl[ \frac{b_{1}}{2\sigma } ( A_{1}\cosh \sqrt{ - \lambda } \zeta + A_{2}\sinh \sqrt{ - \lambda } \zeta ) - c\sqrt{ - \lambda } \biggr], \end{gathered} $$
(5.63)

where \(\zeta = x - w\frac{t^{\alpha }}{\alpha } \) and \(\sigma = A_{1}^{2} - A_{2}^{2}\).

In particular, by taking \(A_{1} = 0\) and \(A_{2} \ne 0\) in (5.63), we attain the following solitary wave solution:

$$ \begin{gathered} u_{1_{1}} (x, t)=w+ \frac{b_{1}}{A_{2}} \operatorname{sech} \sqrt{-\lambda } \zeta , \\ v_{1_{1}} ( x, t ) =- \frac{b_{1}^{2}}{4 A_{2}^{2}} + \frac{b_{1}}{A_{2}} \tanh \sqrt{-\lambda } \zeta \biggl[ \frac{b_{1}}{2 A_{2}} \tanh \sqrt{-\lambda } \zeta +c \sqrt{-\lambda } \operatorname{sech} \sqrt{-\lambda } \zeta \biggr]. \end{gathered} $$
(5.64)

When \(A_{2} = 0\) and \(A_{1} \ne 0\), we obtain the previously mentioned wave solutions

$$ \begin{gathered} u_{1_{2}} ( x, t ) = w + \frac{b_{1}}{A_{1}} \operatorname{csch} \sqrt{ - \lambda } \zeta , \\ v_{1_{2}} ( x, t ) = \frac{b_{1}^{2}}{4A_{1}^{2}} - \frac{b_{1}}{A_{1}}\coth \sqrt{ - \lambda } \zeta \biggl[ \frac{b_{1}}{2A_{1}}\coth \sqrt{ - \lambda } \zeta - c\sqrt{ - \lambda } \operatorname{csch} \sqrt{ - \lambda } \zeta \biggr]. \end{gathered} $$
(5.65)

Result 2

$$ \begin{gathered} a_{0} = w,\qquad a_{1} = a_{1},\qquad b_{1} = \pm a_{1}\sqrt{ - \frac{\lambda ^{2}\sigma + \mu ^{2}}{\lambda }},\\ \alpha _{0} = - a_{1}\lambda ( a_{1} - c ),\qquad \alpha _{1} = 0,\qquad\alpha _{2} = - a_{1} ( a_{1} - c ), \\ \beta _{1} = a_{1}\mu ( a_{1} - c ),\qquad \beta _{2} = \mp a_{1} ( a_{1} - c )\sqrt{ - \frac{\lambda ^{2}\sigma + \mu ^{2}}{\lambda }}, \\ b = a_{1}^{2} - c^{2},\qquad w = w,\qquad \zeta _{1} = - \frac{1}{2} \bigl( w^{2} + \lambda a_{1}^{2} \bigr),\qquad \zeta _{2} = 0. \end{gathered} $$
(5.66)

From (4.4), (5.61), and (5.66) with (5.3) we find the following hyperbolic function solutions of (5.2):

$$ \begin{gathered} u_{2} ( x, t ) = w + \frac{a_{1}}{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda } \\ \hphantom{u_{2} ( x, t ) = }{}\times \biggl[ A_{1}\sqrt{ - \lambda } \cosh \sqrt{ - \lambda } \zeta + A_{2}\sqrt{ - \lambda } \sinh \sqrt{ - \lambda } \zeta \pm \frac{\sqrt{\lambda ^{2}\sigma + \mu ^{2}}}{\sqrt{ - \lambda }} \biggr], \\ v_{2} ( x, t ) = - a_{1}\lambda ( a_{1} - c ) + \frac{a_{1}\mu ( a_{1} - c )}{A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda } \\ \hphantom{v_{2} ( x, t ) =}{}+ \frac{a_{1} ( a_{1} - c ) ( A_{1}\cosh \sqrt{ - \lambda } \zeta + A_{2}\sinh \sqrt{ - \lambda } \zeta )}{ ( A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda )^{2}} \\ \hphantom{v_{2} ( x, t ) =}{}\times \bigl[ A_{1}\lambda \cosh \sqrt{ - \lambda } \zeta + A_{2}\lambda \sinh \sqrt{ - \lambda } \zeta \mp \sqrt{\lambda ^{2}\sigma + \mu ^{2}} \bigr], \end{gathered} $$
(5.67)

where \(\zeta = x - w\frac{t^{\alpha }}{\alpha } \).

In particular, if we substitute \(A_{1} = 0\), \(\mu = 0\), and \(A_{2} > 0\) into (5.67), we get the following wave solutions:

$$ \begin{gathered} u_{2_{1}} (x, t)=w+ a_{1} \sqrt{- \lambda } [ \tanh \sqrt{-\lambda } \zeta \pm i \operatorname{sech} \sqrt{-\lambda } \zeta ], \\ v_{2_{1}} ( x, t ) =- a_{1} \lambda ( a_{1} -c ) + a_{1} \lambda ( a_{1} -c ) \tanh \sqrt{-\lambda } \zeta [ \tanh \sqrt{-\lambda } \zeta \pm i \operatorname{sech} \sqrt{-\lambda } \zeta ]. \end{gathered} $$
(5.68)

On the other hand, if we introduce \(A_{2} = 0\), \(\mu = 0\), and \(A_{1} > 0\), then we get the following wave solutions:

$$ \begin{gathered} u_{2_{2}} ( x, t ) = w + a_{1} \sqrt{ - \lambda } ( \coth \sqrt{ - \lambda } \zeta \pm \operatorname{csch} \sqrt{ - \lambda } \zeta ), \\ v_{2_{2}} ( x, t ) = - a_{1}\lambda ( a_{1} - c ) + a_{1}\lambda ( a_{1} - c )\coth \sqrt{ - \lambda } \zeta ( \coth \sqrt{ - \lambda } \zeta \pm \operatorname{csch} \sqrt{ - \lambda } \zeta ). \end{gathered} $$
(5.69)

Result 3

$$ \begin{gathered} a_{0} = a_{0},\qquad a_{1} = 0,\qquad b_{1} = - \frac{2 ( a_{0} - w ) ( \lambda ^{2}\sigma + \mu ^{2} )}{\lambda \mu },\\ \alpha _{0} = \frac{ ( a_{0} - w )^{2} ( \lambda ^{2}\sigma + 2\mu ^{2} )}{\mu ^{2}},\qquad\alpha _{1} = 0, \\ \alpha _{2} = \frac{2 ( a_{0} - w )^{2} ( \lambda ^{2}\sigma + \mu ^{2} )}{\lambda \mu ^{2}},\qquad \beta _{1} = - \frac{2 ( a_{0} - w )^{2} ( \lambda ^{2}\sigma + \mu ^{2} )}{\lambda \mu }, \\ \beta _{2} = - \frac{2c ( a_{0} - w ) ( \lambda ^{2}\sigma + \mu ^{2} )}{\lambda \mu },\qquad b = - \frac{ ( a_{0} - w )^{2} ( \lambda ^{2}\sigma + \mu ^{2} ) + c^{2}\lambda \mu ^{2}}{\lambda \mu ^{2}}, \\ w = w,\qquad \zeta _{1} = - \frac{a_{0} ( a_{0} - 2w ) ( 2\lambda ^{2}\sigma - \mu ^{2} ) + 2\lambda ^{2}\sigma w^{2}}{2\mu ^{2}},\qquad \zeta _{2} = - \frac{\lambda ^{2}\sigma ( a_{0} - w )^{3}}{\mu ^{2}}. \end{gathered} $$
(5.70)

For the values of the parameters organized in (5.70), we carry out other hyperbolic function solutions of (5.2) given in the underneath:

$$ \begin{gathered} u_{3} ( x, t ) = a_{0} - \frac{2 ( a_{0} - w ) ( \lambda ^{2}\sigma + \mu ^{2} )}{A_{1}\lambda \mu \sinh \sqrt{ - \lambda } \zeta + A_{2}\lambda \mu \cosh \sqrt{ - \lambda } \zeta + \mu ^{2}}, \\ v_{3} ( x, t ) = \frac{1}{\mu ^{2}} ( a_{0} - w )^{2} \bigl( \lambda ^{2}\sigma + 2\mu ^{2} \bigr) \\ \hphantom{v_{3} ( x, t ) =}{}- \frac{2 ( a_{0} - w )^{2} ( \lambda ^{2}\sigma + \mu ^{2} )}{A_{1}\lambda \mu \sinh \sqrt{ - \lambda } \zeta + A_{2}\lambda \mu \cosh \sqrt{ - \lambda } \zeta + \mu ^{2}} \\ \hphantom{v_{3} ( x, t ) =}{}- \frac{2}{\mu } \frac{ ( a_{0} - w ) ( \lambda ^{2}\sigma + \mu ^{2} ) ( A_{1}\cosh \sqrt{ - \lambda } \zeta + A_{2}\sinh \sqrt{ - \lambda } \zeta )}{ ( A_{1}\sinh \sqrt{ - \lambda } \zeta + A_{2}\cosh \sqrt{ - \lambda } \zeta + \mu / \lambda )^{2}} \\ \hphantom{v_{3} ( x, t ) =}{}\times \biggl[ \frac{1}{\mu } ( a_{0} - w ) ( A_{1} \cosh \sqrt{ - \lambda } \zeta + A_{2}\sinh \sqrt{ - \lambda } \zeta ) - \frac{c}{\sqrt{ - \lambda }} \biggr], \end{gathered} $$
(5.71)

where \(\zeta = x - w\frac{t^{\alpha }}{\alpha } \).

Similarly, by taking special values of the parameters we might attain many other solitary wave solutions, but for simplicity, the solutions are not designated here.

Case 2: When \(\lambda > 0\), in this case, solving the system of algebraic equations with Maple software package, we obtain three different sets of results.

Result 1

$$ \begin{gathered} a_{0} = w,\qquad a_{1} = 0,\qquad b_{1} = b_{1},\qquad \alpha _{0} = - \frac{b_{1}^{2}}{4\sigma },\qquad \alpha _{1} = 0, \\ \alpha _{2} = - \frac{b_{1}^{2}}{2\lambda \sigma },\qquad \beta _{1} = 0,\qquad \beta _{2} = b_{1}c,\qquad b = \frac{ - 4c^{2}\lambda \sigma + b_{1}^{2}}{4\lambda \sigma }, \\ w = w,\qquad \mu = 0,\qquad \zeta _{1} = \frac{ - 2\sigma w^{2} + b_{1}^{2}}{4\sigma },\qquad \zeta _{2} = 0. \end{gathered} $$
(5.72)

For these values, the trigonometric function solutions of (5.2) are:

$$ \begin{gathered} u_{1} ( x, t ) = w + \frac{b_{1}}{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta }, \\ v_{1} ( x, t ) = - \frac{b_{1}^{2}}{4\sigma } - \frac{A_{1}b_{1}\cos \sqrt{\lambda } \zeta - A_{2}b_{1}\sin \sqrt{\lambda } \zeta }{ ( A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta )^{2}} \\ \hphantom{v_{1} ( x, t ) =}{}\times \biggl[ \frac{b_{1}}{2\sigma } ( A_{1}\cos \sqrt{\lambda } \zeta - A_{2}\sin \sqrt{\lambda } \zeta ) - c\sqrt{\lambda } \biggr], \end{gathered} $$
(5.73)

where \(\zeta = x - w\frac{t^{\alpha }}{\alpha } \) and \(\sigma = A_{1}^{2} + A_{2}^{2}\).

In particular, by taking \(A_{1} = 0\) and \(A_{2} \ne 0\) in (5.73), we determine the following periodic wave solutions:

$$ \begin{gathered} u_{1_{1}} ( x, t ) = w + \frac{b_{1}}{A_{2}} \sec \sqrt{\lambda } \zeta , \\ v_{1_{1}} ( x, t ) = - \frac{b_{1}^{2}}{4A_{2}^{2}} - \frac{b_{1}}{A_{2}}\tan \sqrt{\lambda } \zeta \biggl[ \frac{b_{1}}{2A_{2}}\tan \sqrt{\lambda } \zeta + c \sqrt{\lambda } \sec \sqrt{\lambda } \zeta \biggr], \end{gathered} $$
(5.74)

Alternatively, taking \(A_{2} = 0\) and \(A_{1} \ne 0\), we determine the following periodic wave solutions:

$$ \begin{gathered} u_{1_{2}} ( x, t ) = w + \frac{b_{1}}{A_{1}} \csc \sqrt{\lambda } \zeta , \\ v_{1_{2}} ( x, t ) = - \frac{b_{1}^{2}}{4A_{1}^{2}} - \frac{b_{1}}{A_{1}}\cot \sqrt{\lambda } \zeta \biggl[ \frac{b_{1}}{2A_{1}}\cot \sqrt{\lambda } \zeta - c \sqrt{\lambda } \csc \sqrt{\lambda } \zeta \biggr]. \end{gathered} $$
(5.75)

Result 2

$$ \begin{gathered} a_{0} = w,\qquad a_{1} = a_{1},\qquad b_{1} = \pm a_{1}\sqrt{ \frac{\lambda ^{2}\sigma - \mu ^{2}}{\lambda }},\\ \alpha _{0} = - a_{1}\lambda ( a_{1} - c ),\qquad \alpha _{1} = 0,\qquad\alpha _{2} = - a_{1} ( a_{1} - c ), \\ \beta _{1} = a_{1}\mu ( a_{1} - c ),\qquad \beta _{2} = \mp a_{1} ( a_{1} - c )\sqrt{ \frac{\lambda ^{2}\sigma - \mu ^{2}}{\lambda }}, \\ b = a_{1}^{2} - c^{2},\qquad w = w,\qquad \zeta _{1} = - \frac{1}{2} \bigl( w^{2} + \lambda a_{1}^{2} \bigr),\qquad \zeta _{2} = 0. \end{gathered} $$
(5.76)

In this case, the trigonometric function solutions of (5.2) are:

$$ \begin{gathered} u_{2} ( x, t ) = w + \frac{a_{1}}{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda } \\ \hphantom{u_{2} ( x, t ) =}{}\times \biggl[ A_{1}\sqrt{\lambda } \cos \sqrt{\lambda } \zeta - A_{2}\sqrt{\lambda } \sin \sqrt{\lambda } \zeta \pm \frac{\sqrt{\lambda ^{2}\sigma - \mu ^{2}}}{\sqrt{\lambda }} \biggr], \\ v_{2} ( x, t ) = - a_{1}\lambda ( a_{1} - c ) + \frac{a_{1}\mu ( a_{1} - c )}{A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda }\\ \hphantom{v_{2} ( x, t ) =}{} - \frac{a_{1} ( a_{1} - c ) ( A_{1}\cos \sqrt{\lambda } \zeta - A_{2}\sin \sqrt{\lambda } \zeta )}{ ( A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda )^{2}} \\ \hphantom{v_{2} ( x, t ) =}{}\times \bigl[ A_{1}\lambda \cos \sqrt{\lambda } \zeta - A_{2}\lambda \sin \sqrt{\lambda } \zeta \pm \sqrt{\lambda ^{2}\sigma - \mu ^{2}} \bigr], \end{gathered} $$
(5.77)

where \(\zeta = x - w\frac{t^{\alpha }}{\alpha } \).

For the values \(A_{1} = 0\), \(\mu = 0\), and \(A_{2} > 0\), of the parameters, we get the following periodic wave solutions:

$$ \begin{gathered} u_{2_{1}} (x, t)=w- a_{1} \sqrt{ \lambda } [ \tan \sqrt{\lambda } \zeta \mp \operatorname{sec} \sqrt{\lambda } \zeta ], \\ v_{2_{1}} ( x, t ) =- a_{1} \lambda ( a_{1} -c ) - a_{1} \lambda ( a_{1} -c ) \tan \sqrt{\lambda } \zeta [ \tan \sqrt{\lambda } \zeta \mp \operatorname{sec} \sqrt{\lambda } \zeta ]. \end{gathered} $$
(5.78)

Moreover, for the values \(A_{2} = 0\), \(\mu = 0\), and \(A_{1} > 0\), we get the following periodic wave solutions:

$$ \begin{gathered} u_{2_{2}} ( x, t ) = w + a_{1} \sqrt{\lambda } [ \cot \sqrt{\lambda } \zeta \pm \csc \sqrt{\lambda } \zeta ], \\ v_{2_{2}} ( x, t ) = - a_{1}\lambda ( a_{1} - c ) - a_{1}\lambda ( a_{1} - c )\cot \sqrt{\lambda } \zeta [ \cot \sqrt{\lambda } \zeta \pm \csc \sqrt{\lambda } \zeta ]. \end{gathered} $$
(5.79)

Result 3

$$ \begin{gathered} a_{0} = a_{0},\qquad a_{1} = 0,\qquad b_{1} = \frac{2 ( a_{0} - w ) ( \lambda ^{2}\sigma - \mu ^{2} )}{\lambda \mu },\\ \alpha _{0} = - \frac{ ( a_{0} - w )^{2} ( \lambda ^{2}\sigma - 2\mu ^{2} )}{\mu ^{2}},\qquad \alpha _{1} = 0, \\ \alpha _{2} = - \frac{2 ( a_{0} - w )^{2} ( \lambda ^{2}\sigma - \mu ^{2} )}{\lambda \mu ^{2}},\qquad \beta _{1} = \frac{2 ( a_{0} - w )^{2} ( \lambda ^{2}\sigma - \mu ^{2} )}{\lambda \mu }, \\ \beta _{2} = \frac{2c ( a_{0} - w ) ( \lambda ^{2}\sigma - \mu ^{2} )}{\lambda \mu },\qquad b = \frac{ ( a_{0} - w )^{2} ( \lambda ^{2}\sigma - \mu ^{2} ) - c^{2}\lambda \mu ^{2}}{\lambda \mu ^{2}},\qquad w = w, \\ \zeta _{1} = \frac{a_{0} ( a_{0} - 2w ) ( 2\lambda ^{2}\sigma + \mu ^{2} ) + 2\lambda ^{2}\sigma w^{2}}{2\mu ^{2}},\qquad \zeta _{2} = \frac{\lambda ^{2}\sigma ( a_{0} - w )^{3}}{\mu ^{2}}. \end{gathered} $$
(5.80)

For the values in (5.80), we establish other trigonometric function solutions of (5.2):

$$ \begin{gathered} u_{3} ( x, t ) = a_{0} + \frac{2 ( a_{0} - w ) ( \lambda ^{2}\sigma - \mu ^{2} )}{A_{1}\lambda \mu \sin \sqrt{\lambda } \zeta + A_{2}\lambda \mu \cos \sqrt{\lambda } \zeta + \mu ^{2}}, \\ v_{3} ( x, t ) = - \frac{1}{\mu ^{2}} ( a_{0} - w )^{2} \bigl( \lambda ^{2}\sigma - 2\mu ^{2} \bigr) + \frac{2 ( a_{0} - w )^{2} ( \lambda ^{2}\sigma - \mu ^{2} )}{A_{1}\lambda \mu \sin \sqrt{\lambda } \zeta + A_{2}\lambda \mu \cos \sqrt{\lambda } \zeta + \mu ^{2}} \\ \hphantom{v_{3} ( x, t ) =}{}- \frac{2}{\mu } \frac{ ( a_{0} - w ) ( \lambda ^{2}\sigma - \mu ^{2} ) ( A_{1}\cos \sqrt{\lambda } \zeta - A_{2}\sin \sqrt{\lambda } \zeta )}{ ( A_{1}\sin \sqrt{\lambda } \zeta + A_{2}\cos \sqrt{\lambda } \zeta + \mu / \lambda )^{2}} \\ \hphantom{v_{3} ( x, t ) =}{}\times \biggl[ \frac{1}{\mu } ( a_{0} - w ) ( A_{1} \cos \sqrt{\lambda } \zeta - A_{2}\sin \sqrt{\lambda } \zeta ) - \frac{c}{\sqrt{\lambda }} \biggr], \end{gathered} $$
(5.81)

where \(\zeta = x - w\frac{t^{\alpha }}{\alpha } \).

If the parameters receive diverse definite values, then we might accomplish many other periodic wave solutions, but for succinctness, these solutions are not displayed her.

Case 3 When \(\lambda = 0\), in this case, solving the system of algebraic equations with Maple software package, we get the following results:

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} a_{0} = w,\qquad a_{1} = \pm \frac{b_{1}}{\sqrt{A_{1}^{2} - 2A_{2}\mu }},\qquad b_{1} = b_{1},\qquad \alpha _{0} = 0,\qquad \alpha _{1} = 0, \\ \alpha _{2} = \pm \frac{b_{1}c}{\sqrt{A_{1}^{2} - 2A_{2}\mu }} - \frac{b_{1}^{2}}{A_{1}^{2} - 2A_{2}\mu },\qquad \beta _{1} = \mp \frac{b_{1}\mu c}{\sqrt{A_{1}^{2} - 2A_{2}\mu }} + \frac{b_{1}^{2}\mu }{A_{1}^{2} - 2A_{2}\mu }, \\ \beta _{2} = \mp \frac{b_{1}^{2}}{\sqrt{A_{1}^{2} - 2A_{2}\mu }} + b_{1}c,\qquad b = \frac{b_{1}^{2}}{A_{1}^{2} - 2A_{2}\mu } - c^{2},\\ w = w,\qquad \zeta _{1} = - \frac{1}{2}w^{2},\qquad \zeta _{2} = 0. \end{array}\displaystyle \right \} , \end{aligned}$$
(5.82)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} a_{0} = a_{0},\qquad a_{1} = 0,\qquad b_{1} = \frac{2}{\mu } ( a_{0} - w ) ( A_{1}^{2} - 2A_{2}\mu ), \\ \alpha _{0} = ( a_{0} - w )^{2},\qquad \alpha _{1} = 0, \qquad \alpha _{2} = - \frac{2}{\mu ^{2}} ( a_{0} - w )^{2} ( A_{1}^{2} - 2A_{2}\mu ), \\ \beta _{1} = \frac{2}{\mu } ( a_{0} - w )^{2} ( A_{1}^{2} - 2A_{2}\mu ),\qquad \beta _{2} = \frac{2c}{\mu } ( a_{0} - w ) ( A_{1}^{2} - 2A_{2}\mu ), \\ b = \frac{1}{\mu ^{2}} ( a_{0} - w )^{2} ( A_{1}^{2} - 2A_{2}\mu ) - c^{2},\qquad w = w,\\ \zeta _{1} = \frac{3}{2}a_{0}^{2} - 3a_{0}w + w^{2},\qquad \zeta _{2} = ( a_{0} - w )^{3}. \end{array}\displaystyle \right \} , \end{aligned}$$
(5.83)

In this case, the rational function solutions of (5.2) are

$$ \begin{gathered} u_{1} ( x, t ) = w + \frac{b_{1}}{\frac{\mu }{2}\zeta ^{2} + A_{1}\zeta + A_{2}} \times \biggl[ 1 \pm \frac{\mu \zeta + A_{1}}{\sqrt{A_{1}^{2} - 2A_{2}\mu }} \biggr], \\ v_{1} ( x, t ) = \frac{b_{1}\mu }{\frac{\mu }{2}\zeta ^{2} + A_{1}\zeta + A_{2}} \times \biggl[ \frac{b_{1}}{A_{1}^{2} - 2A_{2}\mu } \mp \frac{c}{\sqrt{A_{1}^{2} - 2A_{2}\mu }} \biggr] \\ \hphantom{v_{1} ( x, t ) = }{}+ \frac{b_{1} ( \mu \zeta + A_{1} )}{ ( \frac{\mu }{2}\zeta ^{2} + A_{1}\zeta + A_{2} )^{2}} \times \biggl[ c \mp \frac{b_{1} - c\mu \zeta - A_{1}c}{\sqrt{A_{1}^{2} - 2A_{2}\mu }} - \frac{b_{1} ( \mu \zeta + A_{1} )}{A_{1}^{2} - 2A_{2}\mu } \biggr], \end{gathered} $$
(5.84)

where \(\zeta =x-w \frac{t^{\alpha }}{\alpha }\); and

$$ \begin{gathered} u_{2} ( x, t ) = a_{0} + \frac{2}{\mu } \frac{ ( a_{0} - w ) ( A_{1}^{2} - 2A_{2}\mu )}{\frac{\mu }{2}\zeta ^{2} + A_{1}\zeta + A_{2}}, \\ v_{2} ( x, t ) = ( a_{0} - w )^{2} + \frac{2}{\mu } \frac{ ( a_{0} - w )^{2} ( A_{1}^{2} - 2A_{2}\mu )}{\frac{\mu }{2}\zeta ^{2} + A_{1}\zeta + A_{2}} \\ \hphantom{v_{2} ( x, t ) =}{}- \frac{2}{\mu } \frac{ ( a_{0} - w ) ( A_{1}^{2} - 2A_{2}\mu ) ( \mu \zeta + A_{1} )}{ ( \frac{\mu }{2}\zeta ^{2} + A_{1}\zeta + A_{2} )^{2}} \times \biggl[ \frac{1}{\mu } ( a_{0} - w ) ( \mu \zeta + A_{1} ) - c \biggr], \end{gathered} $$
(5.85)

where \(\zeta = x - w\frac{t^{\alpha }}{\alpha } \).

The solutions obtained are more general and useful to analyze the shallow water wave profile.

Note: It is worth mentioning that all the solutions derived in this study were confirmed using the Maple software package by returning them to the original equation and found correct.

Physical explanation and graphical representations

In this section, we present some 3-D and 2-D figures of some of the obtained solutions. The figures are carried out by taking suitable values of the parameters to objectify the inward contrivance of the incidents, which are analyzed through the time-fractional coupled BB equations and the coupled WBK equations and are displayed in Figs. 16. For example, Fig. 1 shows the kink-shaped soliton depicted from the solution (5.9) when \(\lambda =7\), \(\mu =12\), \(w=0.5\), \(\xi =x- t^{0.5}\), wherein the fractional order is \(\alpha =0.5\) within the intervals \(-15\leq x\leq 15\) and \(0.1\leq t\leq 15\). Solution (5.15) presents the singular periodic wave for \(\lambda =4\), \(\mu =4.25\), \(w= {1} / {3}\), \(\xi =x- {(20} / {51)} t^{0.85}\) with fractional order \(\alpha =0.85\) and is sketched in Fig. 2 within the limits \(-10\leq x\leq 10\) and \(0.1\leq t\leq 0.2\). The modulus of solution (5.26) signifies the cuspon for \(\lambda =-1\), \(\xi _{1} =0\), \(w= {1} / {2}\), \(\xi =x- {(10} / {19)} t^{0.95}\) with fractional order \(\alpha =0.95\) and is shown in Fig. 3 within the range \(-50\leq x\leq 50\) and \(0.01\leq t\leq 0.5\). Solution (5.51) indicates the kink-shape soliton for \(\lambda =3\), \(\mu =2\), \(c=1\), \(b=1\), \(w=0.5\), \(\xi =x-2 t^{0.5}\) with fractional order \(\alpha =0.5\) and is depicted in Fig. 4 within the intervals \(-90\leq x\leq 90\) and \(0.1\leq t\leq 90\). Solution (5.52) characterizes the singular kink wave for \(\lambda =3\), \(\mu =2\), \(c=1\), \(b=1\), \(w=1\), \(\xi =x-2 t^{0.5}\) with fractional order \(\alpha =0.5\) and is plotted in Fig. 5 within the limits \(-100\leq x\leq 100\) and \(0.1\leq t\leq 1\). Solution (5.64) represents the bell-shape soliton for \(\lambda =-1\), \(b_{1} =1\), \(A_{2} =1\), \(w=1\), \(\xi =x- \frac{20}{19} t^{0.95}\) with fractional order \(\alpha =0.95\) and outlined in Fig. 6 within the range \(-5\leq x\leq 5\) and \(0.1\leq t\leq 0.5\).

Figure 1
figure1

Modulus plot of solution (5.9) which is a kink-shape soliton

Figure 2
figure2

Modulus plot of solution (5.15), which is a singular periodic wave

Figure 3
figure3

Modulus plot of solution (5.26) which is a cuspon

Figure 4
figure4

Modulus plot of solution (5.51) which is a kink-shape soliton

Figure 5
figure5

Modulus plot of solution (5.52) which is a singular kink-shape soliton

Figure 6
figure6

Modulus plot of solution (5.64) which is a bell-shape soliton

Conclusion

By means of the basic \({(G'} / {G} )\)-expansion method and the two-variable \({(G'} / {G}, {1} / {G} )\)-expansion method, in this study, we have ascertained further general solitary wave solutions to the time-fractional coupled Boussinesq–Burger equations and the coupled Whitham—Broer–Kaup system as a linear combination of the exponential, rational, and hyperbolic functions or separately including several free parameters. For definite values of the associated parameters, some well-known solutions are extracted from the broad-ranging solutions, which are available in the literature, and some fresh solutions are derived, which confirm the correctness and validity of the general solutions and the method. We have exposed the graphical representations and discussed the physical significance of the obtained solutions. Every nonlinear equation is distinct and atypical; therefore not all equations can be examined through a single method. The greater the scope of application of a method, the greater the acceptability of that method. Since this study shows that the introduced methods are straightforward, compatible, and powerful mathematical tools for obtaining abundant traveling wave solutions, to test the range of applicability and consistency, the method can be implemented to other types of nonlinear fractional differential systems to analyze closed-form soliton solutions, and this is the concern of further research. Numerical solutions to these equations can also be explored in the future by following the effective schemes discussed in [60, 61].

References

  1. 1.

    Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Connecticut (2006)

    Google Scholar 

  2. 2.

    Solomon, T.H., Weeks, E.R., Swinney, H.L.: Observation of anomalous diffusion and Levy flights in a two-dimensional rotating flow. Phys. Rev. Lett. 71(24), 3975–3978 (1993)

    Article  Google Scholar 

  3. 3.

    Magin, R.L., Abdullah, O., Baleanu, D., Zhou, X.J.: Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation. J. Magn. Reson. 190(2), 255–270 (2008)

    Article  Google Scholar 

  4. 4.

    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  5. 5.

    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications. Gordon & Breach, New York (1993)

    MATH  Google Scholar 

  6. 6.

    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1998)

    MATH  Google Scholar 

  7. 7.

    Yildiz, T.A., Jajarmi, A., Yıldız, B., et al.: New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discrete Contin. Dyn. Syst. 13(3), 407–428 (2020)

    MATH  Article  Google Scholar 

  8. 8.

    Jajarmi, A., Baleanu, D., Sajjadi, S.S., et al.: A new feature of the fractional Euler–Lagrange equations for a coupled oscillator using a nonsingular operator approach. Front. Phys. 7, Article ID 196 (2019)

    Article  Google Scholar 

  9. 9.

    Baleanu, D., Jajarmi, A., Sajjadi, S.S., et al.: A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. Chaos 29(8), 083127 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Jajarmi, A., Arshad, S., Baleanu, D.: A new fractional modelling and control strategy for the outbreak of dengue fever. Physica A 535, 122524 (2019)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Jajarmi, A., Ghanbari, B., Baleanu, D.: A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence. Chaos 29(9), 093111 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Fractional diffusion in inhomogeneous media. J. Phys. A 38(42), L679–L684 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Azevedo, E.N., Sousa, P.L., Souza, R.E., et al.: Concentration-dependent diffusivity and anomalous diffusion: a magnetic resonance imaging study of water ingress in porous zeolite. Phys. Rev. E 73(1), Article ID 011204 (2006)

    Article  Google Scholar 

  14. 14.

    Sun, H.G., Chen, W., Chen, Y.Q.: Variable order fractional differential operators in anomalous diffusion modeling. Physica A 388(21), 4586–4592 (2009)

    Article  Google Scholar 

  15. 15.

    Ingman, D., Suzdalnitsky, J.: Application of differential operator with servo-order function in model of viscoelastic deformation process. J. Eng. Mech. 131(7), 763–767 (2005)

    Article  Google Scholar 

  16. 16.

    Sun, H.G., Chen, Y.Q., Chen, W.: Random order fractional differential equation models. Signal Process. 91(3), 525–530 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Pedro, H.T.C., Kobayashi, M.H., Pereira, J.M.C., Coimbra, C.F.M.: Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere. J. Vib. Control 14(9–10), 1659–1672 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Cloot, A.H., Botha, J.P.: A generalized groundwater flow equation using the concept of non-integer order. Water SA 32(1), 1–7 (2006)

    Google Scholar 

  19. 19.

    Hameed, H.U., Darus, M., Salah, J.: A note on Caputo’s derivative operator interpretation in economics. J. Appl. Math. 2018, Article ID 1260240 (2018)

    Google Scholar 

  20. 20.

    Akbulut, A., Kaplan, M.: Auxiliary equation method for time-fractional differential equations with conformable derivative. Comput. Math. Appl. 75(3), 876–882 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Bekir, A., Guner, O., Bhrawy, A.H., et al.: Solving nonlinear fractional differential equations using exp-function and \(( {G '} / {G} )\)-expansion methods. Rom. J. Phys. 60, 360–378 (2015)

    Google Scholar 

  22. 22.

    Guner, O., Atik, H.: Soliton solution of fractional-order nonlinear differential equations based on the exp-function method. Optik, Int. J. Light Electron Opt. 127(20), 10076–10083 (2016)

    Article  Google Scholar 

  23. 23.

    Chen, C., Jiang, Y.L.: Simplest equation method for some time-fractional partial differential equations with conformable derivative. Comput. Math. Appl. 75(8), 2978–2988 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Al-Mdallal, Q.M., Syam, M.I.: Sine–cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation. Chaos Solitons Fractals 33(5), 1610–1617 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Rezazadeh, H., Manafian, J., Khodadad, F.S., et al.: Traveling wave solutions for density-dependent conformable fractional diffusion-reaction equation by the first integral method and the improved \(\operatorname{tah}( {\varphi (\xi )} / {2)}\)-expansion method. Opt. Quantum Electron. 50(3), Article ID 121 (2018)

    Article  Google Scholar 

  26. 26.

    Akbar, M.A., Ali, N.H.M., Hussain, Z.: Optical soliton solutions to the \((2+1)\)-dimensional Chaffee–Infante equation and the dimensionless form of the Zakharov equation. Adv. Differ. Equ. 2019, 446 (2019) 1–18

    MathSciNet  Article  Google Scholar 

  27. 27.

    Al-Shawba, A., Gepreel, K., Abdullah, F., Azmia, A.: Abundant closed form solutions of conformable time fractional Sawada–Kotera–Ito equation using \(( {G '} / {G} )\)-expansion method. Results Phys. 9, 337–343 (2018)

    Article  Google Scholar 

  28. 28.

    Al-Shawba, A.A., Abdullah, F.A., Azmi, A.: Travelling wave solutions for fractional Boussinesq equation using modified \(( {G '} / {G} )\)-expansion method. AIP Conf. Proc. 1974, Article ID 020036 (2018)

    Article  Google Scholar 

  29. 29.

    Islam, T., Akbar, M.A., Azad, A.K.: Traveling wave solutions to some nonlinear fractional partial differential equations through the rational \(( {G '} / {G} )\)-expansion method. J. Ocean Eng. Sci. 3(1), 76–81 (2018)

    Article  Google Scholar 

  30. 30.

    Yaşar, E.Y., Giresunlu, I.B.: The \(( {G '} / {G, {1} / {G}} )\)-expansion method for solving nonlinear space-time fractional differential equations. Pramana J. Phys. 87(2), Article ID 17 (2016)

    Article  Google Scholar 

  31. 31.

    Al-Shawba, A.A., Abdullah, F.A., Gepreel, K.A., et al.: Solitary and periodic wave solutions of higher-dimensional conformable time-fractional differential equations using the \(( {G '} / {G, {1} / {G}} )\)-expansion method. Adv. Differ. Equ. 2018(1), 1 (2018)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Inc, M., Yusuf, A., Aliyu, A.I., et al.: Optical soliton solutions for the higher-order dispersive cubic-quintic nonlinear Schrödinger equation. Superlattices Microstruct. 112, 164–179 (2017)

    Article  Google Scholar 

  33. 33.

    Nuruddeen, R., Nass, A.M.: Exact solitary wave solution for the fractional and classical GEW-Burgers equations: an application of Kudryashov method. J. Taibah Univ. Sci. 12(3), 309–314 (2018)

    Article  Google Scholar 

  34. 34.

    Hosseini, K., Ansari, R.: New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method. Waves Random Complex Media 27(4), 628–636 (2017)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Kumar, D., Seadawy, A.R., Joardar, A.K.: Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chin. J. Phys. 56(1), 75–85 (2018)

    Article  Google Scholar 

  36. 36.

    Feng, Q., Meng, F.: Explicit solutions for space–time fractional partial differential equations in mathematical physics by a new generalized fractional Jacobi elliptic equation-based sub-equation method. Optik, Int. J. Light Electron Opt. 127(19), 7450–7458 (2016)

    Article  Google Scholar 

  37. 37.

    Zheng, B.: A new fractional Jacobi elliptic equation method for solving fractional partial differential equations. Adv. Differ. Equ. 2014(1), Article ID 228 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Al Qurashi, M.M., Yusuf, A., Aliyu, A.I., et al.: Optical and other solitons for the fourth-order dispersive nonlinear Schrödinger equation with dual-power law nonlinearity. Superlattices Microstruct. 105, 183–197 (2017)

    Article  Google Scholar 

  39. 39.

    Tchier, F., Yusuf, A., Aliyu, A.I., et al.: Soliton solutions and conservation laws for lossy nonlinear transmission line equation. Superlattices Microstruct. 107, 320–336 (2017)

    Article  Google Scholar 

  40. 40.

    Yusuf, A., Inc, M., Aliyu, A.I., et al.: Optical solitons possessing beta derivative of the Chen–Lee–Liu equation in optical fiber. Front. Phys. 7, Article ID 34 (2019)

    Article  Google Scholar 

  41. 41.

    Khalfallah, M.: Exact traveling wave solutions of the Boussinesq–Burgers equation. Math. Comput. Model. 49, 666–671 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Mhlanga, I.E., Khalique, C.M.: Exact solutions of generalized Boussinesq–Burgers equations and \((2+1)\)-dimensional Davey–Stewartson equations. J. Appl. Math. 2012, Article ID 389017 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Ebadi, G., Yousefzadeh, N., Triki, H., et al.: Envelope solitons, periodic waves, and other solutions to Boussinesq–Burgers equation. Rom. Rep. Phys. 64(4), 915–932 (2012)

    Google Scholar 

  44. 44.

    Iqbal, M.: A fractional Whitham–Broer–Kaup equation and its possible application to tsunami prevention. Therm. Sci. 21, 1847–1855 (2017)

    Article  Google Scholar 

  45. 45.

    El-Sayed, S.M., Kaya, D.: Exact and numerical traveling wave solutions of Whitham–Broer–Kaup equations. Appl. Math. Comput. 167, 1339–1349 (2005)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Arshed, S., Sadia, M.: \(( {G '} / {G}^{2} )\)-Expansion method: new traveling wave solutions for some nonlinear fractional partial differential equations. Opt. Quantum Electron. 50, 123 (2018)

    Article  Google Scholar 

  47. 47.

    Xie, F., Yan, Z., Zhang, H.: Explicit and exact traveling wave solutions of Whitham–Broer–Kaup shallow water equations. Phys. Lett. A 285, 76–80 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Ghehsareh, H.R., Majlesi, A., Zaghian, A.: Lie symmetry analysis and conservation laws for time fractional coupled Whitham–Broer–Kaup equations. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 80, 153–168 (2018)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Saha Ray, S.: A novel method for travelling wave solutions of fractional Whitham–Broer–Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water. Math. Methods Appl. Sci. 38(7), 1352–1368 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Rani, A., Ul-Hassan, Q.M., Ashraf, M., et al.: A novel technique for solving nonlinear WBK equations of fractional-order. J. Sci. Arts 18, 301–316 (2018)

    Google Scholar 

  51. 51.

    Ali, A., Shah, K., Khan, R.A.: Numerical treatment for traveling wave solutions of fractional Whitham–Broer–Kaup equations. Alex. Eng. J. 57, 1991–1998 (2018)

    Article  Google Scholar 

  52. 52.

    Wang, M., Li, X., Zhang, J.: The \(( {G'} / {G} )\)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372(4), 417–423 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Li, L., Li, E., Wang, M.: The \(( {G '} / {G, {1} / {G}} )\)-expansion method and its application to travelling wave solutions of the Zakharov equations. Appl. Math. J. Chin. Univ. 25(4), 454–462 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Khalil, R., Al Horani, M., Yousef, A., et al.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Jarad, F., Uğurlu, E., Abdeljawad, T., et al.: On a new class of fractional operators. Adv. Differ. Equ. 2017, Article ID 247 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Abdeljawad, T., Alzabut, J., Jarad, F.: A generalized Lyapunov-type inequality in the frame of conformable derivatives. Adv. Differ. Equ. 2017(1), Article ID 321 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Yusuf, A., Aliyu, A.I., Baleanu, D.: Conservation laws, soliton-like and stability analysis for the time fractional dispersive long-wave equation. Adv. Differ. Equ. 2018(1), Article ID 319 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    Kumar, S., Kumar, A., Baleanu, D.: Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves. Nonlinear Dyn. 85(2), 699–715 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Khater, M.M., Kumar, D.: New exact solutions for the time fractional coupled Boussinesq–Burger equation and approximate long water wave equation in shallow water. J. Ocean Eng. Sci. 2(3), 223–228 (2017)

    Article  Google Scholar 

  60. 60.

    Hajipour, M., Jajarmi, A., Baleanu, D.: On the accurate discretization of a highly nonlinear boundary value problem. Numer. Algorithms 79(3), 679–695 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  61. 61.

    Hajipour, M., Jajarmi, A., Malek, A., et al.: Positivity-preserving sixth-order implicit finite difference weighted essentially non-oscillatory scheme for the nonlinear heat equation. Appl. Math. Comput. 325, 146–158 (2018)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to express their sincere thanks to the anonymous referees for their valuable comments and suggestions to improve the quality of this paper. The authors would also like to acknowledge Prof. Md. Tariq-Ul-Islam, Department of English, University of Rajshahi, Bangladesh, for his assistance in editing the English language and grammatical errors. the

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This work is supported by the Publication Fee Funding Grant and Bridging Grant Scheme (304/PMATHS/6316285) by Research Creativity and Management Office (RCMO) Universiti Sains Malaysia.

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Correspondence to M. Ali Akbar.

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Al-Shawba, A.A., Abdullah, F.A., Azmi, A. et al. Reliable methods to study some nonlinear conformable systems in shallow water. Adv Differ Equ 2020, 232 (2020). https://doi.org/10.1186/s13662-020-02686-x

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Keywords

  • Traveling wave solutions; \(( {G'} /{G)}\)-expansion method
  • Two-variable \(( {G'} /{G, {1} / {G)}}\)-expansion method
  • Time-fractional Boussinesq–Burger equations
  • Time-fractional Whitham–Broer–Kaup equations
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