Fractional Consistent Riccati Expansion Method and Soliton-Cnoidal Solutions for the Time-Fractional Extended Shallow Water Wave Equation in (2 + 1)-Dimension

Abstract

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, $G^{\prime }/G$-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations.

1. Introduction

In the last few years, fractional calculus, which includes derivatives and integrals of arbitrary orders, has become an important and popular research area. With the rapid development of fractional calculus, many different fractional derivatives have been proposed, such as Riemann–Liouville [1,2], Gerasimov–Caputo [3,4], and Atangana–Baleanu derivative [5]. We should mention that the Russian (Soviet) mathematician A.N. Gerasimov introduced the concept of fractional derivatives 20 years prior to Caputo. Unfortunately, the original works of A.N. Gerasimov were published shortly after World War II in Russian and are not available to general readers. But we can find more information in [4,6]. Therefore, it would be more appropriate to call the Caputo derivative the “Gerasimov–Caputo derivative”.

Fractional nonlinear evolutional equations (FNEEs) with the different fractional derivatives have been widely used in many fields such as epidemiology of disasters [7,8], physics of fluids [9,10,11,12], engineering and science [13], and economics and finance [14,15]. It is very critical to find solutions for these equations. Explicit solutions are not only helpful in revealing the internal mechanism of natural phenomena but also beneficial in developing various calculation methods and programs. For example, fractional soliton, which is a new concept in mathematics and depends on the time variable, space variable, and the fractional order, can describe the wave distributions observed in fluid dynamics, plasma, and elastic media [16,17,18]. To search for explicit solutions of FNEEs, many effective methods have been presented, such as the Lie symmetry method [19,20,21,22,23], fractional sub-equation method [24,25,26], $G^{\prime }/G$-expansion method [27], exp-function method [28], and so on [29,30,31,32]. Among those, the fractional complex transform offers a transformation between FNEEs and NEEs [10,29,30]. Then, methods which are used to find the exact solutions of NEEs, such as the Hirota bilinear method [33], finite symmetry group method [34], consistent Riccati expansion method [35,36], Bernoulli and its improved version [37,38], and so on, can be applied to extract the exact solutions of FNEEs.

As mentioned above, searching for explicit solutions and developing new methods are very important. In this paper, we will propose a new fractional consistent Riccati expansion (FCRE) method to generate soliton and soliton-cnoidal solutions for FNEEs systematically. The new FCRE method is based on the consistent Riccati expansion method [35], which was proposed to find soliton or soliton-cnoidal solutions of NEEs and has been successfully applied to many NEEs [35,36]. One advantage of the FCRE method is that it can directly generate interaction solutions of two or three different functions. Because real natural phenomena are very complicated, interaction solutions are conducive to better descriptions and explanations of practical phenomena. Another advantage of this approach is that it makes use of the known solutions of the Riccati equation and Jacobian elliptic equation at the same time.

The new method will be applied to the following time-fractional extended shallow water wave (TFESWW) equation in the (2 + 1)-dimension.

$$D_t^{\theta }(u_y)+u_{xxxy}-3u_{xx}{u}_y-3u_x{u}_{xy}+\alpha u_{xx}+\beta u_{yy}+\gamma u_{xy}=0,$$

(1)

where $u=u(x,y,t),$  $\alpha ,\beta$, and $\gamma$ are constants, $D_t^{\theta }(\cdot )$ is the Gerasimov–Caputo time fractional derivative of order $\theta ,\hspace{0.33em}(0<\theta \le 1)$ [3,4]. The Gerasimov–Caputo time fractional derivative is defined as

$$D_t^{\theta }(u_y)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\Gamma (n-\theta )}}{\displaystyle {\int }_0^t{(t-\sigma )}^{n-\theta -1}\frac{{\partial }^n{u}_y}{\partial {\sigma }^n}d\sigma ,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}0\le n-1<\theta <n,\hspace{0.33em}t>\sigma \ge 0,}\\ \hspace{0.33em}{\displaystyle \frac{{\partial }^n{u}_y}{\partial t^n}},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}\theta =n.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

The TFESWW Equation (1) can be used to model the propagation of water waves in oceanography and atmospheric fields, and it includes a lot of KdV-type equations as its particular cases.

When $\theta =1,$ the TFESWW Equation (1) becomes the new extended shallow water wave (ESWW) equation in the (2 + 1)-dimension.

$$u_{yt}+u_{xxxy}-3u_{xx}{u}_y-3u_x{u}_{xy}+\alpha u_{xx}+\beta u_{yy}+\gamma u_{xy}=0.$$

(2)

The ESWW Equation (2) was proposed by Wazwaz in 2022. Soliton solutions, lump solutions, and Painlevé integrability have been studied in [39]. Breather solutions, lump-kink, and two-kink solutions of (2) have been derived using the Hirota bilinear method [40].

When $\theta =1,\beta =0,$ the TFESWW Equation (1) becomes the following extended shallow water wave equation [41]:

$$u_{yt}+u_{xxxy}-3u_{xx}{u}_y-3u_x{u}_{xy}+\alpha u_{xx}+\gamma u_{xy}=0.$$

(3)

When $\theta =1,\alpha =0,\beta =0,$ the TFESWW Equation (1) becomes the following extended shallow water wave equation:

$$u_{yt}+u_{xxxy}-3u_{xx}{u}_y-3u_x{u}_{xy}+\gamma u_{xy}=0.$$

(4)

Closed-form traveling wave solutions have been derived by the exp $(-\gamma (g))$ expansion method and the generalized projective Riccati equation method [42].

When $\theta =1,\alpha =0,\beta =0,\gamma =0,$ the TFESWW Equation (1) becomes the Boiti–Leon–Manna–Pempinelli (BLMP) equation by Gilson [43]:

$$u_{yt}+u_{xxxy}-3u_{xx}{u}_y-3u_x{u}_{xy}=0.$$

(5)

It is also called the (2 + 1)-dimensional shallow water wave equation by Clarkson and Mansfield [44,45]. Analytic solutions for the (2 + 1)-dimensional time-fractional BLMP equation in the sense of the ABR fractional operator have been found through the modified Khater method [46]. Shock wave solutions of the (2 + 1)-dimensional time-fractional BLMP equation in the sense of Riesz–Caputo type fractional operator have been derived by He’s semi inverse method [47].

When $y=x,$ (5) can be reduced to the potential KdV equation:

$$u_t+u_{xxx}-3u_x^2=0.$$

(6)

When $y=x,$ under the transformation $\psi =u_x,$ (5) can also be changed to the well-known KdV equation:

$${\psi }_t+{\psi }_{xxx}-6\psi {\psi }_x=0.$$

(7)

The ESWW Equation (2) is also a particular case of the new extended shallow water wave equation in the (3 + 1)-dimension [39]:

$$u_{yt}+u_{xxxy}-3u_{xx}{u}_y-3u_x{u}_{xy}+\alpha u_{xx}+\beta u_{yy}+\gamma u_{xy}+\delta u_{yz}=0.$$

(8)

Equation (8) is a KdV-type equation and can model the propagation of water waves in (3 + 1)-dimensional incompressible fluid. It changes to the (2 + 1)-dimensional ESWW Equation (2) when setting $\delta =0,$ and more results can be found in [48,49,50] and the references therein.

To our knowledge, explicit solutions for the TFESWW Equation (1) have not been reported. Soliton and soliton-cnoidal solutions for (1) will be investigated by the new proposed FCRE method. The paper is organized as follows. Brief descriptions of the new proposed FCRE method will be given in Section 2. The FCRE method is used to solve the TFESWW Equation (1) in Section 3, and plenty of soliton and soliton-cnoidal solutions will be obtained. Section 4 is devoted to discussion of the results in this paper. In Section 5, some conclusions and future directions of the paper are presented.

2. Brief Descriptions of the New Proposed FCRE Method

In the following, we will explain the FCRE method and give the main steps.

Step 1: For a time-fractional NEE given by

$$G_1(\mathrm{x},y,t,u,u_x,u_y,D_t^{\theta }(u),u_{xx},u_{xy},u_{yy},D_t^{\theta }(u_x),D_t^{\theta }(u_y),\dots \dots )=0,\hspace{0.33em}\hspace{0.33em}0<\theta \le 1$$

(9)

where $x,y$, and $t$ are three independent variables, $D_t^{\theta }(.)$ is the Gerasimov–Caputo time fractional derivative of order $\theta$ [3,4]. Substituting the following fractional complex transform

$$u=u(x,y,T),T={\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}},$$

(10)

into (9), an integer-order NEE can be obtained as follows:

$$G_2(\mathrm{x},y,T,u,u_x,u_y,u_T,u_{xx},u_{xy},u_{xT},u_{yy},u_{yT},\dots \dots )=0,\hspace{0.33em}$$

(11)

where $\Gamma$ represents the Gamma function.

Step 2: Suppose that the solution of (11) has the following form:

$$u={\displaystyle {\displaystyle \sum _{j=0}^J}{\widehat{u}}_j{R}^j(W)},$$

(12)

where ${\widehat{u}}_j(j=0,1,2,\cdots ,J)$ and $W$ are undetermined functions concerning $x,y$, and $T.$ The positive integer $J$ is determined by balancing the derivative term of the highest order with the nonlinear term of the highest order in (11). The function $R(W)$ is a solution of the following Riccati equation:

$$R_W=A+M{R}^2,R_W=A+M{R}^2,$$

(13)

with $A$ and $M$ being constants. Exact solutions of (13) have been reported in many references, such as [51].

Step 3: Substituting (12) with (13) into (11), we can obtain a system of overdetermined equations composed by the coefficients of different $R^j(W).$  ${\widehat{u}}_j$ and $W$ will be determined by solving the system of equations. Since the overdetermined equations are usually difficult to solve, we introduce two hypotheses of $W$:

Case I:  $W=k_{1}x+k_{2}y+k_{3}T+k_0,$ where $k_0,k_1,k_2$, and $k_3$ are undetermined constants.

Case II:  $W=k_{1}x+k_{2}y+k_{3}T+k_0+F(\eta ),\eta =n_{1}x+n_{2}y+n_{3}T+n_0,$ with $n_0,n_1,n_2$, and $n_3$ being constants. $F_{\eta }$ denotes the derivative of $F$ with respect to $\eta ,$ and $F_{\eta }$ satisfies the following Jacobian elliptic function:

$$F_{\eta \eta }=\sqrt{C_0+C_1{F}_{\eta }+C_2{(F_{\eta })}^2+C_3{(F_{\eta })}^3+C_4{(F_{\eta })}^4},$$

(14)

where $C_0,C_1,C_2,C_3$, and $C_4$ are constants. When $C_0,C_1,C_2,C_3$, and $C_4$ take different values, the solutions of (14) include soliton solutions, combined-soliton solutions, triangular periodic solutions, Jacobian elliptic function solutions, combined Jacobian elliptic function solutions and rational function solutions, and they have been summarized in [52]. For example, when taking the values $C_1=0,C_3=0,C_0={\displaystyle \frac{k^2-1}{2}},C_2=2-k^2,C_4=-2,$ (14) has a solution

$$\hspace{0.33em}{F}_{\eta }={\displaystyle \frac{\mathrm{DN}\left(\eta ,\mathrm{k}\right)}{\sqrt{2}}},\hspace{0.33em}\hspace{0.33em}$$

integrating the above equation with respect to $\eta$, we obtain

$$F={\displaystyle \frac{\mathrm{AM}\left(\eta ,\mathrm{k}\right)}{\sqrt{2}}},$$

(15)

where $\mathrm{DN}(\eta ,k)$ and $\mathrm{AM}(\eta ,k)$ are Jacobian elliptic functions, and they are defined as follows:

$$\mathrm{DN}(\eta ,k)=\mathrm{JacobiDN}(\eta ,k),\mathrm{AM}(\eta ,k)=\mathrm{Jacobi}\mathrm{AM}(\eta ,k),$$

where $k$ is the modulus of the Jacobian elliptic function. Other Jacobian elliptic functions $\mathrm{CN}(\eta ,k)$ and $\mathrm{SN}(\eta ,k),$ which will be used in this paper, can be defined as follows:

$$\mathrm{CN}(\eta ,k)=\mathrm{JacobiCN}(\eta ,k),\mathrm{SN}(\eta ,k)=\mathrm{JacobiSN}(\eta ,k).$$

Step 4: Replacing $T$ with ${\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}$, exact solutions of (9) will be derived.

Remark 1.

The two hypotheses of $W$ are helpful for finding ${\widehat{u}}_j$ from the overdetermined equations. Also, they are used to obtain interaction solutions such as soliton-cnoidal solutions. Other hypotheses of $W$ deserve further study.

3. Application of the FCRE Method to the TFESWW Equation (1)

In the present analysis, we will apply the steps of the FCRE method to obtain exact solutions for the TFESWW Equation (1). To reduce (1) to an integer-order NEE, we introduce the following fractional complex transform:

$$u=u(x,y,T),T={\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}.$$

(16)

Substituting (16) into (1), we obtain the following (2 + 1)-dimensional ESWW equation:

$$u_{yT}+u_{xxxy}-3u_{xx}{u}_y-3u_x{u}_{xy}+\alpha u_{xx}+\beta u_{yy}+\gamma u_{xy}=0.$$

(17)

Equation (17) is an integer-order NEE, and we first seek its exact solutions. If we replace $T$ with $t,$ (17) is the same as (2).

Suppose that the solution of (17) has the form of (12), and we obtain $n=1$ by balancing the highest nonlinearity and dispersive term, so the solution of (17) has the following form:

$$u=L+mR(W),$$

(18)

where $L,m$, and $W$ are functions of $x,y$, and $T,$  $R(W)$ is a solution of the Riccati Equation (13).

Substituting (18) with the Riccati Equation (13) into (17) and collecting the coefficients of different $R^j(W),$ we have

$$12m{M}^3{W}_y{W_x}^2(2M{W}_x-m)R^5(W)+h_4{R}^4(W)+h_3{R}^3(W)+h_2{R}^2(W)+h_{1}R(W)+h_0=0,$$

(19)

where $h_i\hspace{0.33em}(i=0,1,2,3,4)$ are functions of $m,L,W$, and their derivatives.

Setting the coefficients of $R^5(W)$ in (19) to zero, we can obtain

$$m=2M{W}_x.$$

(20)

Substituting (20) into (19), we have

$$h_3{R}^3(W)+h_2{R}^2(W)+h_{1}R(W)+h_0=0,$$

(21)

and the expressions of $h_i\hspace{0.33em}(i=0,1,2,3)$ are as follows.

$$\begin{array}{l}{h}_0=L_{xxxy}+\alpha L_{xx}+\beta L_{yy}-3L_y{L}_{xx}-6AM{W}_y{W}_x{L}_{xx}+2AM\beta W_x{W}_{yy}-12AM{W}_x{W}_{xy}{L}_x\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+4AM\beta W_y{W}_{xy}+4AM\gamma W_x{W}_{xy}+2AM\gamma W_y{W}_{xx}-18AM{W}_x{W}_{xx}{L}_y+6AM\alpha W_x{W}_{xx}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-6AM{W}_y{W}_{xx}{L}_x-24A^2{M}^2{W}_y{W_x}^2{W}_{xx}-8A^2{M}^2{W_x}^3{W}_{xy}+2AM{W}_{yt}{W}_x-6AM{W_x}^2{L}_{xy}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+2AM{W}_{\mathit{xy}}{W}_t+2AM{W}_y{W}_{xt}+12AM{W}_{xxy}{W}_{xx}+8AM{W}_{xxxy}{W}_x+8AM{W}_{xxx}{W}_{xy}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+2AM{W}_{xxxx}{W}_y+\gamma L_{xy}-3L_x{L}_{xy}+L_{yt},\\ h_1=-12A{M}^2{W}_x{W}_{xy}{W}_{xx}+4A{M}^2{W}_y{W}_x{W}_{xxx}-16A^2{M}^3{W}_y{W_x}^4+4A{M}^2\alpha {W_x}^3-12A{M}^2{W_x}^3{L}_y\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+2M{W}_{xxxxy}+2M{W}_{xyt}-6M{W}_{xx}{L}_{xy}-6M{W}_{xy}{L}_{xx}+2\beta M{W}_{xyy}+2\alpha M{W}_{xxx}-6M{W}_{xxx}{L}_y\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+2M\gamma W_{xxy}-6M{W}_{xxy}{L}_x+4A{M}^2\beta {W_y}^2{W}_x+4A{M}^2{W}_y{W}_x{W}_t+4A{M}^2\gamma {W_x}^2{W}_y\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-12A{M}^2{W}_y{W_x}^2{L}_x+12A{M}^2{W_x}^2{W}_{xxy},\\ h_2=-24A{M}^3{W}_y{W_x}^2{W}_{xx}+2M^2{W}_{xy}{W}_t+2M^2{W}_x{W}_{yt}-6M^2{W_x}^2{L}_{xy}+2M^2{W}_y{W}_{xt}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-4M^2{W}_{xy}{W}_{xxx}+8M^2{W}_{xxxy}{W}_x+2M^2{W}_{xxxx}{W}_y+2M^2\beta W_x{W}_{yy}+4M^2\gamma W_x{W}_{xy}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-12M^2{W}_x{W}_{xy}{L}_x+4M^2\beta W_y{W}_{xy}-8A{M}^3{W_x}^3{W}_{xy}-6M^2{W}_y{W}_{xx}{L}_x+2M^2\gamma W_y{W}_{xx}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-18M^2{W}_x{W}_{xx}{L}_y+6M^2\alpha W_x{W}_{xx}-6M^2{W}_y{W}_{xx}{L}_{xx},\\ h_3=4M^3{W}_y{W}_x{W}_t+4M^3\beta {W_y}^2{W}_x-12M^3{W}_y{W_x}^2{L}_x-16A{M}^4{W}_y{W_x}^4+4M^3\gamma {W_x}^2{W}_y\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-12M^3{W}_x{W}_{xy}{W}_{xx}+4M^3{W}_x{W}_y{W}_{xxx}-12M^3{W_x}^3{L}_y+4M^3\alpha {W_x}^3+12M^3{W_x}^2{W}_{xxy}.\end{array}$$

(22)

From the above analysis, the following nonauto $\mathrm{B}\ddot{\mathrm{a}}\mathrm{c}\mathrm{k}\mathrm{l}\mathrm{u}\mathrm{n}\mathrm{d}$ transform theorem has been proven.

Theorem 1.

If $L$ and $W$ are solutions of (22), and $R(W)$ is a solution of (13), then

$$u=L+mR(W),$$

is the solution of the (2 + 1)-dimensional ESWW Equation (17), with $m$ being determined by (20).

In the following paragraphs, we will apply Theorem 1 and the two hypotheses of $W$ to seek solutions of (17).

3.1. Soliton Solutions and Combined-Soliton Solutions

Taking $W=k_{1}x+k_{2}y+k_{3}T+k_0,$  $m$, and $L$ can be obtained by solving (20) and (22), respectively, and the results are as follows.

$$\begin{array}{l}L={\displaystyle \frac{N_2{k}_1}{k_2}}\ln \left(\tanh (\xi )-1\right)-{\displaystyle \frac{N_2{k}_1}{k_2}}\ln \left(\tanh (\xi )+1\right)+{\displaystyle \frac{2N_2{k}_1}{k_2}}\tanh (\xi )+f(T)\\ +{\displaystyle \frac{x}{6N_2{k_1}^2{k_2}^2}}\left(-N_3{k}_1{k_2}^3-4AM{N}_2{k_1}^4{k_2}^2+2N_2{k_1}^2{k_2}^2\gamma +N_2{k}_1{k_2}^2{k}_3-8{N_2}^3{k_1}^4\right)\\ +{\displaystyle \frac{y}{6N_2{k_1}^2{k_2}^2}}\left(2\beta N_2{k_2}^4+N_3{k_2}^4-4AM{N}_2{k_1}^3{k_2}^3+N_2{k}_3{k_2}^3+2\alpha N_2{k_1}^2{k_2}^2+8{N_2}^3{k_1}^3{k}_2\right),\end{array}$$

(23a)

$$m=2M{k}_1,$$

(23b)

where $\xi =-{\displaystyle \frac{N_2{k}_1}{k_2}}x+N_{2}y+N_{3}T+N_1$, $k_0,k_1,k_2,k_3$, and $N_1,N_2,N_3$ are constants, and $f(T)$ is an arbitrary function.

3.1.1. One-Soliton Solution

When $A=1,M=-1,$ the Riccati Equation (13) has two soliton solutions

$$R_1=\tanh (W),R_2=\coth (W).$$

(24)

Applying Theorem 1, two soliton solutions for the ESWW Equation (17) can be obtained:

$$u_1=L+m\tanh (W),$$

(25)

$$u_2=L+m\coth (W),$$

(26)

where  $m$ and $L$ are determined by (23).

3.1.2. Combined-Soliton Solution

When $A={\displaystyle \frac{1}{2}},M=-{\displaystyle \frac{1}{2}},$ the Riccati Equation (13) has two combined-soliton solutions:

$$R_3={\displaystyle \frac{\tanh (W)}{1+\mathrm{sech}(W)}}, R_4={\displaystyle \frac{\tanh (W)}{1-\mathrm{sech}(W)}}.$$

(27)

Applying Theorem 1, two combined-soliton solutions for the ESWW Equation (17) can be obtained:

$$u_3=L+m{\displaystyle \frac{\tanh (W)}{1+\mathrm{sech}(W)}},$$

(28)

$$u_4=L+m{\displaystyle \frac{\tanh (W)}{1-\mathrm{sech}(W)}},$$

(29)

where  $L$ and $m$ are determined by (23).

3.1.3. Rational Function Solution

When $A=0,$ the Riccati Equation (13) has a rational function solution

$$R_5={\displaystyle \frac{-1}{MW+N_0}}.$$

(30)

Applying Theorem 1, a rational function solution for the ESWW Equation (17) can be obtained:

$$u_5=L+m{\displaystyle \frac{-1}{MW+N_0}},$$

(31)

where  $L$ and $m$ are determined by (23) with $A=0,$ and $N_0$ is an arbitrary constant.

3.2. Soliton-Cnoidal Solutions and Combined-Soliton Solutions

In this subsection, we will apply the second hypothesis of $W$ to derive exact solutions of (17). We set

$$W=k_{1}x+k_{2}y+k_{3}T+k_0+F(\eta ),$$

(32)

where $\eta =n_{1}x+n_{2}y+n_{3}T+n_0,$ with $n_0,n_1,n_2$, and $n_3$ being constants. For simplicity, we suppose $F_{\eta }=\varphi ,$ where $F_{\eta }$ denotes the derivative of $F$ with respect to $\eta ,$ and $\varphi$ satisfies the following first-order Jacobian elliptic equation:

$${\varphi }^{\prime }=\sqrt{C_0+C_1\varphi +C_2{\varphi }^2+C_3{\varphi }^3+C_4{\varphi }^4},$$

(33)

where $C_0,C_1,C_2,C_3$, and $C_4$ are constants. Substituting (32) with $F_{\eta }=\varphi$ and (33) into (22) and collecting the different coefficients, we have the following expression:

$${\displaystyle \sum _{i=1}^{J_1}{H}_{si}{\varphi }^i\sqrt{C_0+C_1\varphi +C_2{\varphi }^2+C_3{\varphi }^3+C_4{\varphi }^4}}+{\displaystyle \sum _{j=1}^{J_2}{H}_{sj}\frac{{\varphi }^j}{\sqrt{C_0+C_1\varphi +C_2{\varphi }^2+C_3{\varphi }^3+C_4{\varphi }^4}}}+{\displaystyle \sum _{l=1}^{J_3}{H}_{sl}{\varphi }^l}=0,$$

where $H_{si},H_{sj}$ and $H_{sl}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left(s=1,2,3,4\right)$ are functions of $k_0,k_1,k_2,k_3,n_0,n_1,n_2,n_3,C_0,C_1,$  $C_2,C_3,C_4,L$, and its derivatives. It is very tedious to write all the expressions of $H_{si},$  $H_{sj}$, and $H_{sl}.$ For simplicity, we only list several interesting results which are obtained from them.

3.2.1. $A=1,M=-1,\mathrm{R}=\tanh (W)$, or $\mathrm{R}=\coth (W)$

Case 1.

In this case, the undetermined constants and functions are as follows:

$$C_1=0,C_3=0,C_4=-2,C_0={\displaystyle \frac{1}{4}},C_2=-{\displaystyle \frac{7}{4}},n_2=0,k_1=0,k_3=-k_2\beta ,$$

$$\begin{gathered} \varphi =-{\displaystyle \frac{3\mathrm{SN}\left(\eta ,\frac{\sqrt{3}}{2}\right)}{2}}{\displaystyle \frac{\mathrm{CN}\left(\eta ,\frac{\sqrt{3}}{2}\right)}{2{\left(\mathrm{SN}\left(\eta ,\frac{\sqrt{3}}{2}\right)\right)}^2+1}},\hspace{0.33em}\hspace{0.33em}F=\sqrt{2}\arctan \left(\sqrt{2}\mathrm{DN}\left(\eta ,{\displaystyle \frac{\sqrt{3}}{2}}\right)\right), \\ L=-{\displaystyle \frac{7}{12}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T), \end{gathered}$$

(34a)

$$m={\displaystyle \frac{3n_1\mathrm{CN}\left(\eta ,\frac{\sqrt{3}}{2}\right)}{1+2{\left(\mathrm{DN}\left(\eta ,\frac{\sqrt{3}}{2}\right)\right)}^2}}\mathrm{SN}\left(\eta ,{\displaystyle \frac{\sqrt{3}}{2}}\right),$$

(34b)

$$W=k_{2}y-k_2\beta T+k_0+\sqrt{2}\arctan \left(\sqrt{2}\mathrm{DN}\left(\eta ,{\displaystyle \frac{\sqrt{3}}{2}}\right)\right),$$

(34c)

$$\eta =n_{1}x+n_{3}T+n_0,$$

(35)

Two soliton-cnoidal solutions of the ESWW Equation (17) can be derived:

$$u_6=L+m\tanh (W),$$

(36a)

$$u_7=L+m\coth (W),$$

(36b)

where  $k_0,k_2,n_0,n_1$, and  $n_3$ are constants, and  $k_2{n}_1\ne 0,$  $f(T)$ is an arbitrary function of  $T,$  $L,m$, and  $W$ are determined by (34),  $\eta$ is determined by (35), and  $\mathrm{CN}(\eta ,{\displaystyle \frac{\sqrt{3}}{2}}),\mathrm{SN}(\eta ,{\displaystyle \frac{\sqrt{3}}{2}})$, and  $\mathrm{DN}(\eta ,{\displaystyle \frac{\sqrt{3}}{2}})$ are Jacobian elliptic functions with the modulus  ${\displaystyle \frac{\sqrt{3}}{2}}$.

Case 2.

In this case, the undetermined constants and functions are as follows:

$$C_1=0,C_3=0,C_4=-2,C_0={\displaystyle \frac{k^2-1}{2}},C_2=2-k^2,n_2=0,k_1=0,k_3=-k_2\beta ,$$

$$\hspace{0.33em}\varphi ={\displaystyle \frac{\mathrm{DN}\left(\eta ,\mathrm{k}\right)}{\sqrt{2}}},\hspace{0.33em}\hspace{0.33em}F={\displaystyle \frac{\mathrm{AM}\left(\eta ,\mathrm{k}\right)}{\sqrt{2}}},$$

(37)

$$L=-{\displaystyle \frac{1}{3}}{n}_1^2{k}^{2}x+{\displaystyle \frac{2}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T),$$

(38a)

$$m=-\sqrt{2}{n}_1\mathrm{DN}\left(\eta ,k\right),$$

(38b)

$$W=k_{2}y-k_2\beta T+k_0+{\displaystyle \frac{\sqrt{2}}{2}}\mathrm{AM}\left(\eta ,\mathrm{k}\right),$$

(38c)

Two soliton-cnoidal solutions of the ESWW Equation (17) can be obtained:

$$u_8=-{\displaystyle \frac{1}{3}}{n}_1^2{k}^{2}x+{\displaystyle \frac{2}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T)-\sqrt{2}{n}_1\mathrm{DN}\left(\eta ,k\right)\tanh (W),$$

(39a)

$$u_9=-{\displaystyle \frac{1}{3}}{n}_1^2{k}^{2}x+{\displaystyle \frac{2}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T)-\sqrt{2}{n}_1\mathrm{DN}\left(\eta ,k\right)\coth (W),$$

(39b)

where  $\eta$ and  $W$ are determined by (35) and (38c), respectively.

Case 3.

In this case, the undetermined constants and functions are as follows:

$$C_1=0,C_3=0,C_4=-2,C_0={\displaystyle \frac{k^2(1-k^2)}{2}},C_2=2k^2-1,n_2=0,k_1=0,k_3=-k_2\beta ,$$

$$\hspace{0.33em}\varphi ={\displaystyle \frac{k\mathrm{CN}\left(\eta ,\mathrm{k}\right)}{\sqrt{2}}},\hspace{0.33em}\hspace{0.33em}F={\displaystyle \frac{1}{\sqrt{2}}}\arctan \left(k{\displaystyle \frac{\mathrm{SN}\left(\eta ,\mathrm{k}\right)}{\mathrm{DN}\left(\eta ,\mathrm{k}\right)}}\right),$$

(40)

$$L={\displaystyle \frac{2}{3}}{n}_1^2{k}^{2}x-{\displaystyle \frac{1}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T),$$

(41a)

$$m=-\sqrt{2}{n}_{1}k\mathrm{CN}(\eta ,k),$$

(41b)

$$W=k_{2}y-k_2\beta t+k_0+{\displaystyle \frac{1}{2}}\sqrt{2}\arctan \left({\displaystyle \frac{k\mathrm{SN}(\eta ,k)}{\mathrm{DN}(\eta ,k)}}\right),$$

(41c)

Two soliton-cnoidal solutions of the ESWW Equation (17) can be derived:

$$u_{10}=L+m\tanh (W),$$

(42a)

$$u_{11}=L+m\coth (W),$$

(42b)

where  $\eta$ is determined by (35),  $L,m$, and  $W$ are determined by (41).

As we know, Jacobian elliptic functions have close connections with hyperbolic functions and trigonometric functions. When  $k\to 1,$

$$\mathrm{SN}(\eta ,k)\to \tanh (\eta ),\mathrm{CN}(\eta ,k)\to \mathrm{sech}(\eta ),\mathrm{DN}(\eta ,k)\to \mathrm{sech}(\eta ).$$

(43)

When  $k\to 0,$

$$\mathrm{SN}(\eta ,k)\to \sin (\eta ),\mathrm{CN}(\eta ,k)\to \cos (\eta ),\mathrm{DN}(\eta ,k)\to 1.$$

(44)

Case 4.

Taking advantage of the properties (43), we can obtain two new solutions from (40)–(42) in Case 3.

When  $k\to 1,$  $\varphi ={\displaystyle \frac{k\mathrm{CN}\left(\eta ,\mathrm{k}\right)}{\sqrt{2}}}\to {\displaystyle \frac{\mathrm{sech}(\eta )}{\sqrt{2}}},$

$$F={\displaystyle \int \varphi d\eta =}{\displaystyle \int \frac{\mathrm{sech}(\eta )}{\sqrt{2}}d\eta }={\displaystyle \frac{1}{2}}\sqrt{2}\arctan \left(\sinh (\eta )\right).$$

Taking  $k=1$ in (41) and (42), we obtain two inverse trigonometric function and hyperbolic functions interaction solutions for the ESWW Equation (17):

$$\begin{array}{l}{u}_{12}={\displaystyle \frac{1}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T)\\ -{\displaystyle \frac{\sqrt{2}{n}_1\cosh (\eta )}{1+{\left(\sinh (\eta )\right)}^2}}\tanh \left(k_{2}y-k_2\beta t+k_0+{\displaystyle \frac{1}{2}}\sqrt{2}\arctan \left(\sinh (\eta )\right)\right),\end{array}$$

(45a)

$$\begin{array}{l}{u}_{13}={\displaystyle \frac{1}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T)\\ -{\displaystyle \frac{\sqrt{2}{n}_1\cosh (\eta )}{1+{\left(\sinh (\eta )\right)}^2}}\coth \left(k_{2}y-k_2\beta t+k_0+{\displaystyle \frac{1}{2}}\sqrt{2}\arctan \left(\sinh (\eta )\right)\right),\end{array}$$

(45b)

with  $\eta =n_{1}x+n_{3}T+n_0.$

We should point out that if we take  $k=1$ in (37)–(39), the same results will be obtained.

Case 5.

When  $k\to 0,$  $\varphi ={\displaystyle \frac{k\mathrm{CN}\left(\eta ,\mathrm{k}\right)}{\sqrt{2}}}\to 0,$  $F$ is a constant, we take  $F=0$ for simplicity. From (40)–(42), we obtain a rational function solution:

$$u_{14}=-{\displaystyle \frac{1}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T).$$

(46)

Case 6.

When $k\to 0,$  $\hspace{0.33em}\varphi ={\displaystyle \frac{\mathrm{DN}\left(\eta ,\mathrm{k}\right)}{\sqrt{2}}}\to {\displaystyle \frac{1}{\sqrt{2}}},\hspace{0.33em}\hspace{0.33em}F={\displaystyle \int \varphi d\eta =}{\displaystyle \int \frac{1}{\sqrt{2}}d\eta }={\displaystyle \frac{\eta }{\sqrt{2}}}.$  Taking $k=0$ in (37)–(39), we obtain a dark soliton solution and a singular soliton solution:

$$\begin{array}{l}{u}_{15}={\displaystyle \frac{2}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T)\\ -\sqrt{2}{n}_1\tanh \left(k_{2}y-k_2\beta T+k_0+{\displaystyle \frac{1}{2}}\sqrt{2}\left(n_{1}x+n_{3}T+n_0\right)\right),\end{array}$$

(47a)

$$\begin{array}{l}{u}_{16}={\displaystyle \frac{2}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T)\\ -\sqrt{2}{n}_1\coth \left(k_{2}y-k_2\beta T+k_0+{\displaystyle \frac{1}{2}}\sqrt{2}\left(n_{1}x+n_{3}T+n_0\right)\right).\end{array}$$

(47b)

3.2.2. $A={\displaystyle \frac{1}{2}},M=-{\displaystyle \frac{1}{2}},\mathrm{R}={\displaystyle \frac{\tanh (W)}{1\pm \mathrm{sech}(W)}}$

Case 7.

In this case, the undetermined constants and functions are as follows:

$$C_1=0,C_3=0,C_4=-{\displaystyle \frac{1}{2}},C_0=2k^2(1-k^2),C_2=2k^2-1,n_2=0,k_1=0,k_3=-k_2\beta ,$$

$$\hspace{0.33em}\varphi =\sqrt{2}k\mathrm{CN}\left(\eta ,\mathrm{k}\right),\hspace{0.33em}\hspace{0.33em}F=\sqrt{2}\arctan \left(k{\displaystyle \frac{\mathrm{SN}\left(\eta ,\mathrm{k}\right)}{\mathrm{DN}\left(\eta ,\mathrm{k}\right)}}\right),$$

(48)

$$L={\displaystyle \frac{2}{3}}{n}_1^2{k}^{2}x-{\displaystyle \frac{1}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T),$$

(49a)

$$m=-\sqrt{2}{n}_{1}k\mathrm{CN}(\eta ,k),$$

(49b)

$$W=k_{2}y-k_2\beta t+k_0+\sqrt{2}\arctan \left({\displaystyle \frac{k\mathrm{SN}(\eta ,k)}{\mathrm{DN}(\eta ,k)}}\right),$$

(49c)

Two soliton-cnoidal solutions of the ESWW Equation (17) can be derived:

$$u_{17}=L+m{\displaystyle \frac{\tanh (W)}{1+\mathrm{sech}(W)}},$$

(50a)

$$u_{18}=L+m{\displaystyle \frac{\tanh (W)}{1-\mathrm{sech}(W)}},$$

(50b)

where  $\eta =n_{1}x+n_{3}T+n_0,$  $L,m$, and  $W$ are determined by (49).

Case 8.

In this case, the undetermined constants and functions are as follows:

$$C_1=0,C_3=0,C_4=-{\displaystyle \frac{1}{2}},C_0=-2(1-k^2),C_2=2-k^2,n_2=0,k_1=0,k_3=-k_2\beta ,$$

$$\hspace{0.33em}\varphi =\sqrt{2}\mathrm{DN}\left(\eta ,\mathrm{k}\right),\hspace{0.33em}\hspace{0.33em}F=\sqrt{2}\mathrm{AM}\left(\eta ,\mathrm{k}\right),$$

(51)

$$L=-{\displaystyle \frac{1}{3}}{n}_1^2{k}^{2}x+{\displaystyle \frac{2}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T),$$

(52a)

$$m=-\sqrt{2}{n}_1\mathrm{DN}\left(\eta ,k\right),$$

(52b)

$$W=k_{2}y-k_2\beta T+k_0+\sqrt{2}\mathrm{AM}\left(\eta ,\mathrm{k}\right),$$

(52c)

Two soliton-cnoidal solutions of the ESWW Equation (17) can be derived:

$$u_{19}=L+m{\displaystyle \frac{\tanh (W)}{1+\mathrm{sech}(W)}},$$

(53a)

$$u_{20}=L+m{\displaystyle \frac{\tanh (W)}{1-\mathrm{sech}(W)}},$$

(53b)

where  $L,m$ and  $W$ are determined by (52),  $\eta =n_{1}x+n_{3}T+n_0.$

Case 9.

In this case, the undetermined constants and functions are as follows:

$$C_1=0,C_3=0,C_4=-{\displaystyle \frac{1}{2}},C_0=-1,C_2={\displaystyle \frac{3}{2}},n_2=0,k_1=0,k_3=-k_2\beta ,$$

$$\hspace{0.33em}\varphi =\mathrm{ND}\left(\eta ,{\displaystyle \frac{\sqrt{2}}{2}}\right),\hspace{0.33em}\hspace{0.33em}F=\sqrt{2}\arctan \left({\displaystyle \frac{\sqrt{2}\mathrm{SD}\left(\eta ,\frac{\sqrt{2}}{2}\right)}{2\mathrm{CD}\left(\eta ,\frac{\sqrt{2}}{2}\right)}}\right),$$

(54)

$$L=-{\displaystyle \frac{1}{3}}{n}_1^2{k}^{2}x+{\displaystyle \frac{2}{3}}{n}_1^{2}x+{\displaystyle \frac{1}{3}}\gamma x+{\displaystyle \frac{n_3}{3n_1}}x+{\displaystyle \frac{1}{3}}\alpha y-{\displaystyle \frac{1}{3}}\alpha \beta T+f(T),$$

(55a)

$$m={\displaystyle \frac{-\left(n_1\left(2\mathrm{CN}\left(\eta ,\frac{\sqrt{2}}{2}\right)\mathrm{CD}\left(\eta ,\frac{\sqrt{2}}{2}\right)+\mathrm{SD}\left(\eta ,\frac{\sqrt{2}}{2}\right)\mathrm{SN}\left(\eta ,\frac{\sqrt{2}}{2}\right)\right)\right)}{{\left(\mathrm{DN}\left(\eta ,\frac{\sqrt{2}}{2}\right)\right)}^2\left(2{\left(\mathrm{CD}\left(\eta ,\frac{\sqrt{2}}{2}\right)\right)}^2+{\left(\mathrm{SD}\left(\eta ,\frac{\sqrt{2}}{2}\right)\right)}^2\right)}},$$

(55b)

$$W=k_{2}y-k_2\beta T+k_0+\sqrt{2}\arctan \left({\displaystyle \frac{\sqrt{2}\mathrm{SD}\left(\eta ,\frac{\sqrt{2}}{2}\right)}{2\mathrm{CD}\left(\eta ,\frac{\sqrt{2}}{2}\right)}}\right),$$

(55c)

Two soliton-cnoidal solutions of the ESWW Equation (17) can be derived:

$$u_{21}=L+m{\displaystyle \frac{\tanh (W)}{1+\mathrm{sech}(W)}},$$

(56a)

$$u_{22}=L+m{\displaystyle \frac{\tanh (W)}{1-\mathrm{sech}(W)}},$$

(56b)

where  $L,m$, and  $W$ are determined by (55),  $\eta =n_{1}x+n_{3}T+n_0.$

Remark 2.

All the solutions obtained in this paper for the ESWW Equation (17) have been checked by Maple 17 software. If the readers can introduce other appropriate hypotheses of $W$, then $L$ can be solved by (22), and more meaningful solutions can be obtained by Theorem 1.

4. Results and Discussion

In Section 3, we have obtained twenty-two solutions for the ESWW Equation (17). Replacing $T$ with ${\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}$ in $u_i\hspace{0.33em}(i=1,2,\dots ,22)$, twenty-two solutions for the TFESWW Equation (1) can be obtained. For example, when $\alpha =\beta =\gamma =1,k_0=k_2=1,n_0=1,$  $n_1=2,n_3=1,$ we can obtain the following soliton-cnoidal solution for the TFESWW Equation (1) by replacing $T$ with ${\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}$ in (36a).

$$\begin{array}{l}{u}_6=-{\displaystyle \frac{11}{6}}x+{\displaystyle \frac{1}{3}}y-{\displaystyle \frac{1}{3}}{\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}+f\left({\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}\right)+\\ +{\displaystyle \frac{6\mathrm{CN}\left(2x+\frac{t^{\theta }}{\Gamma (1+\theta )}+1,\frac{\sqrt{3}}{2}\right)\mathrm{SN}\left(2x+\frac{t^{\theta }}{\Gamma (1+\theta )}+1,\frac{\sqrt{3}}{2}\right)}{1+2{\left(\mathrm{DN}\left(2x+\frac{t^{\theta }}{\Gamma (1+\theta )}+1,\frac{\sqrt{3}}{2}\right)\right)}^2}}\\ \times \tanh \left(y-{\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}+1+\sqrt{2}\arctan \left(\sqrt{2}\mathrm{DN}\left(2x+{\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}+1,{\displaystyle \frac{\sqrt{3}}{2}}\right)\right)\right),\end{array}$$

(57)

where $\theta$ is the fractional order, $f\left({\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}\right)$ is an arbitrary function of ${\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}$.

When $f\left({\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}\right)=0,$ 3D graphs of (57) with a different fractional order have been plotted in Figure 1.

From (a) to (c) in Figure 1, it can be observed that the periodicity of the solution gradually emerges and becomes apparent as an increase in the fractional order. From (d) to (f), we can see that the periodicity of the solution gradually decreases with the increase in the fractional order. That is to say, the geometric shapes of the solution (57) have changed a lot due to the variation in the fractional order.

When $f\left({\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}\right)=3\sin \left({\displaystyle \frac{t^{\theta }}{\Gamma (1+\theta )}}\right),$ the soliton-cnoidal solution (57) becomes a three-wave interaction solution including a hyperbolic function, Jacobian elliptic function, and trigonometric sine function; 3D graphs are given in Figure 2.

From (a) to (c) in Figure 2, the solitary wave affected by periodic waves can be observed when $\theta$ grows from 0.3 to 0.5. When $\theta$ grows from 0.5 to 0.8, the interaction of the solitary wave and the elliptic wave becomes apparent. From (d) to (f), we can see that with the increase in the fractional order, the influence of the trigonometric sine becomes apparent. When $\theta$ = 0.02, it looks like a standard elliptic periodic wave. When $\theta$ = 0.5, part of the elliptic periodic wave has been deformed. When $\theta$ = 0.98, the effect of the sine function on the three-wave interaction solution can be seen.

Remark 3.

We should point out that when taking  $\alpha =\beta =\gamma =0$ in  $u_i\hspace{0.33em}(i=1,2,\dots ,22)$, analytical solutions can be obtained for the following FNEE:

$$D_t^{\theta }(u_y)+u_{xxxy}-3u_{xx}{u}_y-3u_x{u}_{xy}=0.$$

(58)

It is the time-fractional BLMP equation with the Gerasimov–Caputo derivative. The results derived in this paper are different from those in [46,47] because the fractional derivatives and methods are different. Particularly, the soliton-cnoidal solutions and three-wave interaction solutions for FNEEs obtained in this paper are novel and have not been reported before.

5. Conclusions

A new method named the fractional consistent Riccati expansion (FCRE) method is proposed for systematically searching for soliton and soliton-cnoidal solutions for FNEEs. If we choose different hypotheses of the function $W,$ we can derive different solution structures for the given FNEEs. In this paper, we consider two hypotheses of the function $W.$ The first hypothesis can derive soliton and combined-soliton solutions, and the second can derive soliton-cnoidal solutions or three-wave interaction solutions. The FCRE method here is different from the existent methods, such as the fractional sub-equation method, $G^{\prime }/G$-expansion method, and exp-function method, because it can derive not only soliton solutions but also interaction solutions of soliton, Jacobian elliptic functions, and trigonometric functions. As far as we know, none of the existent methods can directly generate such interaction solutions of FNEEs, so the proposed method is novel.

The validity of this method has been verified by the (2 + 1)-dimensional time-fractional extended shallow water wave (TFESWW) Equation (1), which can describe the dynamical behavior of water waves propagating in the ocean and rivers. Plenty of analytical solutions, including soliton solutions, multiple soliton solutions, soliton-cnoidal interaction solutions, and three-wave interaction solutions, have been derived for (1). It is the first time we have obtained soliton-cnoidal and three-wave interaction solutions for FNEEs. Since the propagation of water waves in the real world is very complicated, it is even impossible to describe the practical phenomena by a single function. Interaction solutions between two or three functions become meaningful and important and also bring a wide range of applications at the physical level. The influence of the fractional order parameter $\theta$ on the dynamic behaviors of the soliton-cnoidal solution and three-wave interaction solution have been illustrated by plenty of graphs.

The FCRE method can also be applied to many other FNEEs, such as the fractional Kadomtsev–Petviashvili-modified-equal-width (KP-MEW) equation [53,54]. In addition, it is interesting to find other hypotheses of the function $W,$ and it will be helpful for generating other meaningful solutions for FNEEs. More about the FCRE method will be investigated in detail in future studies.