New Interaction Patterns for the Truncated M-Fractional Kadomtsev–Petviashvili Equation in High-Dimensional Space

Abstract

A truncated M-fractional Kadomtsev–Petviashvili (KP) equation in high-dimensional space has been proposed. The model provides the oretical support for studying the interaction patterns among waves. According to the attributes of the truncated M-fractional derivative, the truncated M-fractional KP equation can be reduced to the new extended KP equation. New interaction patterns which are compositions of cnoidal functions and soliton or trigonometric functions have been derived by the consistent Riccati expansion method. Applying the simple direct method, the finite symmetry transformation group and Bäcklund transformation have been constructed. Based on the known dark soliton solution and lump solution, new interaction patterns have been derived, including compositions of a dark soliton and an exponential function, compositions of a dark soliton and a trigonometric sine function, and compositions of a lump and a trigonometric sine function. The innovative aspect lies in the way that we find two effective ways to construct new interplay patterns of fractional differential equations.

1. Introduction

Fractional calculus includes fractional derivatives and fractional integrals. After Mandelbrot introduced the concept of fractional dimension [1], research findings on fractional calculus and fractal geometry have increased [2,3,4,5]. Since fractional calculus possesses two unique properties, memory or hereditary property and nonlocal property, it has become a powerful tool to describe various phenomena, such as the patterns of spread of and strategies for alleviating COVID-19 [6], the interaction between tumor cells [7], finance and economics [8,9], heat transfer [10], relaxing medium [11], magneto-radiative gas [12], and so on [13,14,15,16]. In addition, the application of fractional calculus in neural networks is also helpful for the development of artificial intelligence [17]. As a new method, fractional calculus has also been employed to research nonlocal phenomena in quantum mechanics [18,19].

The Kadomtsev–Petviashvili (KP) model

$$u_{xt}+{(u_{xxx}-6u{u}_x)}_x+3u_{yy}=0,$$

is a famous high-dimensional integrable equation and is an extension of the famous KdV model in high-dimensional space [20]. The KP model is a typical completely integrable model, and allows infinitely many conservation laws, multi-soliton solutions, bi-Hamiltonian structure, and a Lax pair [21]. It is widely used in describing the propagation of weakly nonlinear, long waves in dispersive media with small transverse variations (e.g., surface waves in shallow water) and has proven useful for observing wave phenomena in plasmas (e.g., Langmuir waves and ion acoustic waves) [20,21,22]. To better describe real-world phenomena, many extensions of the KP equation have been developed including variable coefficient (2+1)-dimensional KP equation [23], (4+1)-dimensional extension [24], (3+1)- and (2+1)-dimensional KP equations [25], fractional KP equation [26], Camassa-Holm-Kadomtsev-Petviashvili equation [27], and Korteweg-de Vries Kadomtsev-Petviashvili equation [28]. With the rapid development of fractional calculus, fractional forms of the KP extension have attracted attention [26,27,28].

The research objective of this article is the truncated M-fractional (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation.

$$a{D}_{M,T}^{\theta ,\delta }(u_x)-\frac{a^4-6a^2{b}^2+b^4}{16}{u}_{xxxx}-\frac{3(b^2-a^2)}{4}{(u^2)}_{xx}+\alpha u_{xx}+\beta u_{xy}+\eta u_{yy}=0,$$

(1)

where $a,b,\alpha ,\beta$, and $\eta$ are real numbers, $a\ne 0, a\ne \pm b,$ $\theta (0<\theta \le 1)$ is the fractional order, and $D_{M,T}^{\theta ,\delta }$ is the truncated M-fractional derivative operator and is defined as

$$D_{M,T}^{\theta ,\delta }u=\underset{\tau \to 0}{\lim }\frac{u(x,y,T{E}_{\delta }(\tau T^{1-\theta }))-u(x,y,T)}{\tau },\hspace{0.33em}0<\theta <1,\delta >0$$

$E_{\delta }(.)$ is a truncated Mittag-Leffler function of one parameter [29]. The interpretation and features of the operator have also been summarized [30,31]. The kernel function of the M-fractional derivative is based on the Mittag-Leffler function, therefore the M-fractional derivative can describe the decay pattern of memory weights over time more realistically. The truncated M-fractional KP Equation (1) can model the nonlocal interactions of long waves with weak nonlinearity and weak dispersion propagating in viscoelastic shallow water.

When $\theta =1,$ the truncated M-fractional KP Equation (1) becomes the new extended KP equation as follows:

$$a{u}_{xT}-\frac{a^4-6a^2{b}^2+b^4}{16}{u}_{xxxx}-\frac{3(b^2-a^2)}{4}{(u^2)}_{xx}+\alpha u_{xx}+\beta u_{xy}+\eta u_{yy}=0,$$

(2)

which has been proposed by Wazwaz recently. Painlevé integrability, multiple soliton solutions, and lump solutions have been obtained [25]. For (2), analytical solutions, including resonance Y-type solutions, lump solutions, and breather solutions, have been studied [32]. Localized periodic solutions of (2) are constructed using its bilinear form [33]. A dynamical analysis of the soliton wave solutions of (2) has been performed using two analytical techniques [34]. The fractional form of (2) in the sense of the conformable derivative has been studied, and analytical solutions have been derived in [35].To the greatest extent of our knowledge, interaction solutions which are compositions of two or more functions of different types to (2) have not been reported in the previous literature. In addition, the finite symmetry transformation group of (2) has not been studied yet. Our study will fill those research gaps.

Explicit solutions of nonlinear evolution equations can not only help explain interesting physical phenomena but also are useful for improving numerical calculation programs and methods [36,37]. To construct analytical solutions of fractional nonlinear evolution equations, plenty of methods have been developed, such as the exponential function expansion procedure [38], the fractional auxiliary equation method [39,40,41], and the Lie symmetry method [42,43,44]. By utilizing the properties of fractional derivatives, nonlinear evolution equations of fractional orders can also be transformed into nonlinear equations of integer orders [45,46]. Therefore, techniques to obtain explicit solutions of partial differential equations, such as the consistent Riccati expansion method [47,48], the simple direct method [49], the Hirota bilinear method [50], the unified and generalized Bernoulli sub-ODE techniques [51,52], the exp-expansion method [53], the (G′/G)-expansion method [35], the $({\mathrm{G}}^{\prime }/{\mathrm{G}}^2)$-expansion method [54], the elliptic function expansion method [34], and so on, can be used to study analytical solutions of fractional differential equations. However, the solutions derived by most of these methods are single functions or the four arithmetic operations of one or two types of functions. Solutions, which are expressed by the compositions of two or more functions of different types, have richer physical meanings and are worthy of study.

The consistent Riccati expansion method can obtain interaction solutions which are compositions of hyperbolic functions and elliptic functions or trigonometric functions, and solutions of this type are difficult to derive using conventional methods. The simple direct method can help to establish new interplay patterns, for example, interaction solutions among the lump and trigonometric functions. Up to now, the application of the consistent Riccati expansion method and the simple direct method in fractional nonlinear equations is still particularly limited. In this paper, we will apply the two methods to the truncated M-fractional KP Equation (1) and seek its new interaction patterns.

The following is the arrangement of the paper. We give the main steps of the consistent Riccati expansion method and the simple direct method in Section 2. In Section 3, we will extend the consistent Riccati expansion method to (1) and search for its interaction solutions. The finite symmetry transformation group will be constructed via the simple direct method in Section 4. Based on the obtained finite symmetry transformation group, new interaction patterns from the known solutions can be constructed in Section 5. The main conclusions are presented in the last section.

2. Methods

2.1. The Consistent Riccati Expansion Method

The consistent Riccati expansion method can be applied to construct interaction patterns among soliton waves, trigonometric waves, and cnoidal waves [47,48]. Given a truncated M-fractional nonlinear evolution equation

$$G(x,y,T,u,u_x,u_y,D_{M,T}^{\theta ,\delta }u,u_{xx},u_{xy},D_{M,T}^{\theta ,\delta }(u_x),\dots )=0.$$

Making a traveling wave transformation

$$u=u(x,y,t),\hspace{0.33em}\hspace{0.33em}t=\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta },\hspace{0.33em}\delta \in (0,+\infty ),$$

(3)

and taking it into the above fractional equation, then a nonlinear evolution equation can be obtained as follows:

$$\overline{G}(x,y,t,u,u_x,u_y,u_t,u_{xx},u_{xy},u_{xt},\dots )=0.$$

(4)

Suppose $u$ is an exact solution to (4) and can be written as

$$u={\displaystyle \sum _{i=0}^I{\overline{u}}_i{R}^i(W)},$$

(5)

where ${\overline{u}}_i$ and $W$ are unknown functions concerning $x$, $y$, and $t$. $R$ satisfies the Riccati equation

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

(6)

where $A,B$, and $M$ are constants. Analytical solutions of (6) can be found in many references [55,56].

The positive integer $I$ is decided by the homogeneous balance principle. Then, substituting (5) with (6) into (4) and restricting the coefficients of different $R^i(W)$ to zero generates a series of partial differential equations. To obtain the explicit form of ${\overline{u}}_i$, we solve these overdetermined systems of equations. Usually, ${\overline{u}}_i$ depends on $W$ and its derivatives, then solutions of (4) can be constructed. More details will be illustrated in Section 3.

2.2. The Simple Direct Method

The simple direct method is often employed to construct Bäcklund transformations of nonlinear evolution equations [49]. For a given nonlinear evolution equation as (4), we suppose that a solution of (4) is

$$u={\tilde{u}}_0+{\tilde{u}}_{1}U(X,Y,V),$$

where ${\tilde{u}}_0,{\tilde{u}}_1,X,Y,V$ are unknown functions of $x,y$, and $t.$ Restricting $U$ to satisfy the following equation:

$$\overline{G}(X,Y,V,U,U_X,U_Y,U_V,U_{XX},U_{XY},U_{XV},\dots )=0,$$

(7)

then the explicit form of ${\tilde{u}}_0,{\tilde{u}}_1,X,Y$, and $V$ can be obtained. Equations (4) and (7) are identical in form; however, their independent and dependent variables are not the same. For more details about the method, readers can refer to [49].

3. Interaction Patterns Among Elliptic Functions and Hyperbolic or Trigonometric Functions

In the following, we will apply the consistent Riccati expansion technique to construct interaction patterns of (1). To reduce the truncated M-fractional KP Equation (1) to a partial differential equation, we take the transformation (3) into (1). The fractional Equation (1) can be changed to the new extended KP equation as follows:

$$a{u}_{xt}-\frac{a^4-6a^2{b}^2+b^4}{16}{u}_{xxxx}-\frac{3(b^2-a^2)}{4}{(u^2)}_{xx}+\alpha u_{xx}+\beta u_{xy}+\eta u_{yy}=0.$$

(8)

Equation (8) is a new type of the KP extension and has been studied by different methods [25,32,33,34,35]; however, interaction solutions have not been studied yet. Next, we will apply the consistent Riccati expansion method to construct interaction solutions of (8).

As stated in Section 2, let the solution of (8) have the form of (5). By applying the homogeneous balance principle in (8), we find that $I=2.$ Taking $u={\overline{u}}_0+{\overline{u}}_{1}R(W)+{\overline{u}}_2{R}^2(W)$ with (6) into (8), coefficients of different $R^i(W)$ can be collected. After complicated calculations, we find that ${\overline{u}}_i$ can be determined by $W.$ The results are as follows:

$$\begin{array}{l}{\overline{u}}_0=\frac{1}{24(a^2-b^2){W_x}^2}\left((8AM+B^2)(a^4-6a^2{b}^2+b^4){W_x}^4-3(a^4-6a^2{b}^2+b^4){W_{xx}}^2\right.\\ +6B(a^4-6a^2{b}^2+b^4)W_{xx}{W_x}^2+4(a^4-6a^2{b}^2+b^4)W_{xxx}{W}_x-16a{W}_t{W}_x-16\eta {W_y}^2\\ \left.-16\alpha {W_x}^2-16\beta W_x{W}_y\right),\\ {\overline{u}}_1=\frac{M}{2(a^2-b^2)}\left((B{W_x}^2+W_{xx})(a^4-6a^2{b}^2+b^4)\right),\\ {\overline{u}}_2=\frac{M^2(a^4-6a^2{b}^2+b^4)}{2(a^2-b^2)}{W_x}^2.\end{array}$$

(9)

We should point out that the solution construction of (9) depends on the specific form of $W.$ In the next section, we consider the following two different cases.

3.1. Soliton Solutions, Hybrid Soliton Solutions, and Trigonometric Function Solutions

If $W=k_0+k_{1}x+k_{2}y+k_{3}t,$ where $k_0,k_1,k_2$, and $k_3$ are arbitrary constants, the exact solutions of (8) are as follows:

(1)

When $A=\frac{1}{2},B=0,M=-\frac{1}{2},R(W)=\frac{\tanh (W)}{1-\mathrm{sech}(W)},$ then

$$u_1=\frac{-{k_1}^4(a^4-6a^2{b}^2+b^4)-8k_1(a{k}_3+\alpha k_1+\beta k_2)-8\eta {k_2}^2}{12{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\tanh }^2(\mathrm{W})}{8(a^2-b^2){\left(1-\mathrm{sech}(W)\right)}^2}.$$

(2)

When $A=\frac{1}{2},B=0,M=-\frac{1}{2},R(W)=\coth (W)+\mathrm{csch}(W),$ then

$$u_2=\frac{-{k_1}^4(a^4-6a^2{b}^2+b^4)-8k_1(a{k}_3+\alpha k_1+\beta k_2)-8\eta {k_2}^2}{12{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\left(\coth (W)+\mathrm{csch}(W)\right)}^2}{8(a^2-b^2)}.$$

(3)

When $A=\frac{1}{2},B=0,M=\frac{1}{2},R(W)=\csc (W)-\cot (W),$ then

$$u_3=\frac{{k_1}^4(a^4-6a^2{b}^2+b^4)-8k_1(a{k}_3+\alpha k_1+\beta k_2)-8\eta {k_2}^2}{12{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\left(\csc (W)-\cot (W)\right)}^2}{8(a^2-b^2)}.$$

(4)

When $A=\frac{1}{2},B=0,M=\frac{1}{2},R(W)=\tan (W)-\sec (W),$ then

$$u_4=\frac{{k_1}^4(a^4-6a^2{b}^2+b^4)-8k_1(a{k}_3+\alpha k_1+\beta k_2)-8\eta {k_2}^2}{12{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\left(\tan (W)-\sec (W)\right)}^2}{8(a^2-b^2)}.$$

(5)

When $A=-\frac{1}{2},B=0,M=-\frac{1}{2},R(W)=\frac{\cot (W)}{1\pm \csc (W)},$ then

$$u_5=\frac{{k_1}^4(a^4-6a^2{b}^2+b^4)-8k_1(a{k}_3+\alpha k_1+\beta k_2)-8\eta {k_2}^2}{12{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\cot }^2(W)}{8(a^2-b^2){\left(1\pm \csc (W)\right)}^2}.$$

(6)

When $A=-\frac{1}{2},B=0,M=-\frac{1}{2},R(W)=\cot (W)\pm \csc (W),$ then

$$u_6=\frac{{k_1}^4(a^4-6a^2{b}^2+b^4)-8k_1(a{k}_3+\alpha k_1+\beta k_2)-8\eta {k_2}^2}{12{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\left(\cot (W)\pm \csc (W)\right)}^2}{8(a^2-b^2)}.$$

(7)

When $A=1,B=0,M=-1,R(W)=\coth (W),$ then

$$u_7=\frac{-{k_1}^4(a^4-6a^2{b}^2+b^4)-2k_1(a{k}_3+\alpha k_1+\beta k_2)-2\eta {k_2}^2}{3{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\coth }^2(W)}{2(a^2-b^2)}.$$

(8)

When $A=1,B=0,M=-1,R(W)=\tanh (W),$ then

$$u_8=\frac{-{k_1}^4(a^4-6a^2{b}^2+b^4)-2k_1(a{k}_3+\alpha k_1+\beta k_2)-2\eta {k_2}^2}{3{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\tanh }^2(W)}{2(a^2-b^2)}.$$

(9)

When $A=1,B=-2,M=2,R(W)=\frac{\tan (W)}{1+\tan (W)},$ then

$$u_9=\frac{5{k_1}^4(a^4-6a^2{b}^2+b^4)-4k_1(a{k}_3+\alpha k_1+\beta k_2)-4\eta {k_2}^2}{6{k_1}^2(a^2-b^2)}-\frac{2{k_1}^2(a^4-6a^2{b}^2+b^4)\tan (W)}{(a^2-b^2){\left(1+\tan (W)\right)}^2}.$$

(10)

When $A=-1,B=2,M=-2,R(W)=\frac{\cot (W)}{1+\cot (W)},$ then

$$u_{10}=\frac{5{k_1}^4(a^4-6a^2{b}^2+b^4)-4k_1(a{k}_3+\alpha k_1+\beta k_2)-4\eta {k_2}^2}{6{k_1}^2(a^2-b^2)}-\frac{2{k_1}^2(a^4-6a^2{b}^2+b^4)\cot (W)}{(a^2-b^2){\left(1+\cot (W)\right)}^2}.$$

(11)

When $A=1,B=0,M=1,R(W)=\tan (W),$ then

$$u_{11}=\frac{{k_1}^4(a^4-6a^2{b}^2+b^4)-2k_1(a{k}_3+\alpha k_1+\beta k_2)-2\eta {k_2}^2}{3{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\tan }^2(W)}{2(a^2-b^2)}.$$

(12)

When $A=-1,B=0,M=-1,R(W)=\cot (W),$ then

$$u_{12}=\frac{{k_1}^4(a^4-6a^2{b}^2+b^4)-2k_1(a{k}_3+\alpha k_1+\beta k_2)-2\eta {k_2}^2}{3{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\cot }^2(W)}{2(a^2-b^2)}.$$

(13)

When $A=1,B=2,M=2,R(W)=\frac{\tan (W)}{1-\tan (W)},$ then

$$u_{13}=\frac{5(a^4-6a^2{b}^2+b^4){k_1}^4-4k_1(a{k}_3+\alpha k_1+\beta k_2)-4\eta {k_2}^2}{6{k_1}^2(a^2-b^2)}+\frac{2{k_1}^2(a^4-6a^2{b}^2+b^4)\tan (W)}{(a^2-b^2){\left(1-\tan (W)\right)}^2}.$$

(14)

When $A=-1,B=-2,M=-2,R(W)=\frac{\cot (W)}{1-\cot (W)},$ then

$$u_{14}=\frac{5(a^4-6a^2{b}^2+b^4){k_1}^4-4k_1(a{k}_3+\alpha k_1+\beta k_2)-4\eta {k_2}^2}{6{k_1}^2(a^2-b^2)}+\frac{2{k_1}^2(a^4-6a^2{b}^2+b^4)\cot (W)}{(a^2-b^2){\left(1-\cot (W)\right)}^2}.$$

3.2. Interaction Solutions Among Cnoidal Waves, Trigonometric Waves, and Soliton Waves

In this case, we suppose that

$$W=k_{1}x+k_{2}y+k_{3}t+k_0+F(\xi ),\hspace{0.33em}\xi =n_{1}x+n_{2}y+n_{3}t+n_0,$$

(10)

where $k_i$ and $n_i(i=0,1,2,3)$ are arbitrary constants and $F_{\xi }$ satisfies the following ordinary differential equation

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

(11)

With $C_j\hspace{0.33em}(j=0,1,2,3,4)$ being constants. If $C_j\hspace{0.33em}(j=0,1,2,3,4)$ take different values, the explicit solutions of (11) have been studied in [54].

3.2.1. $A=1, B=0, M=-1$

In this case, the solutions of (6) are $R(W)=\tanh (W),$ and $R(W)=\coth (W).$

When $C_1=0,C_3=0,C_0=\frac{k^2}{4},C_2=-1-k^2,C_4=4,$ (11) has aJacobian elliptic function solution $\hspace{0.33em}{F}_{\xi }\left(\xi \right)=\frac{k}{2}\mathrm{sn}\left(\xi \right),$ then

$$F(\xi )=\frac{1}{2}\ln \left(\mathrm{dn}\left(\xi \right)-k\mathrm{cn}\left(\xi \right)\right),$$

where $\mathrm{sn}\left(\xi \right)=\mathrm{JacobiSN}(\xi , \mathrm{k}),\mathrm{cn}\left(\xi \right)=\mathrm{JacobiCN}(\xi , \mathrm{k})$ and $\mathrm{dn}\left(\xi \right)=\mathrm{JacobiDN}(\xi , \mathrm{k})$ are Jacobian elliptic functions and $k\hspace{0.33em}$ is a constant and $0<k<1.$

Taking (10) into (9), the two interaction solutions among cnoidal waves and soliton waves to (8) are as follows:

$$\begin{array}{l}{u}_{15}=\frac{1}{24(a^2-b^2){(k{n}_1\mathrm{sn}(\xi )+2k_1)}^2}\left(48b^4{k_1}^4{\tanh }^2(W)\right.+48a^4{k_1}^4{\tanh }^2(W)-32b^4{k_1}^4\\ -64\eta {k_2}^2+3b^4{k}^4{n_1}^4\mathrm{sn}4(\xi ){\tanh }^2(W)+3a^4{k}^4{n_1}^4{\mathrm{sn}}^4(\xi ){\tanh }^2(W)-32\beta k{n}_1{k}_2\mathrm{sn}(\xi )\\ -3a^4{k}^2{n_1}^4{\mathrm{cn}}^2(\xi ){\mathrm{dn}}^2(\xi )-3b^4{k}^2{n_1}^2{\mathrm{dn}}^4(\xi ){\mathrm{cn}}^2(\xi )-16\beta k^2{n}_1{n}_2{\mathrm{sn}}^2(\xi )-64\eta k{k}_2{n}_2\mathrm{sn}(\xi )\\ -64k{a}^4{k_1}^3{n}^1\mathrm{sn}(\xi )-48k^2{a}^4{k_1}^2{n_1}^2{\mathrm{sn}}^2(\xi )-16a^4{k}^3{n_1}^3{k}_1{\mathrm{sn}}^3(\xi )-4k^4{a}^4{n_1}^4{\mathrm{sn}}^2(\xi ){\mathrm{cn}}^2(\xi )\\ -4k^2{a}^4{n_1}^4{\mathrm{dn}}^2(\xi ){\mathrm{sn}}^2(\xi )+12k^4{a}^2{b}^2{n_1}^4{\mathrm{sn}}^4(\xi )-32ak{n}_3{k}_1\mathrm{sn}(\xi )-16a{k}^2{n}_1{n}_3{\mathrm{sn}}^2(\xi )\\ -32ak{n}_1{k}_3\mathrm{sn}(\xi )-64{k_1}^3{b}^{4}k{n}_1\mathrm{sn}(\xi )-48{k_1}^2{b}^4{k}^2{n_1}^2{\mathrm{sn}}^2(\xi )-4b^4{k}^4{n_1}^4{\mathrm{cn}}^2(\xi ){\mathrm{sn}}^2(\xi )\\ -16b^4{k}_1{k}^3{n_1}^3{\mathrm{sn}}^3(\xi )-4b^4{k}^2{n_1}^4{\mathrm{sn}}^2(\xi ){\mathrm{dn}}^2(\xi )-64\alpha k{k}_1{n}_1\mathrm{sn}(\xi )-32\beta k{k}_1{n}_2\mathrm{sn}(\xi )\\ +192a^2{b}^2{k_1}^4-64a{k}_1{k}_3-2b^4{k}^4{n_1}^4{\mathrm{sn}}^4(\xi )-288a^2{b}^2{k_1}^4{\tanh }^2(W)-2a^4{k}^4{n_1}^4{\mathrm{sn}}^4(\xi )\\ -16\alpha k^2{n_1}^2\mathrm{sn}2(\xi )-16\eta k^2{n_2}^2{\mathrm{sn}}^2(\xi )+144k^2{n_1}^3{a}^2{b}^2{k}_1\mathrm{sn}(\xi )\mathrm{cn}(\xi )\mathrm{dn}(\xi )\tanh (W)\\ +144k{n_1}^2{a}^2{b}^2{k_1}^2\mathrm{cn}(\xi )\mathrm{dn}(\xi )\tanh (W)+36k^3{n_1}^4{a}^2{b}^2\mathrm{dn}(\xi )\mathrm{cn}(\xi ){\mathrm{sn}}^2(\xi )\tanh (W)\\ -24k^2{n_1}^3{k}_1(a^4+b^4)\mathrm{sn}(\xi )\mathrm{cn}(\xi )\mathrm{dn}(\xi )\tanh (W)+48k{n_1}^3{a}^2{b}^2{k}_1\mathrm{sn}(\xi )(k^2{\mathrm{cn}}^2(\xi )+{\mathrm{dn}}^2(\xi ))\\ -6k^3{n_1}^4(a^4+b^4)\mathrm{cn}(\xi )\mathrm{dn}(\xi ){\mathrm{sn}}^2(\xi )\tanh (W)-24k{n_1}^2(a^4+b^4){k_1}^2\mathrm{cn}(\xi )\mathrm{dn}(\xi )\tanh (W)\\ -144k_1{a}^2{b}^{2}k{n}_1\mathrm{sn}(\xi ){\tanh }^2(W)(k^2{n_1}^2{\mathrm{sn}}^2(\xi )+3k_{1}k{n}_1\mathrm{sn}(\xi )+4{k_1}^2)-64\beta k_1{k}_2\\ +96k_1{a}^2{b}^2{k}^2{n_1}^2{\mathrm{sn}}^2(\xi )\left(3k_1+k{n}_1\mathrm{sn}(\xi )\right)+24a^2{b}^2{k}^2{n_1}^4{\mathrm{sn}}^2(\xi )(k^2{\mathrm{cn}}^2(\xi )+{\mathrm{dn}}^2(\xi ))\\ -8k_1{b}^4{k}^3{n_1}^3\mathrm{sn}(\xi ){\mathrm{cn}}^2(\xi )-8k{k}_1{b}^4{n_1}^3\mathrm{sn}(\xi ){\mathrm{dn}}^2(\xi )+18a^2{b}^2{k}^2{n_1}^4{\mathrm{cn}}^2(\xi ){\mathrm{dn}}^2(\xi )\\ +96k(a^4+b^4){k_1}^3{n}_1\mathrm{sn}(\xi ){\tanh }^2(W)+72k^2(a^4+b^4){k_1}^2{n_1}^2{\mathrm{sn}}^2(\xi ){\tanh }^2(W)\\ +24k^3(a^4+b^4)k_1{n_1}^3\mathrm{sn}3(\xi ){\tanh }^2(W)-18k^4{a}^2{b}^2{n_1}^4{\mathrm{sn}}^4(\xi ){\tanh }^2(W)-32a^4{k_1}^4\\ -8k_1{k}^3{a}^4{n_1}^3\mathrm{sn}(\xi ){\mathrm{cn}}^2(\xi )-8k{a}^4{k}_1{n_1}^3\mathrm{sn}(\xi ){\mathrm{dn}}^2(\xi )\left.+384a^2{b}^2{k_1}^{3}k{n}_1\mathrm{sn}(\xi )-64\alpha {k_1}^2\right),\end{array}$$

and

$$\begin{array}{l}{u}_{16}=\frac{1}{24(a^2-b^2){(k{n}_1\mathrm{sn}(\xi )+2k_1)}^2}\left(48{k_1}^4{\coth }^2(W)(a^4+b^4)\right.-32{k_1}^4(a^4+b^4)\\ +3(a^4+b^4){n_1}^4{k}^4{\mathrm{sn}}^4(\xi ){\coth }^2(W)-32ak\mathrm{sn}(\xi )\left(n_1{k}_3+n_3{k}_1\right)-16a{k}^2{n}_1{n}_3{\mathrm{sn}}^2(\xi )\\ -64{k_1}^3(a^4+b^4)k{n}_1\mathrm{sn}(\xi )-3k^2{n_1}^4(a^4+b^4){\mathrm{dn}}^2(\xi ){\mathrm{cn}}^2(\xi )-48k^2(a^4+b^4){k_1}^2{n_1}^2{\mathrm{sn}}^2(\xi )\\ -16k_1{k}^3(a^4+b^4){n_1}^3{\mathrm{sn}}^3(\xi )-4k^4(a^4+b^4){n_1}^4{\mathrm{sn}}^2(\xi ){\mathrm{cn}}^2(\xi )+192a^2{b}^2{k_1}^4-64a{k}_1{k}_3\\ -4k^2(a^4+b^4){n_1}^4{\mathrm{sn}}^2(\xi ){\mathrm{dn}}^2(\xi )-64\alpha k{k}_1{n}_1\mathrm{sn}(\xi )-32\beta k\mathrm{sn}(\xi )(k_1{n}_2+k_2{n}_1)\\ -16\beta k^2{n}_1{n}_2{\mathrm{sn}}^2(\xi )-64\eta k_{2}k{n}_2\mathrm{sn}(\xi )+12k^4{a}^2{b}^2{n_1}^4{\mathrm{sn}}^4(\xi )-288{k_1}^4{a}^2{b}^2{\coth }^2(W)\\ -2a^4{k}^4{n_1}^4{\mathrm{sn}}^4(\xi )-2b^4{k}^4{n_1}^4{\mathrm{sn}}^4(\xi )-16\alpha k^2{n_1}^2{\mathrm{sn}}^2(\xi )-16\eta k^2{n_2}^2{\mathrm{sn}}^2(\xi )\\ +144k{n_1}^2{a}^2{b}^2{k_1}^2\mathrm{cn}(\xi )\mathrm{dn}(\xi )\coth (W)+18a^2{b}^2{k}^2{n_1}^4{\mathrm{cn}}^2(\xi ){\mathrm{dn}}^2(\xi )-64\eta {k_2}^2\\ -24k^2{n_1}^3(a^4+b^4-6a^2{b}^2)k_1\mathrm{cn}(\xi )\mathrm{dn}(\xi )\mathrm{sn}(\xi )\coth (W)-64\beta k_1{k}_2-64\alpha {k_1}^2\\ +48k{n_1}^3{a}^2{b}^2{k}_1\mathrm{sn}(\xi )(k^2{\mathrm{cn}}^2\xi +d{n}^2\xi )-24k{n_1}^2(a^4+b^4){k_1}^2\mathrm{cn}(\xi )\mathrm{dn}(\xi )\coth (W)\\ -6k^3{n_1}^4\mathrm{cn}(\xi )\mathrm{dn}(\xi ){\mathrm{sn}}^2(\xi )\coth (W)(a^4+b^4-6a^2{b}^2)+384a^2{b}^2{k_1}^{3}k{n}_1\mathrm{sn}(\xi )\\ -144k_1{a}^2{b}^{2}k{n}_1\mathrm{sn}(\xi ){\coth }^2(W)(k^2{n_1}^2{\mathrm{sn}}^2(\xi )+3k_{1}k{n}_1\mathrm{sn}(\xi )+4{k_1}^2)\\ +24a^2{b}^2{k}^2{n_1}^2{\mathrm{sn}}^2(\xi )(k^2{n_1}^2{\mathrm{cn}}^2(\xi )+4k_{1}k{n}_1\mathrm{sn}(\xi )+12{k_1}^2+{n_1}^2{\mathrm{dn}}^2(\xi ))\\ -8k_1{b}^{4}k{n_1}^3\mathrm{sn}(\xi )(k^2{\mathrm{cn}}^2(\xi )+{\mathrm{dn}}^2(\xi ))+96(a^4+b^4){k_1}^{3}n1k\mathrm{sn}(\xi ){\coth }^2(W)\\ +72(a^4+b^4){k_1}^2{n_1}^2{k}^2{\mathrm{sn}}^2(\xi ){\coth }^2(W)+24(a^4+b^4)k_1{n_1}^3{k}^3{\mathrm{sn}}^3(\xi ){\coth }^2(W)\\ -18a^2{b}^2{n_1}^4{k}^4{\mathrm{sn}}^4(\xi ){\coth }^2(W)-\left.8k_{1}k{a}^4{n_1}^3\mathrm{sn}(\xi )({\mathrm{dn}}^2(\xi )+k^2{\mathrm{cn}}^2(\xi )\right),\end{array}$$

where

$$W=k_{1}x+k_{2}y+k_{3}t+k_0+\frac{1}{2}\ln \left(\mathrm{dn}\left(n_{1}x+n_{2}y+n_{3}t+n_0\right)-k\mathrm{cn}\left(n_{1}x+n_{2}y+n_{3}t+n_0\right)\right),\hspace{0.33em}$$

with $k_i$ and $n_i(i=0,1,2,3)$ being arbitrary constants.

3.2.2. $A=1, B=0, M=1$

In this case, (6) possesses a trigonometric function solution $R(W)=\tan (W).$ Under the condition $C_1=0,C_3=0,C_0=\frac{k^2}{4}(1-k^2),C_2=2k^2-1,C_4=-4,$ a solution of (11) is $\hspace{0.33em}{F}_{\xi }\left(\xi \right)=\frac{k}{2}\mathrm{cn}\left(\xi \right),$ then

$$F(\xi )=\frac{1}{2}\arctan \left(\frac{k\mathrm{sn}\left(\xi \right)}{\mathrm{dn}\left(\xi \right)}\right),$$

so an interaction solution among cnoidal waves and trigonometric function waves to (8) can be obtained as follows:

$$\begin{array}{l}{u}_{17}=\frac{1}{24(a^2-b^2){\left(k{n}_1\mathrm{cn}(\xi )+2k_1\right)}^2}\left((a^4-6a^2{b}^2+b^4)(48{k_1}^2{k}^2{n_1}^2+2k^4{n_1}^4)\right.\\ -16k^2{n}_1(a{n}_3+\beta n_2)+32{k_1}^4(a^4-6a^2{b}^2+b^4)+3k^4{n_1}^4{\tan }^2(W)(a^4-6a^2{b}^2+b^4)\\ +k^2{n_1}^4{\mathrm{sn}}^2(\xi )(a^4-6a^2{b}^2+b^4)+k^4{n_1}^4{\mathrm{sn}}^2(\xi )(a^4-6a^2{b}^2+b^4)(4-3{\mathrm{sn}}^2(\xi ))\\ +16k^2{\mathrm{sn}}^2(\xi )(\eta {n_2}^2+\alpha {n_1}^2)-24k_1{k}^3{n_1}^3{\tan }^2(W)\mathrm{cn}(\xi ){\mathrm{sn}}^2(\xi )(a^4-6a^2{b}^2+b^4)\\ +(a^4-6a^2{b}^2+b^4)\left(48{k_1}^4{\tan }^2(W)+6k^3{n_1}^4\mathrm{dn}(\xi )\mathrm{sn}(\xi )\tan (W)({\mathrm{sn}}^2(\xi )-1)\right)\\ +(a^4-6a^2{b}^2+b^4)\left(72{k_1}^2{k}^2{n_1}^2{\tan }^2(W)-24k{n_1}^2{k_1}^2\tan (W)\mathrm{dn}(\xi )\mathrm{sn}(\xi )\right)\\ +(a^4-6a^2{b}^2+b^4)\left(3k^4{n_1}^4{\tan }^2(W){\mathrm{sn}}^2(\xi )({\mathrm{sn}}^2(\xi )-2)-8k{n_1}^3{k}_1\mathrm{cn}(\xi )\right)\\ +16k{k}_1{n}_1(k^2{n_1}^2+4{k_1}^2)\mathrm{cn}(\xi )(a^4-6a^2{b}^2+b^4)-48k^2{k_1}^2{n_1}^2{\mathrm{sn}}^2(\xi )(a^4-6a^2{b}^2+b^4)\\ +16k^2{n}_1{\mathrm{sn}}^2(\xi )(a{n}_3+n_2\beta )-32ak\mathrm{cn}(\xi )(k_1{n}_3+n_1{k}_3)-64k\mathrm{cn}(\xi )(\eta k_2{n}_2+\alpha k_1{n}_1)\\ -32k\beta \mathrm{cn}(\xi )(k_1{n}_2+n_1{k}_2)-24k^2{n_1}^3{k}_1\mathrm{dn}(\xi )\mathrm{sn}(\xi )\mathrm{cn}(\xi )\tan (W)(a^4-6a^2{b}^2+b^4)\\ -72k^2{k_1}^2{n_1}^2{\mathrm{sn}}^2(\xi ){\tan }^2(W)(a^4-6a^2{b}^2+b^4)-64k_1(\beta k_2+a{k}_3)-16k^2(\eta {n_2}^2+\alpha {n_1}^2)\\ +24k{k}_1{n}_1(k^2{n_1}^2+4{k_1}^2)\mathrm{cn}(\xi ){\tan }^2(W)(a^4-6a^2{b}^2+b^4)-4k^2{n_1}^4(a^4-6a^2{b}^2+b^4)\\ -64\eta {k_2}^2\left.-64\alpha {k_1}^2\right),\end{array}$$

where $W=k_{1}x+k_{2}y+k_{3}t+k_0+\frac{1}{2}\arctan \left(\frac{k\mathrm{sn}\left(n_{1}x+n_{2}y+n_{3}t+n_0\right)}{\mathrm{dn}\left(n_{1}x+n_{2}y+n_{3}t+n_0\right)}\right).\hspace{0.33em}$

We should point out that the solutions $u_{15},u_{16}$, and $u_{17}$ are new interaction patterns, which are the composition of several types of functions.

Remark 1.

In the solutions $u_i(i=1,2,\dots ,17)$ obtained above, if we replace $t$ with $\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta },$ solutions for the truncated M-fractional KP Equation (1) can be derived.

4. Finite Symmetry Transformation Group via the Simple Direct Method

Usually, the Lie point symmetry group (Lie group for short) is a transformation group that can transform a solution of an equation into another solution of the equation. However, constructing solutions by Lie group involves complex group theory. In order to avoid group theory, the simple direct method was proposed [49]. Using this method, we can directly derive new solutions from old ones. The idea and main steps can be found in many studies [49,57].

4.1. Finite Symmetry Transformation Group

The main idea of the simple direct method has been elaborated in Section 2. We suppose that a solution of (8) is

$$u={\tilde{u}}_0+{\tilde{u}}_{1}U(X,Y,V),$$

(12)

where ${\tilde{u}}_0,{\tilde{u}}_1,X,Y,V$ are unknown functions of $x,y$, and $t$, and they can be solved by restricting $U$ to satisfy an equation in the same form as (8) as follows:

$$U_{XXXX}=\frac{\left(24(a^2-b^2)U+16\alpha \right)U_{XX}+16a{U}_{XV}+16\beta U_{XY}+16\eta U_{YY}+24(a^2-b^2){(U_X)}^2}{a^4-6a^2{b}^2+b^4}.$$

(13)

Here, (13) is the same equation as (8) with different independent variables and dependent variables. Taking (12) into (8), and replacing $U_{XXXX}$ with (13), we will have

$$-\frac{{\tilde{u}}_1}{16}\left({V_x}^4{U}_{VVVV}+{Y_x}^4{U}_{YYYY}\right)\left(a^4-6a^2{b}^2+b^4\right)+{\Omega }_1\left(X,Y,V,U_X,U_Y,U_V,\dots \dots \right)=0,$$

(14)

where ${\Omega }_1\left(X,Y,V,U_X,U_Y,U_V,\dots \dots \right)$ has nothing to do with $U_{VVVV}$ and $U_{YYYY}.$ Solving (14), the result is

$$V_x=0,\hspace{0.33em}\hspace{0.33em}{Y}_x=0.$$

(15)

Taking the above results into (14), we have

$$2{\tilde{u}}_1\eta Y_y{V}_y{U}_{YV}+{\tilde{u}}_1\eta {V_y}^2{U}_{VV}+{\Omega }_2\left(X,Y,V,U_X,U_Y,U_V,\dots \dots \right)=0,$$

(16)

where ${\Omega }_2\left(X,Y,V,U_X,U_Y,U_V,\dots \dots \right)$ has nothing to do with $U_{YV}$ and $U_{VV}.$ Then,

$$V_y=0.$$

(17)

Taking the above result into (16), we will get

$$\begin{array}{l}{\tilde{u}}_{1x}{V}_t{U}_V+(2X_x{\tilde{u}}_{1x}+3{\tilde{u}}_1{X}_{xx}){X_x}^2(a^4-6a^2{b}^2+b^4)U_{XXX}-X_x{\tilde{u}}_{1}a\left({X_x}^3-V_t\right)U_{XV}\\ +{\Omega }_3\left(X,Y,V,U_X,U_Y,\dots \dots \right)=0,\end{array}$$

(18)

where ${\Omega }_3\left(X,Y,V,U_X,U_Y,\dots \dots \right)$ has nothing to do with $U_V,U_{XV}$ and $U_{XXX}.$ It can be derived that

$$V=r(t),X={r_t}^{\frac{1}{3}}x+f_1(y,t),{\tilde{u}}_1=\frac{f_2(y,t)}{\sqrt{r_t}},$$

(19)

where $r(t),f_1(y,t)$, and $f_2(y,t)$ are arbitrary differentiable functions. After completing the above calculation step by step, we have

$$\begin{array}{l}{f}_2\left(y,t\right)\eta \left({Y_y}^2{r_t}^{-\frac{1}{2}}-{r_t}^{\frac{5}{6}}\right)=0,\\ \frac{3(a^2-b^2)f_2(y,t)}{2{r_t}^{\frac{13}{3}}}\left(f_2(y,t){r_t}^4-{r_t}^{\frac{31}{6}}\right)=0,\end{array}$$

$$-\frac{1}{6{r_t}^{\frac{5}{6}}}\left(\alpha f_2(y,t){r_t}^{\frac{5}{3}}-6a{f}_{1t}{r_t}^{\frac{2}{3}}-6\beta f_{1t}{r_t}^{\frac{2}{3}}-6\eta {f_{1y}}^2{r_t}^{\frac{1}{3}}-9r_t{\tilde{u}}_0{a}^2+9{\tilde{u}}_0{b}^2{r}_t-2ax{r}_{tt}-6\alpha r_t\right)=0,$$

$$\frac{f_2(y,t)}{{r_t}^{\frac{7}{6}}}\left(\frac{2}{3}ay{r_t}^{\frac{2}{3}}{r}_{tt}+a{r}_t{f}_{3t}(t)-\beta {r_t}^2+2\eta {r_t}^{\frac{4}{3}}f1y(y,t)+\beta {r_t}^{\frac{5}{3}}\right)=0.$$

From the above system, it can be derived that

$$\begin{array}{l}{\tilde{u}}_0=-\frac{1}{54\eta (a^2-b^2){r_t}^{\frac{8}{3}}}\left(12a\eta x{r}_{tt}{r_t}^{\frac{5}{3}}-6a\beta y{r}_{tt}{r_t}^{\frac{5}{3}}+18a^{2}y{\psi }_t{r}_{tt}{r}_t-18a^{2}y{\psi }_{tt}{r_t}^2\right.\\ +8a^2{y}^2{r_{tt}}^2{r_t}^{\frac{2}{3}}-6a^2{y}^2{r}_{ttt}{r_t}^{\frac{5}{3}}+36a\eta {\tau }_t{r_t}^{\frac{7}{3}}-9{\beta }^2{r_t}^{\frac{8}{3}}+9{\beta }^2{r_t}^{\frac{10}{3}}-18a\beta {\psi }_t{r_t}^{\frac{7}{3}}\\ \left.+9a^2{{\psi }_t}^2{r_t}^{\frac{4}{3}}+36\eta \alpha {r_t}^{\frac{8}{3}}-36{r_t}^{\frac{10}{3}}\alpha \eta \right),\\ {\tilde{u}}_1={r_t}^{\frac{2}{3}},\\ X=\frac{6\eta x{r}_t+3\beta y{r_t}^{\frac{4}{3}}-3\beta y{r}_t-3ay{\psi }_t{r_t}^{\frac{1}{3}}-a{y}^2{r}_{tt}+6\tau \eta {r_t}^{\frac{2}{3}}}{6\eta {r_t}^{\frac{2}{3}}},Y={r_t}^{\frac{2}{3}}y+\psi ,V=r,\end{array}$$

(20)

where $r,\Psi$, and $\tau$ are all arbitrary functions of $t$.

With the help of (20), we obtain the finite symmetry transformation group as follows:

$$\begin{array}{l}u=-\frac{1}{54\eta {r_t}^{\frac{11}{3}}(a^2-b^2)}\left(8a^2{r_{tt}}^2{y}^2{r_t}^{\frac{3}{5}}-36\alpha \beta {r_t}^{\frac{11}{3}}+9{\beta }^2{r_t}^{\frac{11}{3}}-6a^2{r}_{ttt}{y}^2{r_t}^{\frac{8}{3}}-6a\beta y{r}_{tt}{r_t}^{\frac{8}{3}}\right.\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-12a\eta r_{tt}x{r_t}^{\frac{8}{3}}-18a\beta {\psi }_t{r_t}^{\frac{10}{3}}+36\alpha \eta {r_t}^{\frac{11}{3}}-9{\beta }^2{r_t}^{\frac{11}{3}}+36a\eta {\tau }_t{r_t}^{\frac{10}{3}}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left.-18a^2{\psi }_{tt}y{r_t}^3+18a^2{\psi }_{tt}y{r}_{tt}{{\mathrm{r}}_t}^2+9a^2{{\psi }_t}^2{r_t}^{\frac{7}{3}}\right)+{r_t}^{\frac{2}{3}}U(X,Y,V),\end{array}$$

(21)

From the above analysis, we can draw the following conclusion:

Theorem 1.

Suppose that $U\left(x,y,t\right)$ is an exact solution of (8), then $u$ expressed by (21) is also a solution of (8) with $X,Y$, and $V$ being determined by (20). (21) is also a Bäcklund transformation of (8).

4.2. The Lie Point Symmetry

The transformation group represented by Theorem 1 is the non-Lie transformation group, and it includes the classical Lie symmetry as a special case. To build the connection between Lie symmetry and Theorem 1, we introduce

$$r(t)=t+\epsilon f(t),\psi (t)=\epsilon g(t),\tau (t)=\epsilon h(t),$$

(22)

with $f,g$, and $h$ being arbitrary functions of $t$. So (21) will be reduced to

$$u=U+\epsilon \sigma \left(U\right),$$

where $\sigma \left(U\right)$ is the Lie symmetry and

$$\begin{array}{l}\sigma (U)=\left(\frac{f^{\prime }(t)\beta y-f^{″}(t)a{y}^2}{6\eta }+\frac{x{f}^{\prime }(t)}{3}-\frac{g^{\prime }(t)ay}{2\eta }+h(t)\right)U_x+\left(\frac{2}{3}{f}^{\prime }(t)y+g(t)\right)U_y+f(t)U_t\\ +\frac{-f^{‴}(t)a^2{y}^2}{9\eta (a^2-b^2)}+\frac{a(2\eta x-\beta y)f^{″}(t)}{9\eta (a^2-b^2)}-\frac{a^{2}y{g}^{″}(t)}{3\eta (a^2-b^2)}+\frac{(6b^{2}U-4\alpha )\eta +{\beta }^2-6a^{2}U\eta }{9\eta (a^2-b^2)}{f}^{\prime }(t)\\ -\frac{a\beta g^{\prime }(t)}{3\eta (a^2-b^2)}+\frac{2a{h}^{\prime }(t)}{3(a^2-b^2)}.\end{array}$$

(23)

The vector field of the above symmetry can be expressed as

$$\begin{array}{l}v=\left(\frac{f^{\prime }(t)\beta y-f^{″}(t)a{y}^2}{6\eta }+\frac{x{f}^{\prime }(t)}{3}-\frac{g^{\prime }(t)ay}{2\eta }+h(t)\right)\frac{\partial }{\partial x}+\left(\frac{2}{3}{f}^{\prime }(t)y+g(t)\right)\frac{\partial }{\partial y}+f(t)\frac{\partial }{\partial t}\\ -\left(\frac{-f^{‴}(t)a^2{y}^2}{9\eta (a^2-b^2)}+\frac{a(2\eta x-\beta y)f^{″}(t)}{9\eta (a^2-b^2)}-\frac{a^{2}y{g}^{″}(t)}{3\eta (a^2-b^2)}+\frac{(6b^{2}U-4\alpha )\eta +{\beta }^2-6a^{2}U\eta }{9\eta (a^2-b^2)}{f}^{\prime }(t)\right.\\ \left.-\frac{a\beta g^{\prime }(t)}{3\eta (a^2-b^2)}+\frac{2a{h}^{\prime }(t)}{3(a^2-b^2)}\right)\frac{\partial }{\partial U}.\end{array}$$

(24)

The above vector field can also be obtained by the standard Lie group method [34].

5. New Interaction Patterns

In Section 4, we have given a Bäcklund transformation of (8), which is expressed by Theorem 1. Utilizing the Bäcklund transformation, new interaction patterns can be derived from the known solutions.

5.1. New Interaction Patterns Based on the Dark Soliton Solution

In Section 3, we have obtained a dark soliton solution $u_8$ for (8). Taking the dark soliton solution $u_8$ as a seed solution, a new solution for (8) can be obtained from the Bäcklund transformation as follows:

$$\begin{array}{l}{u}_{18}=-\frac{1}{54\eta {r_t}^{\frac{11}{3}}(a^2-b^2)}\left(8a^2{r_{tt}}^2{y}^2{r_t}^{\frac{3}{5}}-36\alpha \beta {r_t}^{\frac{11}{3}}+9{\beta }^2{r_t}^{\frac{11}{3}}-6a^2{r}_{ttt}{y}^2{r_t}^{\frac{8}{3}}-6a\beta y{r}_{tt}{r_t}^{\frac{8}{3}}-12a\eta r_{tt}x{r_t}^{\frac{8}{3}}\right.\\ \left.-18a\beta {\psi }_t{r_t}^{\frac{10}{3}}+36\alpha \eta {r_t}^{\frac{11}{3}}-9{\beta }^2{r_t}^{\frac{11}{3}}+36a\eta {\tau }_t{r_t}^{\frac{10}{3}}-18a^2{\psi }_{tt}y{r_t}^3+18a^2{\psi }_{tt}y{r}_{tt}{{\mathrm{r}}_t}^2+9a^2{{\psi }_t}^2{r_t}^{\frac{7}{3}}\right)+\\ {r_t}^{\frac{2}{3}}\left(\frac{-{k_1}^4(a^4-6a^2{b}^2+b^4)-2k_1(a{k}_3+\alpha k_1+\beta k_2)-2\eta {k_2}^2}{3{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\tanh }^2(W)}{2(a^2-b^2)}\right),\end{array}$$

(25)

where

$$W=k_0+k_1\left(\frac{6\eta x{r}_t+3\beta y{r_t}^{\frac{4}{3}}-3\beta y{r}_t-3ay{\psi }_t{r_t}^{\frac{1}{3}}-a{y}^2{r}_{tt}+6\tau \eta {r_t}^{\frac{2}{3}}}{6\eta {r_t}^{\frac{2}{3}}}\right)+k_2\left({r_t}^{\frac{2}{3}}y+\psi \right)+k_{3}r,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}$$

with $r,\Psi$, and $\tau$ being arbitrary functions of $t$. Since there are three arbitrary functions $r,\Psi$, and $\tau$, (25) includes many interaction patterns. In the following, we illustrate several interesting patterns.

5.1.1. Fractional Soliton Solution

When $r=t,\Psi =0,\tau =0,$ the above solution is precisely the same with $u_8.$ If we substitute $t=\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }$ into $u_8$, a fractional soliton solution for the truncated M-fractional KP Equation (1) can be obtained as follows:

$$u_8=\frac{-{k_1}^4(a^4-6a^2{b}^2+b^4)-2k_1(a{k}_3+\alpha k_1+\beta k_2)-2\eta {k_2}^2}{3{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\tanh }^2(W)}{2(a^2-b^2)},$$

(26)

where $W=k_0+k_{1}x+k_{2}y+k_3\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta },$ $k_0,k_1,k_2$, and $k_3$ are arbitrary constants.

5.1.2. Interaction Solution Between the Exponential Function and the Dark Soliton

When $r=e^t,\Psi =0,\tau =0,$ the solution (25) becomes

$$\begin{array}{l}{u}_{19}=-\frac{1}{54\eta (a^2-b^2)}(36\alpha \eta e^{\frac{2}{3}t}-9{\beta }^2{e}^{\frac{2}{3}t}-2a^2{y}^2-12a\eta x+6a\beta y-36\alpha \eta +9{\beta }^2)\\ +e^{\frac{2}{3}t}\left(\frac{-{k_1}^4(a^4-6a^2{b}^2+b^4)-2k_1(a{k}_3+\alpha k_1+\beta k_2)-2\eta {k_2}^2}{3{k_1}^2(a^2-b^2)}\right.\\ \left.+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\tanh }^2(W)}{2(a^2-b^2)}\right),\end{array}$$

(27)

where

$$W=k_0+k_1\left(\frac{-a{e}^{\frac{t}{3}}{y}^2+3\beta y{e}^{\frac{2t}{3}}+6\eta e^{\frac{t}{3}}x-3\beta e^{\frac{t}{3}}y}{6\eta }\right)+k_2\left(e^{\frac{2t}{3}}y\right)+k_3{e}^t.$$

This is a new interaction solution between the exponential function and the dark soliton. If we substitute $t=\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }$ into (27), a new interaction solution for the truncated M-fractional KP Equation (1) can be obtained as follows:

$$\begin{array}{l}{u}_{19}=-\frac{1}{54\eta (a^2-b^2)}\left(36\alpha \eta e^{\frac{2}{3}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}-9{\beta }^2{e}^{\frac{2}{3}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}-2a^2{y}^2-12a\eta x+6a\beta y-36\alpha \eta +9{\beta }^2\right)\\ +e^{\frac{2}{3}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}\left(\frac{-{k_1}^4(a^4-6a^2{b}^2+b^4)-2k_1(a{k}_3+\alpha k_1+\beta k_2)-2\eta {k_2}^2}{3{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\tanh }^2(W)}{2(a^2-b^2)}\right),\end{array}$$

(28)

where

$$W=k_0+k_1\left(\frac{\left(-a{y}^2+6\eta x-3\beta y\right)e^{\frac{1}{3}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}+3\beta y{e}^{\frac{2}{3}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}}{6\eta }\right)+k_2\left(e^{\frac{2}{3}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}y\right)+k_3{e}^{\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}$$

The above solution (28) is the composition of the soliton and the exponential function. When $k_1=k_2=k_3=1,k_0=0,a=6,b=2,\alpha =1,\beta =1,\eta =2,$ graphs of (28) can be obtained. From Figure 1, we can see that as the fractional derivative $\theta$ decreases, the geometric appearance of (28) has hardly changed, but the position of the solution moves more and more in the negative direction of the x-axis.

5.1.3. Interaction Solution Between the Trigonometric Sine Function and the Dark Soliton

When $r=\sin (t),\Psi =0,\tau =0,$ the solution (25) becomes

$$\begin{array}{l}{u}_{20}=-\frac{1}{54\eta (a^2-b^2){\cos }^2(t)}\left(12a\eta x\sin (t)\cos (t)-6a\beta y\sin (t)\cos (t)-8a^2{y}^2{\sin }^2(t)\right.\\ \left.-9{\beta }^2{\cos }^{\frac{8}{3}}(t)+9{\beta }^2{\cos }^2(t)-36\alpha \eta {\cos }^2(t)+36\alpha \eta {\cos }^{\frac{8}{3}}(t)-6a^2{y}^2{\cos }^2(t)\right)+{\left(\cos (t)\right)}^{\frac{2}{3}}\times \\ \left(\frac{-{k_1}^4(a^4-6a^2{b}^2+b^4)-2k_1(a{k}_3+\alpha k_1+\beta k_2)-2\eta {k_2}^2}{3{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\tanh }^2(W)}{2(a^2-b^2)}\right),\end{array}$$

(29)

where

$$W=k_0+k_1\left(\frac{3\beta y{\cos }^{\frac{4}{3}}(t)+6\eta x\cos (t)+a{y}^2\sin (t)-3\beta y\cos (t)}{6\eta {\cos }^{\frac{2}{3}}(t)}\right)+k_2\left({\cos }^{\frac{2}{3}}(t)y\right)+k_3\sin (t).$$

This is a new interaction solution between the trigonometric sine function and the dark soliton.

If we substitute $t=\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }$ into (29), a new interaction solution for the truncated M-fractional KP Equation (1) can be obtained as follows:

$$\begin{array}{l}{u}_{20}=-\frac{1}{54\eta (a^2-b^2){\cos }^2\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}\left(\left(12a\eta x-6a\beta y\right)\sin \left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)\cos \left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)\right.\\ \left.-8a^2{y}^2{\sin }^2\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)+\left(9{\beta }^2-6a^2{y}^2-36\alpha \eta \right){\cos }^2\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)+\left(-9{\beta }^2+36\alpha \eta \right){\cos }^{\frac{8}{3}}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)\right)\\ +{\cos }^{\frac{2}{3}}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)\left(\frac{-{k_1}^4(a^4-6a^2{b}^2+b^4)-2k_1(a{k}_3+\alpha k_1+\beta k_2)-2\eta {k_2}^2}{3{k_1}^2(a^2-b^2)}+\frac{{k_1}^2(a^4-6a^2{b}^2+b^4){\tanh }^2(W)}{2(a^2-b^2)}\right),\end{array}$$

(30)

where

$$\begin{array}{l}W=k_0+k_1\left(\frac{3\beta y{\cos }^{\frac{4}{3}}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)+(6\eta x-3\beta y)\cos \left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)+a{y}^2\sin \left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}{6\eta {\cos }^{\frac{2}{3}}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}\right)\\ +k_2\left({\cos }^{\frac{2}{3}}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)y\right)+k_3\sin \left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right).\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}$$

The above solution (30) is the composition of the soliton and the trigonometric function. When $k_1=k_2=k_3=1,k_0=0,a=6,b=2,\alpha =1,\beta =1,\eta =2,$ graphs of (30) can be obtained. From Figure 2, we can observe the influence of the trigonometricsine function on the dark soliton. As the fractional derivative $\theta$ decreases, the position of the solution moves more and more in the negative direction of the x-axis.

5.2. New Interaction Patterns Based on the Lump Solution

The lump solution, which has crucial physical significance and wide applications, is a solution of a special type. A lump solution for (8) has been obtained in [25], and the expression is

$$u_{21}=-\frac{a^4-6a^2{b}^2+b^4}{2(a^2-b^2)}\frac{2({a_1}^2+{a_5}^2)f-4{(a_{1}g+a_{5}h)}^2}{f^2},$$

(31)

where

$$\begin{array}{l}g=\frac{a_6\sqrt{\alpha \eta }}{\alpha }x-\frac{a{a}_3}{\alpha }y+a_{3}t+\frac{a_6{a}_8\alpha }{a{a}_3},h=\frac{a{a}_3\sqrt{\alpha \eta }}{{\alpha }^2}x+a_{6}y-\frac{a_6\alpha }{a}t+a_8,f=g^2+h^2+a_9,\\ a_9=\frac{2\eta \left({\alpha }^2{a_6}^2(a^4-6a^2{b}^2+b^4)\left({\alpha }^2{a_6}^2+a^2{a_3}^2\right)\right)}{16{\alpha }^4},\end{array}$$

(32)

with $a_3,a_6$, and $a_8$ being arbitrary real parameters.

When $a=6,b=2,\eta =2,\alpha =\beta =1,$ the following lump solution of (8) is

$$u_{22}=1008\times \frac{24\sqrt{2}x-t^2+12yt+72x^2-36y^2-6044}{{(24\sqrt{2}x+t^2-12yt+72x^2+36y^2+6052)}^2}.$$

(33)

5.2.1. Fractional Lump Solution

If we substitute $t=\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }$ into (33), a fractional lump solution for the truncated M-fractional KP Equation (1) is as follows:

$$u=1008\times \frac{24\sqrt{2}x-{\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}^2+12y\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)+72x^2-36y^2-6044}{(24\sqrt{2}x+{\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)}^2-12y\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)+72x^2+36y^2+6052{)}^2}.$$

(34)

When $\theta$ and $\delta$ take different values, graphs of (34) can be obtained. From Figure 3 and Figure 4, we find that the fractional derivative $\theta$ has a small impact on the shape and width of the fractional lump solution (34). When $\theta$ is fixed, the position of the fractional lump moves more and more in the positive direction of the y-axis as $\delta$ increases.

From Figure 1, Figure 2, Figure 3 and Figure 4, we find that two parameter values $\theta$ and $\delta$ in the truncated M-fractional derivative both have an impact on the position of the solutions. When fixing one of them, the shape and width of the solutions have remained almost unchanged and the position will change.

5.2.2. Interaction Solution Between Trigonometric Sine Function and the Lump Soliton

If we take the lump solution $u_{21}$ in (31) as a seed solution, the following new solution for (8) can be obtained from Theorem 1.

$$\begin{array}{l}{u}_{23}=-\frac{1}{54\eta {r_t}^{\frac{11}{3}}(a^2-b^2)}\left(8a^2{r_{tt}}^2{y}^2{r_t}^{\frac{3}{5}}-36\alpha \beta {r_t}^{\frac{11}{3}}+9{\beta }^2{r_t}^{\frac{11}{3}}-6a^2{r}_{ttt}{y}^2{r_t}^{\frac{8}{3}}-6a\beta y{r}_{tt}{r_t}^{\frac{8}{3}}-12a\eta r_{tt}x{r_t}^{\frac{8}{3}}\right.\\ \hspace{0.17em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left.-18a\beta {\psi }_t{r_t}^{\frac{10}{3}}+36\alpha \eta {r_t}^{\frac{11}{3}}-9{\beta }^2{r_t}^{\frac{11}{3}}+36a\eta {\tau }_t{r_t}^{\frac{10}{3}}-18a^2{\psi }_{tt}y{r_t}^3+18a^2{\psi }_{tt}y{r}_{tt}{{\mathrm{r}}_t}^2+9a^2{{\psi }_t}^2{r_t}^{\frac{7}{3}}\right)\\ \hspace{0.17em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+{r_t}^{\frac{2}{3}}\left(-\frac{a^4-6a^2{b}^2+b^4}{2(a^2-b^2)}\frac{2({a_1}^2+{a_5}^2)f-4{(a_{1}g+a_{5}h)}^2}{f^2}\right),\\ \end{array}$$

(35)

where

$$g=\frac{a_6\sqrt{\alpha \eta }}{\alpha }X-\frac{a{a}_3}{\alpha }Y+a_{3}r+\frac{a_6{a}_8\alpha }{a{a}_3},h=\frac{a{a}_3\sqrt{\alpha \eta }}{{\alpha }^2}X+a_{6}Y-\frac{a_6\alpha }{a}r+a_8,f=g^2+h^2+a_9,$$

with $a_9$ being determined by (32),$X$ and $Y$ are expressed by (20).

When $r=\sin (t),\Psi =0,\tau =0,$ the above solution (35) becomes

$$\begin{array}{l}{u}_{24}=-\frac{1}{54\eta (a^2-b^2){\cos }^2(t)}\left(12a\eta x\sin (t)\cos (t)-6a\beta y\sin (t)\cos (t)-9{\beta }^2{\cos }^{\frac{8}{3}}(t)\right.\\ \left.-8a^2{y}^2{\sin }^2(t)+9{\beta }^2{\cos }^2(t)-36\alpha \eta {\cos }^2(t)+36\alpha \eta {\cos }^{\frac{8}{3}}(t)-6a^2{y}^2{\cos }^2(t)\right)\\ +{\left(\cos (t)\right)}^{\frac{2}{3}}\left(-\frac{a^4-6a^2{b}^2+b^4}{2(a^2-b^2)}\frac{2({a_1}^2+{a_5}^2)f-4{(a_{1}g+a_{5}h)}^2}{f^2}\right),\end{array}$$

(36)

where

$$g=\frac{a_6\sqrt{\alpha \eta }}{\alpha }X-\frac{a{a}_3}{\alpha }Y+a_{3}r+\frac{a_6{a}_8\alpha }{a{a}_3},h=\frac{a{a}_3\sqrt{\alpha \eta }}{{\alpha }^2}X+a_{6}Y-\frac{a_6\alpha }{a}r+a_8,f=g^2+h^2+a_9,$$

(37)

$$X=\frac{3\beta y{\cos }^{\frac{4}{3}}\left(t\right)+(6\eta x-3\beta y)\cos \left(t\right)+a{y}^2\sin \left(t\right)}{6\eta {\cos }^{\frac{2}{3}}\left(\frac{\Gamma (1+\delta )}{\theta }{T}^{\theta }\right)},Y={\cos }^{\frac{2}{3}}\left(t\right)y,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}$$

(38)

with $a_9=\frac{2\eta \left({\alpha }^2{a_6}^2(a^4-6a^2{b}^2+b^4)\left({\alpha }^2{a_6}^2+a^2{a_3}^2\right)\right)}{16{\alpha }^4},$ $a_3,a_6$, and $a_8$ being arbitrary real parameters. This is a new interaction pattern between the trigonometric sine function and the lump soliton.

Remark 2.

In the above analysis, we obtain several uncommon interaction patterns from old solutions $u_8$ and $u_{21}.$ From (25) and (35), more interaction patterns can be obtained if the three arbitrary functions $r,\mathrm{\Psi }$ , and $\tau$ become other specific functions. If we take other known solutions as seed solutions, other interaction patterns can be obtained by Theorem 1. For example, taking the breather wave solution in [32,33] as an old solution, new interaction solutions with the breather wave solution can be obtained.

6. Conclusions

Recently, there has been a continuous interest in water surface waves, which can be analogs of optical and quantum wave systems. In this paper, the truncated M-fractional KP equation, which can model complex wave propagations, has been studied. Interaction patterns among elliptic functions and hyperbolic functions or trigonometric functions have been constructed by the consistent Riccati expansion method. Solutions of this type are difficult to derive using other conventional methods. Up to now, there have been many analytical methods, including the exp-expansion method, the (G′/G)-expansion method, the (G′/G²)-expansion method, the elliptic function expansion method, the unified and generalized Bernoulli sub-ODE techniques, and so on, which can derive various exact solutions of nonlinear systems. Compared with these methods, the consistent Riccati expansion method can construct solutions which are composite functions of two or more functions. For example, the solution $u_{17},$ which is a composition of three types of functions: hyperbolic functions, inverse trigonometric tangent functions, and elliptic functions. However, solutions derived by other methods are single functions or the four arithmetic operations of one or two types of functions. For example, a solution in [34] has been obtained by the elliptic function expansion method and it has the form

$$u_7(x,y,t)=L_0+\frac{\sinh {\left((-t\omega +c_1)\sqrt{{\alpha }_1}\right)}^2{L}_2{\alpha }_0}{{\alpha }_1}+\frac{L_1\sqrt{{\alpha }_0}\tanh \left((-t\omega +c_1)\sqrt{{\alpha }_1}\right)}{\mathrm{sech}\left((-t\omega +c_1)\sqrt{{\alpha }_1}\right){\alpha }_1}.$$

(39)

The solution belongs to the four arithmetic operations of the same type of functions and does not belong to composite functions of two or more functions of different types. So, the advantage of the consistent Riccati expansion method lies in that it can derive solutions that are composite functions of two or more functions of different types.

Applying the simple direct method, the finite symmetry transformation group and Bäcklund transformation have been obtained. Based on Theorem 1, Lie point symmetries have been derived and they are exactly the same as that in [34]. Based on the Bäcklund transformation, new interaction patterns can be obtained from old seed solutions. In summary, the fractional soliton solution, fractional lump solution, and several interesting interaction patterns among the trigonometric sine function, exponential functions, dark soliton, and the lump soliton have been constructed. Some solutions have also been illustrated with several representative graphs.

Using the analytical techniques of the (G′/G)-expansion and modified Kudryashov methods, many exact solutions for the fractional form of (2) with the conformable derivative have been obtained in [35]. After careful analysis and comparison, we find that the solutions in [35] are single functions or the four arithmetic operations of the same type of functions. They are also not composite functions of two or more functions of different types.

This article has two new contributions. The first one is that we obtain the fractional soliton solution (26), the fractional lump solution (34), solutions $u_{15}$ and $u_{16}$ (which are a composition of the cnoidal function and soliton function), solution $u_{17}$ (which is a composition of the cnoidal function and trigonometric solution), solution (28) (which isa composition of the dark and exponential function), the trigonometric sine function and the dark soliton interaction solution (30), and the trigonometric sine function and the lump interaction solution (36) for the first time. More importantly, we find that the consistent Riccati expansion method and the simple direct method are effective approaches to obtaining new interaction patterns which are composite functions of two or more functions of different types.

The methods used in this paper will be applied to more high-dimensional nonlinear systems in the future. For example, the (3+1)-dimensional form of Equation (1) and (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili equation with fractional derivatives.