1 Introduction

Fluid mechanics deals with the underlying mechanisms of liquids, gases or plasmas, and the forces on them [1,2,3,4,5,6,7,8]. It has the applications in a wide range of disciplines, such as oceanography, astrophysics, meteorology, and biomedical engineering [9,10,11,12,13,14,15,16,17]. For the insight into the fluid mechanics problems, people have focused their attention on the analytic solutions of the nonlinear evolution equations (NLEEs) to describe the nonlinear waves [18,19,20,21,22,23,24,25,26,27]. For example, soliton solutions have been derived for the (\(2+1\))-dimensional Korteweg–de Vries (KdV) equation [28, 29], lump solutions have been obtained for the extended Kadomtsev–Petviashvili (KP) equation [32, 33], rogue wave solutions have been constructed for the B-type KP equation [34,35,36,37], and periodic wave solutions have been studied for the (\(2+1\))-dimensional extended shallow water wave equation [38]. Methods for deriving the analytic solutions of the NLEEs including the inverse scattering transform, Pfaffian technique, Lie symmetry method and Hirota–Riemann method have been proposed [39,40,41,42,43,44,45,46]. Among them, the Pfaffian technique has been used to construct the soliton solutions and the Hirota–Riemann method has been utilized to derive the periodic wave solutions of the NLEEs [47,48,49,50,51,52].

Ref. [53] has considered the (\(2+1\))-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada (gCDGKS) equation,

$$\begin{aligned}&36u_t+\left( u_{xxxx}+15uu_{xx}+15u^3\right) _x-\alpha \partial _x^{-1} u_{yy}\nonumber \\&\quad -\gamma \left( u_{xxy}+3uu_y+3u_x\partial _x^{-1} u_{y}\right) =0, \end{aligned}$$
(1)

where \(u = u(x, y, t)\) is the differentiable function with respect to the variables x, y and t, \(\alpha \) and \(\gamma \) are the real constants, the subscripts represent the partial derivatives, and \(\partial _x^{-1}\) represents the integral with respect to x. Soliton solutions for Eq. (1) have been constructed via the Hirota bilinear method, and lump solutions for Eq. (1) have been derived via the symbolic computation [53]. In fluid mechanics, special cases for Eq. (1) are given as follows:

  • When \(\alpha =\gamma =5\), Eq. (1) has been reduced to the (\(2+1\))-dimensional fifth-order KdV equation in fluid mechanics [54,55,56],

    $$\begin{aligned}&36u_t+\left( u_{xxxx}+15uu_{xx}+15u^3\right) _x-5 \partial _x^{-1} u_{yy}\nonumber \\&\quad -5 \left( u_{xxy}+3uu_y+3u_x\partial _x^{-1} u_{y}\right) =0. \end{aligned}$$
    (2)

    Periodic solitary wave solutions for Eq. (2) have been constructed via the Hirota bilinear method [54]. Quasi-periodic solutions for Eq. (2) have been derived in terms of the Riemann theta functions [55]. Lump-type and rogue wave solutions for Eq. (2) have been obtained via the symbolic computation [56].

  • When \(\alpha =\gamma =5, t=36T^{'}\) and \(u_y=0\), Eq. (1) has been reduced to the Sawada–Kotera equation for the long waves in shallow water under the gravity [57,58,59,60,61],

    $$\begin{aligned}&u_{T^{'}}+u_{xxxxx}+15u_x u_{xx}+15uu_{xxx}\nonumber \\&\quad +45u^2u_x=0. \end{aligned}$$
    (3)

    Eq. (3) has also been seen in lattice dynamics, quantum mechanics and nonlinear optics [58]. Soliton solutions for Eq. (3) have been constructed via the Hirota bilinear method [59]. Periodic and rational solutions for Eq. (3) have been constructed via the (\(G^{'}\)/G)-expansion method [60]. Traveling waves with different frequencies and velocities for Eq. (3) have been constructed via the three wave method [61].

Through the dependent transformation [53],

$$\begin{aligned} u=2\left( \ln f\right) _{xx}, \end{aligned}$$
(4)

where f is a real function of x, y and t, Eq. (1) has been written as the bilinear form [53],

$$\begin{aligned} \left( 36D_xD_t+D_x^6-\alpha D_y^2-\gamma D_x^3D_y\right) f\cdot f=0, \end{aligned}$$
(5)

where the bilinear operators \(D_x, D_y\) and \(D_t\) are defined by [62]

$$\begin{aligned}&D_x^lD_y^mD_t^n\theta (x, y, t)\cdot \vartheta (x^{'}, y^{'}, t^{'})\nonumber \\&\quad \equiv \left( \frac{\partial }{\partial x}-\frac{\partial }{\partial {x^{'}}}\right) ^l\left( \frac{\partial }{\partial y}-\frac{\partial }{\partial {y^{'}}}\right) ^m\left( \frac{\partial }{\partial t}-\frac{\partial }{\partial {t^{'}}}\right) ^n\nonumber \\&\quad ~~~~~\theta (x, y, t)\vartheta (x^{'}, y^{'}, t^{'})|_{x^{'}=x,~y^{'}=y,~t^{'}=t}, \end{aligned}$$
(6)

with \(\theta (x, y, t)\) being a differentiable function of x, y and t, \(\vartheta (x^{'}, y^{'}, t^{'})\) being a differentiable function of the independent variables \(x^{'}, y^{'}\) and \(t^{'}\), and l, m and n being the non-negative integers.

On the other hand, the Nth-order Pfaffian, i.e.,

\((1, 2, \ldots , 2N)\), has the following expansion [62]:

$$\begin{aligned}&(1, 2, \ldots , 2N)\nonumber \\&\qquad =\sum \limits _{j=2}^{2N}(-1)^{j}(1, j)\quad (2, 3, \ldots , \hat{j}, \ldots , 2N), \end{aligned}$$
(7)

where \(\hat{j}\) means that the element j is omitted, \((2, 3, \ldots , \hat{j}, \ldots , 2N)\) is the \((N-1)\)th-order Pfaffian, (r, j) is the antisymmetric element of the Pfaffian and defined as

$$\begin{aligned} (r, j)=c_{rj}+\int ^x D_x\phi _{r}\cdot \phi _{j}\mathrm{d}x, \end{aligned}$$
(8)

r, j and N are the positive integers, \(\phi _{r}\)’s and \(\phi _{j}\)’s are the real functions of x, y and t, and \(c_{rj}\) is a constant satisfying the condition \(c_{rj}=-c_{jr}\). Pfaffian has been said to possess the following properties [62]:

$$\begin{aligned}&(\alpha _{1},\alpha _{2},\ldots ,\alpha _{2N},1,2,\ldots ,2N)(1,2,\ldots ,2N)\nonumber \\&\quad =\sum \limits _{j=2}^{2N}(-1)^{j}(\alpha _{1},\alpha _{j},1,2,\ldots ,2N)\nonumber \\&\quad (\alpha _{2},\alpha _{3},\ldots ,\hat{\alpha }_{j},\ldots ,\alpha _{2N},1,2,\ldots ,2N), \end{aligned}$$
(9)

where \(\alpha _{j}\)’s are the real numbers, and \(\hat{\alpha }_{j}\) means that the element \(\alpha _{j}\) is omitted.

$$\begin{aligned}&(d_{0},d_{1},d_{2},d_{3},\bullet )(\bullet )-(d_{0},d_{1},\bullet )(d_{2},d_{3},\bullet )\nonumber \\&+\,(d_{0},d_{2},\bullet )(d_{1},d_{3},\bullet )-(d_{0},d_{3},\bullet )(d_{1},d_{2},\bullet )=0,\nonumber \\&(d_{n},r)=\frac{\partial ^{n}\phi _{r}}{\partial x^{n}},~(d_{m},d_{n})=0,\nonumber \\&(m, n=0, 1, 2,\ldots , 2N-1), \end{aligned}$$
(10)

where \((\bullet )=(1, 2, \ldots , 2N)\).

However, to our knowledge, soliton solutions via the Pfaffian technique and periodic wave solutions via the Hirota–Riemann method for Eq. (1) have not been investigated. In Sect. 2, the Nth-order Pfaffian solutions for Eq. (1) will be constructed via the Pfaffian technique, and soliton solutions for Eq. (1) will be derived via the Nth-order Pfaffian solutions. In Sect. 3, periodic wave solutions for Eq. (1) will be obtained via the Hirota–Riemann method, and asymptotic behaviors of the periodic wave solutions will be given. In Sect. 4, our conclusions will be presented.

2 Pfaffian solutions for Eq. (1)

In this section, we would like to construct the Pfaffian solutions for Eq. (1) via the Pfaffian technique. To derive the Nth-order Pfaffian \((1, 2, \ldots , 2N)\) satisfying Bilinear Form (5), we can set the differentiable functions \(\phi _{r}\)’s and \(\phi _{j}\)’s in Eq. (8) satisfying the following conditions:

$$\begin{aligned} \frac{\partial \phi _r}{\partial y}= & {} \frac{5}{\gamma }\frac{\partial ^3\phi _r}{\partial x^3},~~ \frac{\partial \phi _r}{\partial t}=\frac{1}{4}\frac{\partial ^5\phi _r}{\partial x^5},~~ \frac{\partial \phi _j}{\partial y}=\frac{5}{\gamma }\frac{\partial ^3\phi _j}{\partial x^3},\nonumber \\ \frac{\partial \phi _j}{\partial t}= & {} \frac{1}{4}\frac{\partial ^5\phi _j}{\partial x^5},~~\alpha =\frac{\gamma ^2}{5}, \end{aligned}$$
(11)

then we have

$$\begin{aligned} \frac{\partial (r,j)}{\partial x}= & {} \frac{\partial \phi _{r}}{\partial x}\phi _{j}-\frac{\partial \phi _{j}}{\partial x}\phi _{r}\nonumber \\= & {} (d_{1},r)(d_{0},j)-(d_{0},r)(d_{1},j)\nonumber \\= & {} (d_{0},d_{1},r,j),\nonumber \\ \frac{\partial (r,j)}{\partial y}= & {} \int \left( \frac{\partial ^{2} \phi _{r}}{\partial x\partial y}\phi _{j}+\frac{\partial \phi _{r}}{\partial x}\frac{\partial \phi _{j}}{\partial y}\right. \nonumber \\&-\left. \frac{\partial ^{2} \phi _{j}}{\partial x\partial y}\phi _{r}-\frac{\partial \phi _{j}}{\partial x}\frac{\partial \phi _{r}}{\partial y}\right) \mathrm{d}x\nonumber \\= & {} \frac{5}{\gamma }[(d_{0},d_{3},r,j)-2(d_{1},d_{2},r,j)],\nonumber \\ \frac{\partial (r,j)}{\partial t}= & {} \int \left( \frac{\partial ^{2} \phi _{r}}{\partial x\partial t}\phi _{j}+\frac{\partial \phi _{r}}{\partial x}\frac{\partial \phi _{j}}{\partial t}\right. \nonumber \\&-\left. \frac{\partial ^{2} \phi _{j}}{\partial x\partial t}\phi _{r}-\frac{\partial \phi _{j}}{\partial x}\frac{\partial \phi _{r}}{\partial t}\right) \mathrm{d}x\nonumber \\= & {} \frac{1}{4}\Big [2(d_{2},d_{3},r,j)-2(d_{1},d_{4},r,j)\nonumber \\&+\,(d_{0},d_{5},r,j)\Big ]. \end{aligned}$$
(12)

According to Eqs. (12), the following differential conditions can be derived:

(13)
(14)

Combining Eqs. (9) and (10) with Eqs. (13) and (14), we obtain

$$\begin{aligned}&\left( 36D_xD_t+D_x^6-\alpha D_y^2-\gamma D_x^3D_y\right) \tau _{N}\cdot \tau _{N}\nonumber \\&\quad =\frac{2}{5}(\gamma ^2 \tau _{N,y}^2-\gamma ^2 \tau _{N} \tau _{N,yy}-15 \gamma \tau _{N,xy} \tau _{N,xx}\nonumber \\&\qquad +\,15 \gamma \tau _{N,x} \tau _{N,xxy}+5 \gamma \tau _{N,y} \tau _{N,xxx}\nonumber \\&\qquad -\,5 \gamma \tau _{N} \tau _{N,xxxy}-50 \tau _{N,xxx}^2-180 \tau _{N,t} \tau _{N,x}\nonumber \\&\qquad +\,180 \tau _{N} \tau _{N,xt}+75 \tau _{N,xx} \tau _{N,xxxx}\nonumber \\&\qquad -\,30 \tau _{N,x} \tau _{N,xxxxx}+5 \tau _{N} \tau _{N,xxxxxx})\nonumber \\&\quad =90\Big [(d_{0},d_{1},d_{2},d_{3},\bullet )(\bullet )-(d_{0},d_{1},\bullet )(d_{2},d_{3},\bullet )\nonumber \\&\qquad +(d_{0},d_{2},\bullet )(d_{1},d_{3},\bullet )-(d_{0},d_{3},\bullet )(d_{1},d_{2},\bullet )\Big ]\nonumber \\&\quad =0. \end{aligned}$$
(15)

Thus, we find that \(f=\tau _{N}\) satisfies Bilinear Form (5) and the Nth-order Pfaffian solutions for Eq. (1) can be derived as

$$\begin{aligned} u=2(\ln \tau _{N})_{xx}. \end{aligned}$$
(16)

To construct the soliton solutions for Eq. (1) via the Nth-Order Pfaffian Solutions (16), we can set \(\phi _{r}\)’s and \(\phi _{j}\)’s in Conditions (11) as

$$\begin{aligned} \phi _{r}= & {} e^{k_{r}x+\frac{5k_{r}^3}{\gamma }y+\frac{k_{r}^5}{4}t},\nonumber \\ \phi _{j}= & {} e^{k_{j}x+\frac{5k_{j}^3}{\gamma }y+\frac{k_{j}^5}{4}t}, \end{aligned}$$
(17)

where \(k_{r}\)’s and \(k_{j}\)’s are real constants. Motivated by Ref. [62], we set \(c_{12}=c_{34}=1\), \(c_{13}=c_{14}=c_{23}=c_{24}=0\), and obtain

$$\begin{aligned} (r,j)=c_{rj}+\frac{k_{r}-k_{j}}{k_{r}+k_{j}}\phi _{r}\phi _{j}. \end{aligned}$$
(18)

Hereby, when \(N=1\) and 2 in the Nth-Order Pfaffian Solutions (16), the one- and two-soliton solutions for Eq. (1) can be expressed as

$$\begin{aligned} u= & {} 2(\ln \tau _1)_{xx}, \end{aligned}$$
(19)
$$\begin{aligned} u= & {} 2(\ln \tau _2)_{xx}, \end{aligned}$$
(20)

with

$$\begin{aligned}&\tau _{1}=(1,2)=1+A_1e^{\xi _{1}+\xi _{2}},\nonumber \\&\tau _{2}=(1,2,3,4)\nonumber \\&\quad \,\, =(1,2)(3,4)-(1,3)(2,4)+(1,4)(2,3)\nonumber \\&\quad \,\,=1+A_1e^{\xi _{1}+\xi _{2}}+A_2e^{\xi _{3}+\xi _{4}}+A_{12}e^{\xi _{1}+\xi _{2}+\xi _{3}+\xi _{4}},\nonumber \\&A_1=\frac{H_{1}-H_{2}}{H_{1}+H_{2}},~~~A_2=\frac{H_{3}-H_{4}}{H_{3}+H_{4}},\nonumber \\&\xi _{\varrho }=H_\varrho x+S_\varrho y+J_\varrho t,\nonumber \\&A_{12}=\frac{(H_{1}-H_{4})(H_{2}-H_{3})}{(H_{1}+H_{4})(H_{2}+H_{3})}+\frac{(H_{1}-H_{2})(H_{3}-H_{4})}{(H_{1}+H_{2})(H_{3}+H_{4})}\nonumber \\&\qquad \quad \,\,-\frac{(H_{1}-H_{3})(H_{2}-H_{4})}{(H_{1}+H_{3})(H_{2}+H_{4})},\nonumber \\&H_\varrho =k_{\varrho },~S_\varrho =\frac{5k_{\varrho }^3}{\gamma },~J_\varrho =\frac{k_{\varrho }^5}{4},\quad (\varrho =1,2,3,4).\nonumber \\ \end{aligned}$$
(21)

Equation (19) indicates that the amplitude of the one soliton is irrelevant to \(\gamma \), the velocity along the x direction of the one soliton is independent of \(\gamma \), while the velocity along the y direction is proportional to \(\gamma \). Figure 1 shows the propagation of the one soliton, and we notice that the one soliton keeps its amplitude and velocity invariant. Figure 2 shows the interaction between the two solitons, and we find that the total amplitude of the interaction region is lower than that of any soliton.

Fig. 1
figure 1

One soliton via Solutions (19) with \(k_1=0.6,~k_2=0.4\) and \(\gamma =1.2\)

Full size image
Fig. 2
figure 2

Interaction between the two solitons via Solutions (20) with \(k_1=-0.52,~k_2=-0.5,~k_3=-0.35,~k_4=-0.24\) and \(\gamma =1.2\)

Full size image

3 Periodic wave solutions for Eq. (1)

In this section, we will utilize the Hirota–Riemann method [63] to construct the periodic wave solutions for Eq. (1).

3.1 Hirota–Riemann method for the NLEEs

Ref. [63] has considered a generalized (N+1)-dimensional NLEE:

$$\begin{aligned} \mathscr {F}\left( u, u_t, u_{x_1}, u_{x_2}, u_{x_N},\ldots \right) =0, \end{aligned}$$
(22)

where \(\mathscr {F}\) is a polynomial function and \(x_1, x_2,\ldots , x_N\) are the space variables. Using the Hirota bilinear method and the dependent variable transformation,

$$\begin{aligned} u=u_0+p \partial _{x_N}^q\ln \vartheta \left( \zeta , \lambda \right) , \end{aligned}$$
(23)

where \(\partial _{x_N}^q\) represents the \(q-th\) order partial derivatives with respect to \(x_N\), \(\vartheta \left( \zeta , \lambda \right) \) is the Riemann theta function, \(\zeta =(\zeta _1, \zeta _2, \ldots , \zeta _N)^T\) (the superscript T signifies the vector transpose), \(i\lambda = (i\lambda _{\mu \iota })\) is a positive definite and real-valued symmetric \(N \times N\) matrix. \(\zeta _\mu =Q_\mu x+B_\mu y+R_\mu t+\epsilon _\mu , (\mu ,\iota =1, 2,\ldots , N)\), p, q, N are the positive integers, and Q’s, B’s, R’s, \(\epsilon \)’s and \(u_0\) are all the real constants; Ref. [63] obtains the bilinear form for Eq. (22) as

$$\begin{aligned} \mathscr {F}\left( D_{x_1}, D_{x_2}, \ldots , D_{x_N}, D_{t}, c\right) \vartheta \left( \zeta , \lambda \right) \cdot \vartheta \left( \zeta , \lambda \right) =0,\nonumber \\ \end{aligned}$$
(24)

where c is an integration constant and must not be dropped in our present periodic case because the elliptic functions generally do not satisfy the equations with the zero integration constants. Then, the multi-periodic wave solutions for Eq. (22) can be constructed via the Riemann theta function,

$$\begin{aligned} \vartheta \left( \zeta , \lambda \right) =\sum _{\eta \in \mathbb {Z}^2}e^{\pi i\langle \eta \lambda , \eta \rangle +2\pi i\langle \zeta , \eta \rangle }, \end{aligned}$$
(25)

where \(i=\sqrt{-1}\), the integer value vector \(\eta =(\eta _1, \eta _2,\ldots , \eta _N)^T\in \mathbb {Z}^N\), \(\zeta =(\zeta _1, \zeta _2,\ldots , \zeta _N)^T\in \mathbb {C}^N\), \(\mathbb {Z}\) denotes the integer number, where \(\mathbb {C}\) denotes the complex number. In this paper, taking the matrix \(\lambda \) to be pure imaginary matrix yields Riemann Theta Function (25) real-valued. For two vectors \(f=(f_1, f_2,\ldots , f_N)^T\) and \(g=(g_1, g_2,\ldots , g_N)^T\), their inner product is defined by

$$\begin{aligned} \langle f, g\rangle =f_1g_1+f_2g_2+\cdots +f_Ng_N. \end{aligned}$$
(26)

3.2 One-periodic wave solutions for Eq. (1)

In order to construct the periodic wave solutions for Eq. (1), we should consider a more generalized bilinear form than Bilinear Form (5) for Eq. (1) by introducing one more widely dependent transformation:

$$\begin{aligned} u=u_0+2\left[ \ln \vartheta (\zeta , \lambda ) \right] _{xx}. \end{aligned}$$
(27)

Substituting Transformation (27) into Eq. (1), we can derive a generalized bilinear form as:

$$\begin{aligned}&\mathscr {L}(D_x, D_y, D_t)\vartheta (\zeta , \lambda )\cdot \vartheta (\zeta , \lambda )\nonumber \\&\quad =\left( 36D_xD_t+D_x^6+u_0D_x^6 \right. \nonumber \\&\left. \qquad -\,\alpha D_y^2-\gamma D_x^3D_y+c\right) \vartheta (\zeta , \lambda )\cdot \vartheta (\zeta , \lambda )\nonumber \\&\quad =0. \end{aligned}$$
(28)

From Riemann Theta Function (25), we derive the one-Riemann theta function as

$$\begin{aligned} \vartheta (\zeta _1, \lambda _1)=\sum _{\eta =-\infty }^{+\infty }e^{\pi i\eta ^2\lambda _{1}+2\pi i\eta \zeta _1}, \end{aligned}$$
(29)

where \(\zeta _1=Q_1x+B_1y+R_1t+\epsilon \), \(\lambda _1\) is a pure imaginary number and meets the condition Im(\(\lambda _1\))>0, and \(\epsilon \) is a real constant. Substituting Eq. (29) into (28), we have

$$\begin{aligned}&\mathscr {L}(D_x, D_y, D_t)\vartheta (\zeta _1, \lambda _1)\cdot \vartheta (\zeta _1, \lambda _1) \nonumber \\&\quad =\sum _{\varpi =-\infty }^{+\infty }\sum _{\eta =-\infty }^{+\infty }\mathscr {L}(D_x, D_y, D_t)\nonumber \\&\qquad e^{\pi i\eta ^2\lambda _1+2\pi i\eta \zeta _1}\cdot e^{\pi i\varpi ^2\lambda _1+2\pi i\varpi \zeta _1}\nonumber \\&\quad =\sum _{\varpi =-\infty }^{+\infty }\sum _{\eta =-\infty }^{+\infty }\mathscr {L}\Big [2i\pi (\eta -\varpi )Q_1, 2i\pi (\eta -\varpi )B_1,\nonumber \\&\qquad 2i\pi (\eta -\varpi )R_1\Big ]e^{\pi i(\varpi ^2+\eta ^2)\lambda _1+2\pi i(\varpi +\eta )\zeta _1}\nonumber \\&\qquad \overset{\varpi ^{'}=\varpi +\eta }{=}\sum _{\varpi ^{'}=-\infty }^{+\infty }\mathscr {\tilde{L}}(\varpi ^{'})e^{2\pi i\varpi ^{'}\zeta _1}, \end{aligned}$$
(30)

with

$$\begin{aligned}&\mathscr {\tilde{L}}(\varpi ^{'})\nonumber \\&\quad =\sum _{\eta {=}-\infty }^{+\infty }\mathscr {L}\Big [2i\pi (2\eta {-}\varpi ^{'})Q_1, 2i\pi (2\eta {-}\varpi ^{'})B_1,\nonumber \\&\qquad \qquad 2i\pi (2\eta -\varpi ^{'})R_1\Big ]e^{\pi i[\eta ^2+(\eta -\varpi ^{'})^2]\lambda _1}\nonumber \\&\quad \overset{\eta =\eta ^{'}+1}{=}\sum _{\eta ^{'}=-\infty }^{+\infty }\mathscr {L}\Big \{2i\pi [2\eta ^{'}-(\varpi ^{'}-2)]Q_1,\nonumber \\&\qquad \qquad 2i\pi [2\eta ^{'}-(\varpi ^{'}-2)]B_1, 2i\pi [2\eta ^{'}-(\varpi ^{'}-2)]R_1\Big \}\nonumber \\&\qquad \qquad e^{\pi i[\eta ^{'2}+(\eta ^{'}-(\varpi ^{'}-2))^2]\lambda _1}\cdot e^{2\pi i(\varpi ^{'}-1)\lambda _1}\nonumber \\&\quad =\mathscr {\tilde{L}}(\varpi ^{'}-2)e^{2\pi i(\varpi ^{'}-1)\lambda _1}\nonumber \\&\quad {=}\cdots {=}{\left\{ \begin{array}{ll}{\mathscr {\tilde{L}}(0)e^{\pi i\varpi ^{'}\lambda _1},} &{} \varpi ^{'} \text{ is } \text{ even },\\ {\mathscr {\tilde{L}}(1)e^{\pi i(\varpi ^{'}+1)\lambda _1},} &{} \varpi ^{'} \text{ is } \text{ odd },\end{array}\right. }~~\varpi ^{'}, \eta ^{'}\in \mathbb {Z},\nonumber \\ \end{aligned}$$
(31)

Equation (31) implies that \(\mathscr {\tilde{L}}(\varpi ^{'})\) for \(\varpi ^{'} \in \mathbb {Z}\) are completely dominated by \(\mathscr {\tilde{L}}(0)\) and \(\mathscr {\tilde{L}}(1)\). If \(\mathscr {\tilde{L}}(0)=\mathscr {\tilde{L}}(1)=0\), then \(\mathscr {L}(D_x, D_y, D_t)\vartheta (\zeta _1, \lambda _1)\cdot \vartheta (\zeta _1, \lambda _1) = 0\).

Based on Bilinear Form (28), the one-periodic waveFootnote 1 solutions can be derived by

$$\begin{aligned} \mathscr {\tilde{L}}(0)= & {} \sum _{\eta =-\infty }^{+\infty }\mathscr {L}\left( 4\eta \pi iQ_1, 4\eta \pi iB_1, 4\eta \pi iR_1\right) e^{2\eta ^2\pi i\lambda _1}\nonumber \\= & {} \sum _{\eta =-\infty }^{+\infty }\Big (-576\eta ^2\pi ^2Q_1R_1-4096\eta ^6\pi ^6Q_1^6\nonumber \\&\quad -4096u_0\eta ^6\pi ^6Q_1^6+16\alpha \eta ^2\pi ^2B_1^2\nonumber \\&\quad -\,256\gamma \eta ^4\pi ^4Q_1^3B_1+c\Big )e^{2i\pi \eta ^2\lambda _1}=0,\nonumber \\ \mathscr {\tilde{L}}(1)= & {} \sum _{\eta =-\infty }^{+\infty }\mathscr {L}\Big [2i\pi (2\eta -1)Q_1, 2i\pi (2\eta -1)B_1,\nonumber \\&\qquad 2i\pi (2\eta -1)R_1\Big ]e^{\pi i(2\eta ^2-2\eta +1)\lambda _1}\nonumber \\= & {} \sum _{\eta =-\infty }^{+\infty }\Big [-144(2\eta -1)^2\pi ^2Q_1R_1\nonumber \\&\quad -\,64(2\eta -1)^6\pi ^6Q_1^6-64u_0(2\eta -1)^6\pi ^6Q_1^6\nonumber \\&\quad +4\alpha (2\eta -1)^2\pi ^2B_1^2-16\gamma (2\eta -1)^4\pi ^4Q_1^3B_1\nonumber \\&\quad +\,c\Big ]e^{\pi i(2\eta ^2-2\eta +1)\lambda _1}=0. \end{aligned}$$
(32)

Through the notations

$$\begin{aligned}&\varDelta =e^{\pi i\lambda _1}, \end{aligned}$$
(33)
$$\begin{aligned}&a_{11}=-\sum _{\eta =-\infty }^{+\infty }576\eta ^2\pi ^2Q_1\varDelta ^{2\eta ^2},\quad a_{12}=\sum _{\eta =-\infty }^{+\infty }\varDelta ^{2\eta ^2},\nonumber \\&a_{21}=-\sum _{\eta =-\infty }^{+\infty }144(2\eta -1)^2\pi ^2Q_1\varDelta ^{2\eta ^2-2\eta +1},\nonumber \\&a_{22}=\sum _{\eta =-\infty }^{+\infty }\varDelta ^{2\eta ^2-2\eta +1},\nonumber \\&b_1=\sum _{\eta =-\infty }^{+\infty }\Big (4096\eta ^6\pi ^6Q_1^6+4096u_0\eta ^6\pi ^6Q_1^6\nonumber \\&\quad -\,16\alpha \eta ^2\pi ^2B_1^2+256\gamma \eta ^4\pi ^4Q_1^3B_1\Big )\varDelta ^{2\eta ^2},\nonumber \\&b_2=\sum _{\eta =-\infty }^{+\infty }\Big [64(2\eta -1)^6\pi ^6Q_1^6+64u_0(2\eta -1)^6\pi ^6Q_1^6\nonumber \\&\quad -\,4\alpha (2\eta -1)^2\pi ^2B_1^2+16\gamma (2\eta -1)^4\pi ^4Q_1^3B_1\Big ]\nonumber \\&\qquad \varDelta ^{2\eta ^2-2\eta +1}, \end{aligned}$$
(34)

Equation (32) can be rewritten as a linear system about \(R_1\) and c, i.e.,

$$\begin{aligned} \begin{aligned}&\left( \begin{array}{cc} a_{11} &{} a_{12} \\ a_{21} &{} a_{22} \\ \end{array} \right) \left( \begin{array}{c} R_1 \\ c \\ \end{array} \right) = \left( \begin{array}{c} b_1 \\ b_2 \\ \end{array} \right) . \end{aligned} \end{aligned}$$
(35)

Solving System (35), we can derive the one-periodic wave solutions for Eq. (1) as

$$\begin{aligned} \begin{aligned}&u=u_0+2\left[ \ln \vartheta (\zeta _1, \lambda _1) \right] _{xx}. \end{aligned} \end{aligned}$$
(36)
Fig. 3
figure 3

One-periodic wave via Solutions (36) with \(\lambda _1=i,~Q_1=0.3,~B_1=0.2\) and \(\alpha =\gamma =u_0=1\)

Full size image

Figure 3 shows that the one-periodic wave can be viewed as a superposition of the overlapping solitary waves, placed one period apart. In the following section, the asymptotic behaviors of One-Periodic Wave Solutions (36) will be studied. Equation (34) can be expanded as

$$\begin{aligned}&a_{11}=-1152\pi ^2Q_1\left( \varDelta ^2+4\varDelta ^8+\cdots +\eta ^2\varDelta ^{2\eta ^2}+\cdots \right) ,\nonumber \\&a_{12}=1+2(\varDelta ^2+\varDelta ^8+\cdots +\varDelta ^{2\eta ^2}+\cdots ),\nonumber \\&a_{21}=-288\pi ^2Q_1[\varDelta +9\varDelta ^5+\cdots \nonumber \\&\quad +\,(2\eta -1)^2\varDelta ^{2\eta ^2-2\eta +1}+\cdots ],\nonumber \\&a_{22}=2(\varDelta +\varDelta ^5+\cdots +\varDelta ^{2\eta ^2-2\eta +1}+\cdots ),\nonumber \\&b_1=2\Big (4096\pi ^6Q_1^6+4096u_0\pi ^6Q_1^6-16\alpha \pi ^2B_1^2\nonumber \\&\quad +\,256\gamma \pi ^4Q_1^3B_1\Big )\varDelta ^{2}\nonumber \\&\quad +\,2\Big (262144\pi ^6Q_1^6+262144u_0\pi ^6Q_1^6\nonumber \\&\quad -64\alpha \pi ^2B_1^2+4096\gamma \pi ^4Q_1^3B_1\Big )\varDelta ^{8}+\cdots \nonumber \\&\quad +\Big (4096\eta ^6\pi ^6Q_1^6+4096u_0\eta ^6\pi ^6Q_1^6\nonumber \\&\quad -16\alpha \eta ^2\pi ^2B_1^2\nonumber \\&\quad +\,256\gamma \eta ^4\pi ^4Q_1^3B_1\Big )\varDelta ^{2\eta ^2}+\cdots ,\nonumber \\&b_2=2\left( 64\pi ^6Q_1^6+64u_0\pi ^6Q_1^6\right. \nonumber \\&\left. \quad -4\alpha \pi ^2B_1^2+16\gamma \pi ^4Q_1^3B_1\right) \varDelta \nonumber \\&\quad +2\Big (46656\pi ^6Q_1^6+46656u_0\pi ^6Q_1^6-36\alpha \pi ^2B_1^2\nonumber \\&\quad +\,1296\gamma \pi ^4Q_1^3B_1\Big )\varDelta ^{5}+\cdots +[64(2\eta -1)^6\pi ^6Q_1^6\nonumber \\&\quad +\,64u_0(2\eta -1)^6\pi ^6Q_1^6-4\alpha (2\eta -1)^2\pi ^2B_1^2\nonumber \\&\quad +\,16\gamma (2\eta -1)^4\pi ^4Q_1^3B_1]\varDelta ^{2\eta ^2-2\eta +1}+\cdots ,\nonumber \\ \end{aligned}$$
(37)

and substituting Eq. (37) into System (35), we have

$$\begin{aligned} \left( \begin{array}{cc} a_{11} &{} a_{12} \\ a_{21} &{} a_{22} \\ \end{array} \right)= & {} \varLambda _0+\varLambda _1\varDelta +\varLambda _2\varDelta ^2+\cdots ,\nonumber \\ \left( \begin{array}{c} b_1 \\ b_2 \\ \end{array} \right)= & {} \varTheta _0+\varTheta _1\varDelta +\varTheta _2\varDelta ^2+\cdots , \end{aligned}$$
(38)

where

$$\begin{aligned}&\varLambda _0=\left( \begin{array}{cc} 0 &{} 1 \\ 0 &{} 0 \\ \end{array} \right) ,~ \varLambda _1=\left( \begin{array}{cc} 0 &{} 0 \\ -288\pi ^2Q_1 &{} 2 \\ \end{array} \right) ,\nonumber \\&\varLambda _2=\left( \begin{array}{cc} -1152\pi ^2Q_1 &{} 2 \\ 0 &{} 0 \\ \end{array} \right) ,~\varLambda _3=\varLambda _4=\mathbf 0 ,\nonumber \\&\varLambda _5=\left( \begin{array}{cc} 0 &{} 0 \\ -2592\pi ^2Q_1 &{} 2 \\ \end{array} \right) , \ldots , \varTheta _0=\varTheta _3=\varTheta _4=\mathbf 0 ,~ \nonumber \\&\nu _1=128\pi ^6Q_1^6+128u_0\pi ^6Q_1^6-8\alpha \pi ^2B_1^2\nonumber \\&\quad +\,32\gamma \pi ^4Q_1^3B_1,\nonumber \\&\nu _2=8192\pi ^6Q_1^6+8192u_0\pi ^6Q_1^6-32\alpha \pi ^2B_1^2\nonumber \\&\quad +\,512\gamma \pi ^4Q_1^3B_1,\nonumber \\&\nu _5=93312\pi ^6Q_1^6+93312u_0\pi ^6Q_1^6-72\alpha \pi ^2B_1^2\nonumber \\&\quad +\,2592\gamma \pi ^4Q_1^3B_1,\nonumber \\&\varTheta _1=\left( \begin{array}{c} 0 \\ \nu _1 \\ \end{array} \right) ,~ \varTheta _2=\left( \begin{array}{c} \nu _2 \\ 0 \\ \end{array} \right) ,~ \varTheta _5=\left( \begin{array}{c} 0 \\ \nu _5 \\ \end{array} \right) ,\nonumber \\&\qquad \cdots . \end{aligned}$$
(39)

Then, \(R_1\) and c in System (35) can be rewritten as

$$\begin{aligned}&\left( \begin{array}{c} R_1 \\ c \\ \end{array} \right) =\varGamma _0+\varGamma _1\varDelta +\varGamma _2\varDelta ^2+\cdots ,\nonumber \\&\varGamma _0=\left( \begin{array}{c} \frac{2\varTheta _0^{[1]}-\varTheta _1^{[2]}}{288\pi ^2Q_1} \\ \varTheta _0^{[1]} \\ \end{array} \right) ,~ \varGamma _1=\left( \begin{array}{c} \frac{2\varTheta _1^{[1]}-(\varTheta _2-\varLambda _2\varGamma _0)^{[2]}}{288\pi ^2Q_1} \\ \varTheta _1^{[1]} \\ \end{array} \right) ,~\nonumber \\&\varGamma _n=\left( \begin{array}{c} \frac{2[\varTheta _{n+1}-\varSigma _{j=2}^n\varLambda _j\varGamma _{n-j}]^{[1]}-[\varTheta _{n+1}-\varSigma _{j=2}^{n+1}\varLambda _j\varGamma _{n-j+1}]^{[2]}}{288\pi ^2Q_1} \\ {[\varTheta _{n+1}-\varSigma _{j=2}^n\varLambda _j\varGamma _{n-j}]^{[1]}} \\ \end{array} \right) ,\nonumber \\&\quad n\ge 2, \end{aligned}$$
(40)

where n is the positive integer, and \(\varTheta ^{[\kappa ]}~ (\kappa = 1, 2)\) denotes the \(\kappa \)-th elements of the two-dimensional vector \(\varTheta \).

From Eq. (40), we have

$$\begin{aligned} \varGamma _0= & {} \left( \begin{array}{c} \frac{16\pi ^4Q_1^6+16u_0\pi ^4Q_1^6-\alpha B_1^2+4\gamma \pi ^2Q_1^3B_1}{-36Q_1} \\ 0 \\ \end{array} \right) ,~\varGamma _1=\mathbf 0 ,\nonumber \\ \varGamma _2= & {} \left( \begin{array}{c} \frac{-32\pi ^4Q_1^6-32u_0\pi ^4Q_1^6+2\alpha B_1^2-8\gamma \pi ^2Q_1^3B_1}{9Q_1} \\ -512\pi ^6Q_1^6-512u_0\pi ^6Q_1^6+32\alpha \pi ^2B_1^2-128\gamma \pi ^4Q_1^3B_1 \\ \end{array} \right) ,\nonumber \\&\cdots . \end{aligned}$$
(41)

Substituting Eqs. (41) into (38) and setting \(\varDelta \rightarrow 0\), we can obtain

$$\begin{aligned} \begin{aligned}&c\rightarrow 0,\\&R_1\rightarrow \frac{16\pi ^4Q_1^6+16u_0\pi ^4Q_1^6-\alpha B_1^2+4\gamma \pi ^2Q_1^3B_1}{-36Q_1}. \end{aligned} \end{aligned}$$
(42)

If we assume

$$\begin{aligned}&u_0=0,\quad Q_1=\frac{k_1+k_2}{2i\pi },\quad B_1=\frac{5k_1^3+5k_2^3}{2\gamma i\pi },\nonumber \\&\epsilon =\frac{-i\pi \lambda +\ln \frac{k_1-k_2}{k_1+k_2}}{2i\pi },\quad \alpha =\frac{\gamma ^2}{5}, \end{aligned}$$
(43)

where \(k_1, k_2, \alpha \) and \(\gamma \) are determined by Eq. (19), we have

$$\begin{aligned} 2i\pi \zeta _1= & {} 2i\pi (Q_1x+B_1y+R_1t+\epsilon )\nonumber \\= & {} (k_1+k_2)x+\frac{5k_1^3+5k_2^3}{\gamma }y+\frac{k_1^5+k_2^5}{4}t\nonumber \\&+\ln \frac{k_1-k_2}{k_1+k_2}-i\pi \lambda _1 \nonumber \\= & {} \xi _{1}+\xi _{2}+\ln \frac{k_1-k_2}{k_1+k_2}-i\pi \lambda _1. \end{aligned}$$
(44)

Combining Eqs. (29) and (44), we further obtain

$$\begin{aligned} \vartheta (\zeta _1, \lambda _1)= & {} \sum _{\eta =-\infty }^{+\infty }e^{\pi i\eta ^2\lambda _1+2\pi i\eta \zeta _1}\nonumber \\= & {} 1+(e^{2\pi i\zeta _1}+e^{-2\pi i\zeta _1})\varDelta +\cdots \nonumber \\= & {} 1+e^{\xi _{1}+\xi _{2}+\ln \frac{k_1-k_2}{k_1+k_2}}+e^{-\left( \xi _{1}+\xi _{2}+\ln \frac{k_1-k_2}{k_1+k_2}\right) }\varDelta ^2+\cdots \nonumber \\&\overset{\varDelta \rightarrow 0}{=}1+\frac{k_1-k_2}{k_1+k_2}e^{\xi _{1}+\xi _{2}}. \end{aligned}$$
(45)

From the above analysis, we find that One-Periodic Wave Solutions (36) approach to One-Soliton Solutions (19) under the limiting condition \(\varDelta \rightarrow 0\) [\(\varDelta \) is defined in (33)].

3.3 Two-periodic wave solutions for Eq. (1)

From Riemann Theta Function (25), we derive the two-Riemann theta function as:

$$\begin{aligned} \vartheta (\zeta , \lambda _2)=\sum _{\eta \in \mathbb {Z}^2}e^{\pi i\langle \lambda _2\eta , \eta \rangle +2\pi i\langle \zeta , \eta \rangle }, \end{aligned}$$
(46)

where \(\eta =(\eta _1, \eta _2)^T\in \mathbb {Z}^2\), \(\zeta =(\zeta _1, \zeta _2)\in \mathbb {C}^2\), \(\mathbb {C}\) denotes the complex number, \(\zeta _r=Q_rx+B_ry+R_rt+\epsilon _r, r=1, 2\), \(Q_r\)’s, \(B_r\)’s, \(R_r\)’s are all the constants, \(-i\lambda _2\) is a real-valued \(2\times 2\) matrix:

$$\begin{aligned}&\lambda _2=\left( \begin{array}{cc} \lambda _{11} &{} \lambda _{12} \\ \lambda _{12} &{} \lambda _{22} \\ \end{array} \right) ,~\text {Im}(\lambda _{11})>0,~\text {Im}(\lambda _{22})>0,\nonumber \\&\lambda _{12}^2-\lambda _{11}\lambda _{22}>0. \end{aligned}$$
(47)

Substituting Eq. (46) into (28), we can derive

$$\begin{aligned}&\mathscr {L}(D_x, D_y, D_t)\vartheta (\zeta _1, \zeta _2, \lambda _2)\cdot \vartheta (\zeta _1, \zeta _2, \lambda _2)\nonumber \\&\quad =\sum _{\varpi , \eta \in \mathbb {Z}^2}\mathscr {L}\Big (2i\pi \langle \eta -\varpi , Q\rangle , 2i\pi \langle \eta -\varpi , B\rangle ,\nonumber \\&\qquad 2i\pi \langle \eta -\varpi , R\rangle \Big )e^{2\pi i\langle \zeta , \eta +\varpi \rangle +\pi i\left( \langle \lambda _2\eta , \eta \rangle +\langle \lambda _2\varpi , \varpi \rangle \right) }\nonumber \\&\qquad \overset{\varpi ^{'}=\varpi +\eta }{=}\sum _{\varpi ^{'}\in \mathbb {Z}^2}\Big \{\sum _{\eta \in \mathbb {Z}^2}\mathscr {L}\Big (2i\pi \langle 2\eta -\varpi ^{'}, Q\rangle ,\nonumber \\&\qquad 2i\pi \langle 2\eta -\varpi ^{'}, B\rangle , 2i\pi \langle 2\eta -\varpi ^{'}, R\rangle \Big )\nonumber \\&\qquad e^{\pi i\left[ \langle \lambda _2(\eta -\varpi ^{'}), \eta -\varpi ^{'}\rangle +\langle \lambda _2 \eta , \eta \rangle \right] }\Big \}e^{2\pi i\langle \zeta , \varpi ^{'}\rangle }\nonumber \\&\quad =\sum _{\varpi ^{'}\in Z^2}\mathscr {\tilde{L}}\left( \varpi ^{'}\right) e^{2\pi i\langle \zeta , \varpi ^{'}\rangle }, \end{aligned}$$
(48)

where \(Q=(Q_1, Q_2)^T, B=(B_1, B_2)^T, R=(R_1, R_2)^T\) and \(\varpi ^{'}=(\varpi _1^{'}, \varpi _2^{'})^T\). From Eq. (48), and setting \(\eta ^{'}=\eta -\delta _{\sigma , j}, (j=1, 2)\), we can obtain

$$\begin{aligned}&\mathscr {\tilde{L}}\left( \varpi ^{'}\right) \nonumber \\&=\sum _{\eta \in \mathbb {Z}^2}\mathscr {L}\Big (2i\pi \langle 2\eta -\varpi ^{'}, Q\rangle , 2i\pi \langle 2\eta -\varpi ^{'}, B\rangle ,\nonumber \\&\quad 2i\pi \langle 2\eta -\varpi ^{'}, R\rangle \Big )e^{\pi i\left[ \langle \lambda _{2}(\eta -\varpi ^{'}), \eta -\varpi ^{'}\rangle +\langle \lambda _{2}\eta , \eta \rangle \right] }\nonumber \\&=\sum _{\eta \in \mathbb {Z}^2}\mathscr {L}\Big \{2i\pi \sum _{\sigma =1}^{2}[2\eta _\sigma ^{'}-(\varpi _\sigma ^{'}-2\delta _{\sigma , j})]Q_\sigma ,\nonumber \\&\quad 2i\pi \sum _{\sigma =1}^{2}[2\eta _\sigma ^{'}-(\varpi _\sigma ^{'}-2\delta _{\sigma , j})]B_\sigma ,\nonumber \\&\quad 2i\pi \sum _{\sigma =1}^{2}[2\eta _\sigma ^{'}-(\varpi _\sigma ^{'}-2\delta _{\sigma , j})]R_\sigma \Big \}\nonumber \\&e^{\pi i\sum _{\sigma , \varsigma =1}^{2}\left[ (\eta _\sigma ^{'}+\delta _{\sigma , j})(\eta _\varsigma ^{'}+\delta _{\varsigma , j})+(\varpi _\sigma ^{'}-\eta _\sigma ^{'}-\delta _{\sigma , j})(\varpi _\varsigma ^{'}-\eta _\varsigma ^{'}-\delta _{\varsigma , j})\right] \lambda _{\sigma , \varsigma }}\nonumber \\&={\left\{ \begin{array}{ll}{\mathscr {\tilde{L}}(\varpi _1^{'}-2, \varpi _2^{'})e^{2\pi i(\varpi _1^{'}-1)\lambda _{11}+2\pi i\varpi _2^{'}\lambda _{12}},} &{} j=1,\nonumber \\ {\mathscr {\tilde{L}}(\varpi _1^{'}, \varpi _2^{'}-2)e^{2\pi i(\varpi _2^{'}-1)\lambda _{22}+2\pi i\varpi _1^{'}\lambda _{12}},} &{} j=2,\end{array}\right. }\nonumber \\&\quad \varpi ^{'}, \eta ^{'}\in \mathbb {Z}^2, \end{aligned}$$
(49)

where \(\delta _{\sigma , j}\)’s represent the Kronecker’s delta [64]. Equation (49) implies that if \(\mathscr {\tilde{L}}(0, 0)=\mathscr {\tilde{L}}(1, 0)=\mathscr {\tilde{L}}(0, 1)=\mathscr {\tilde{L}}(1, 1)=0\), then \(\mathscr {\tilde{L}}(\varpi _1^{'}, \varpi _2^{'})=0\) for all \(\varpi _1^{'}, \varpi _2^{'}\in \mathbb {Z}^2\), Eq. (46) is the solution for Eq. (28). Setting \(\varPsi _r=(\varPsi _r^{[1]}, \varPsi _r^{[2]})^T, r=1, 2, 3, 4, \varPsi _1=(0, 0)^T, \varPsi _2=(1, 0)^T, \varPsi _3=(0, 1)^T, \varPsi _4=(1, 1)^T\), we have

$$\begin{aligned}&\mathscr {\tilde{L}}(0, 0)\nonumber \\&\quad =\sum _{\eta \in \mathbb {Z}^2}\mathscr {L}\Big (2i\pi \langle 2\eta -\varPsi _1, Q\rangle , 2i\pi \langle 2\eta -\varPsi _1, B\rangle ,\nonumber \\&\quad 2i\pi \langle 2\eta -\varPsi _1, R\rangle \Big )e^{\pi i\left[ \langle \lambda (\eta -\varPsi _1), \eta -\varPsi _1\rangle +\langle \lambda \eta , \eta \rangle \right] }=0,\nonumber \\&\mathscr {\tilde{L}}(1, 0)\nonumber \\&\quad =\sum _{\eta \in \mathbb {Z}^2}\mathscr {L}\Big (2i\pi \langle 2\eta -\varPsi _2, Q\rangle , 2i\pi \langle 2\eta -\varPsi _2, B\rangle ,\nonumber \\&\quad 2i\pi \langle 2\eta -\varPsi _2, R\rangle \Big )e^{\pi i\left[ \langle \lambda (\eta -\varPsi _2), \eta -\varPsi _2\rangle +\langle \lambda \eta , \eta \rangle \right] }=0,\nonumber \\&\quad \mathscr {\tilde{L}}(0, 1)\nonumber \\&\quad =\sum _{\eta \in \mathbb {Z}^2}\mathscr {L}\Big (2i\pi \langle 2\eta -\varPsi _3, Q\rangle , 2i\pi \langle 2\eta -\varPsi _3, B\rangle ,\nonumber \\&\quad 2i\pi \langle 2\eta -\varPsi _3, R\rangle \Big )e^{\pi i\left[ \langle \lambda (\eta -\varPsi _3), \eta -\varPsi _3\rangle +\langle \lambda \eta , \eta \rangle \right] }=0,\nonumber \\&\mathscr {\tilde{L}}(1, 1)\nonumber \\&\quad =\sum _{\eta \in \mathbb {Z}^2}\mathscr {L}\Big (2i\pi \langle 2\eta -\varPsi _4, Q\rangle , 2i\pi \langle 2\eta -\varPsi _4, B\rangle ,\nonumber \\&\quad 2i\pi \langle 2\eta -\varPsi _4, R\rangle \Big )e^{\pi i\left[ \langle \lambda (\eta -\varPsi _4), \eta -\varPsi _4\rangle +\langle \lambda \eta , \eta \rangle \right] }=0.\nonumber \\ \end{aligned}$$
(50)

Combining Eqs. (28) and (50), we derive

$$\begin{aligned}&\sum _{\eta \in \mathbb {Z}^2}\Big [-144\pi ^2\langle 2\eta -\varPsi _r, Q\rangle \langle 2\eta -\varPsi _r, R\rangle \nonumber \\&-\,64\pi ^6\langle 2\eta -\varPsi _r, Q\rangle ^6-64u_0\pi ^6\langle 2\eta -\varPsi _r, Q\rangle ^6\nonumber \\&+\,4\alpha \pi ^2\langle 2\eta -\varPsi _r, B\rangle ^2\nonumber \\&-\,16\gamma \pi ^4\langle 2\eta -\varPsi _r, Q\rangle ^3\langle 2\eta -\varPsi _r, B\rangle +c\Big ]\nonumber \\&e^{\pi i\left[ \langle \lambda (\eta -\varPsi _r), \eta -\varPsi _r\rangle +\langle \lambda \eta , \eta \rangle \right] }=0. \end{aligned}$$
(51)

Accordingly, Eq. (51) can be rewritten as a linear system,

$$\begin{aligned} \begin{aligned}&\left( \begin{array}{cccc} g_{11} &{} g_{12} &{} g_{13} &{} g_{14} \\ g_{21} &{} g_{22} &{} g_{23} &{} g_{24} \\ g_{31} &{} g_{32} &{} g_{33} &{} g_{34} \\ g_{41} &{} g_{42} &{} g_{43} &{} g_{44} \\ \end{array} \right) \left( \begin{array}{c} R_1 \\ R_2 \\ u_0 \\ c \\ \end{array} \right) =\left( \begin{array}{c} q_1 \\ q_2 \\ q_3 \\ q_4\\ \end{array} \right) , \end{aligned} \end{aligned}$$
(52)

with

$$\begin{aligned}&\mathscr {J}_1=e^{\pi i \lambda _{11}},~\mathscr {J}_2=e^{\pi i \lambda _{22}},~\mathscr {J}_3=e^{2\pi i \lambda _{12}},\nonumber \\&G=(g_{rj})_{4\times 4},~q=(q_1, q_2, q_3, q_4)^T, \end{aligned}$$
(53)
$$\begin{aligned}&\mathscr {A}_r(\eta )=\mathscr {J}_1^{\left\{ \eta _1^2+\left( \eta _1-\varPsi _r^{[1]}\right) ^2\right\} }\mathscr {J}_2^{ \left\{ \eta _2^2+\left( \eta _2-\varPsi _r^{[2]}\right) ^2\right\} }\nonumber \\&\quad \quad \mathscr {J}_3^{\left\{ \eta _1\eta _2+\left( \eta _1-\varPsi _r^{[1]}\right) \left( \eta _2-\varPsi _r^{[2]}\right) \right\} },\nonumber \\&g_{r1}=-144\pi ^2\sum _{\eta \in \mathbb {Z}^2}\langle 2\eta -\varPsi _r, Q\rangle \nonumber \\&\quad \qquad \left( 2\eta _1-\varPsi _r^{[1]}\right) \mathscr {A}_r(\eta ),\nonumber \\&g_{r2}=-144\pi ^2\sum _{\eta \in \mathbb {Z}^2}\langle 2\eta -\varPsi _r, Q\rangle \nonumber \\&\quad \left( 2\eta _2-\varPsi _r^{[2]}\right) \mathscr {A}_r(\eta ),\nonumber \\&g_{r3}=-64\pi ^6\sum _{\eta \in \mathbb {Z}^2}\langle 2\eta -\varPsi _r, Q\rangle ^6\mathscr {A}_r(\eta ),~g_{r4}\nonumber \\&\quad =\sum _{\eta \in Z^2}\mathscr {A}_r(\eta ),\nonumber \\&q_r=\sum _{\eta \in \mathbb {Z}^2}\Big (64\pi ^6\langle 2\eta -\varPsi _r, Q\rangle ^6\nonumber \\&\quad \quad -4\alpha \pi ^2\langle 2\eta -\varPsi _r, B\rangle ^2\nonumber \\&\quad \quad +16\gamma \pi ^4\langle 2\eta -\varPsi _r, Q\rangle ^3\nonumber \\&\quad \langle 2\eta -\varPsi _r, B\rangle \Big )\mathscr {A}_r(\eta ). \end{aligned}$$
(54)

Solving System (52), we can derive the two-periodic waveFootnote 2 solutions for Eq. (1) as

$$\begin{aligned} u=u_0+2\left[ \ln \vartheta (\zeta _1, \zeta _2, \lambda ) \right] _{xx}. \end{aligned}$$
(55)
Fig. 4
figure 4

Two-periodic wave via Solutions (55) with \(\lambda _{11}=0.6i,~\lambda _{12}=0.5i,~\lambda _{22}=2i,~Q_1=1,~Q_2=-2.5,~B_1=2,~B_2=2.2\) and \(\alpha =\gamma =u_0=1\)

Full size image

Figure 4 shows that the periodic behaviors for the two-periodic wave exist along the x and y directions, respectively. Similarly, the asymptotic behaviors of Two-Periodic Wave Solutions (55) will be studied. Expansions for the matrices in System (52) can be written as

$$\begin{aligned}&G=\left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) \nonumber \\&+\,\left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 0 \\ -288\pi ^2Q_1 &{} 0 &{} -128\pi ^6Q_1^6 &{} 2 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) \mathscr {J}_1\nonumber \\&\qquad +\,\left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} -288\pi ^2Q_2 &{} -128\pi ^6Q_2^6 &{} 2 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) \mathscr {J}_2\nonumber \\&\qquad +\,\left( \begin{array}{cccc} -1152\pi ^2Q_1 &{} 0 &{} -8192\pi ^6Q_1^6 &{} 2 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) \mathscr {J}_1^2\nonumber \\&\qquad +\,\left( \begin{array}{cccc} 0 &{} -1152\pi ^2Q_2 &{} -8192\pi ^6Q_2^6 &{} 2 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) \mathscr {J}_2^2\nonumber \\&\qquad +\,\left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \beta _1 &{} -\beta _1 &{} \beta _2 &{} 2 \\ \end{array} \right) \mathscr {J}_1\mathscr {J}_2 \nonumber \\&\qquad +\,\left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \beta _3 &{} \beta _3 &{} \beta _4 &{} 2 \\ \end{array} \right) \mathscr {J}_1\mathscr {J}_2\mathscr {J}_3\nonumber \\&\qquad +\,o\left( \mathscr {J}_1^r, \mathscr {J}_2^j, \mathscr {J}_3^l\right) ,~r+j+l\ge 3,\nonumber \\ \end{aligned}$$
(56)
$$\begin{aligned}&\left( \begin{array}{c} R_1 \\ R_2 \\ u_0 \\ c \\ \end{array} \right) =\left( \begin{array}{c} R_1^{(00)} \\ R_2^{(00)} \\ u_0^{(00)} \\ c^{(00)} \\ \end{array} \right) +\left( \begin{array}{c} R_1^{(11)} \\ R_2^{(11)} \\ u_0^{(11)} \\ c^{(11)} \\ \end{array} \right) \mathscr {J}_1+\left( \begin{array}{c} R_1^{(21)} \\ R_2^{(21)} \\ u_0^{(21)} \\ c^{(21)} \\ \end{array} \right) \mathscr {J}_2\nonumber \\&\qquad +\,\left( \begin{array}{c} R_1^{(12)} \\ R_2^{(12)} \\ u_0^{(12)} \\ c^{(12)} \\ \end{array} \right) \mathscr {J}_1^2+\left( \begin{array}{c} R_1^{(22)} \\ R_2^{(22)} \\ u_0^{(22)} \\ c^{(22)} \\ \end{array} \right) \mathscr {J}_2^2\nonumber \\&\qquad +\,\left( \begin{array}{c} R_1^{(2)} \\ R_2^{(2)} \\ u_0^{(2)} \\ c^{(2)} \\ \end{array} \right) \mathscr {J}_1\mathscr {J}_2+\left( \begin{array}{c} R_1^{(3)} \\ R_2^{(3)} \\ u_0^{(3)} \\ c^{(3)} \\ \end{array} \right) \mathscr {J}_1\mathscr {J}_2\mathscr {J}_3\nonumber \\&\qquad +\,o\left( \mathscr {J}_1^r, \mathscr {J}_2^j, \mathscr {J}_3^l\right) ,~r+j+l\ge 3, \end{aligned}$$
(57)
$$\begin{aligned}&q=\left( \begin{array}{c} 0 \\ \rho _1 \\ 0 \\ 0 \\ \end{array} \right) \mathscr {J}_1+\left( \begin{array}{c} 0 \\ 0 \\ \rho _2 \\ 0 \\ \end{array} \right) \mathscr {J}_2+\left( \begin{array}{c} \rho _3 \\ 0 \\ 0 \\ 0 \\ \end{array} \right) \mathscr {J}_1^2\nonumber \\&\qquad +\,\left( \begin{array}{c} \rho _4 \\ 0 \\ 0 \\ 0 \\ \end{array} \right) \mathscr {J}_2^2+\left( \begin{array}{c} 0 \\ 0 \\ 0 \\ \rho _5 \\ \end{array} \right) \mathscr {J}_1\mathscr {J}_2\nonumber \\&\quad \qquad +\,\left( \begin{array}{c} 0 \\ 0 \\ 0 \\ \rho _6 \\ \end{array} \right) \mathscr {J}_1\mathscr {J}_2\mathscr {J}_3+o\left( \mathscr {J}_1^r, \mathscr {J}_2^j, \mathscr {J}_3^l\right) ,\nonumber \\&\quad \qquad r+j+l\ge 3, \end{aligned}$$
(58)

with

$$\begin{aligned} \beta _1= & {} -288\pi ^2(Q_1-Q_2),~\beta _2=-128\pi ^6(Q_1-Q_2)^6,\nonumber \\ \beta _3= & {} -288\pi ^2(Q_1+Q_2),~\beta _4=-128\pi ^6(Q_1+Q_2)^6,\nonumber \\ \rho _1= & {} 8\pi ^2\left( 16\pi ^4Q_1^6-\alpha B_1^2+4\gamma \pi ^2Q_1^3B_1\right) ,\nonumber \\ \rho _2= & {} 8\pi ^2\left( 16\pi ^4Q_2^6-\alpha B_2^2+4\gamma \pi ^2Q_2^3B_2\right) ,\nonumber \\ \rho _3= & {} 32\pi ^2\left( 256\pi ^4Q_1^6-\alpha B_1^2+16\gamma \pi ^2Q_1^3B_1\right) ,\nonumber \\ \rho _4= & {} 32\pi ^2\left( 256\pi ^4Q_2^6-\alpha B_2^2+16\gamma \pi ^2Q_2^3B_2\right) ,\nonumber \\ \rho _5= & {} 8\pi ^2\Big [16\pi ^4(Q_1-Q_2)^6-\alpha (B_1-B_2)^2\nonumber \\&+\,4\gamma \pi ^2(Q_1-Q_2)^3(B_1-B_2)\Big ],\nonumber \\ \rho _6= & {} 8\pi ^2\Big [16\pi ^4(Q_1+Q_2)^6-\alpha (B_1+B_2)^2\nonumber \\&+\,4\gamma \pi ^2(Q_1+Q_2)^3(B_1+B_2)\Big ], \end{aligned}$$
(59)

where \(o\left( \mathscr {J}_1^r, \mathscr {J}_2^j, \mathscr {J}_3^l\right) \) denotes the infinitely small quantity.

Substituting Eqs. (56), (58) and (57) into System (52) and comparing the same order of \(\mathscr {J}_1, \mathscr {J}_2\) and \(\mathscr {J}_3\), we can obtain

$$\begin{aligned}&c^{(00)}=c^{(11)}=c^{(21)}=c^{(2)}=c^{(3)}=0,\nonumber \\&-\,288\pi ^2Q_1R_1^{(00)}-128\pi ^6Q_1^6u_0^{(00)}=\rho _1,\nonumber \\&-\,288\pi ^2Q_2R_2^{(00)}-128\pi ^6Q_2^6u_0^{(00)}=\rho _2,\nonumber \\&c^{(12)}-1152\pi ^2Q_1R_1^{(00)}-8192\pi ^6Q_1^6u_0^{(00)}=\rho _3,\nonumber \\&c^{(22)}-1152\pi ^2Q_2R_2^{(00)} -8192\pi ^6Q_2^6u_0^{(00)}=\rho _4,\nonumber \\&\beta _1R_1^{(00)}-\beta _1R_2^{(00)}+\beta _2u_0^{(00)}=\rho _5,\nonumber \\&\beta _3R_1^{(00)}+\beta _3R_2^{(00)}+\beta _4u_0^{(00)}=\rho _6,\nonumber \\&288\pi ^2Q_2R_2^{(11)}+128\pi ^6Q_2^6u_0^{(11)}=0,\nonumber \\&288\pi ^2Q_1R_1^{(21)}+128\pi ^6Q_1^6u_0^{(21)}=0,\nonumber \\&288\pi ^2Q_1R_1^{(11)}+128\pi ^6Q_1^6u_0^{(11)}=0,\nonumber \\&288\pi ^2Q_2R_2^{(21)}+128\pi ^6Q_2^6u_0^{(21)}=0.\nonumber \\ \end{aligned}$$
(60)

Combining Eqs. (57) and (60), and taking \(u_0^{(00)}=0\), we can notice that

$$\begin{aligned} u_0= & {} o\left( \mathscr {J}_1, \mathscr {J}_2\right) \rightarrow 0,~c\rightarrow 0,\nonumber \\ R_1= & {} \frac{16\pi ^4Q_1^6-\alpha B_1^2+4\gamma \pi ^2Q_1^3B_1}{-36Q_1}+o\left( \mathscr {J}_1, \mathscr {J}_2\right) \nonumber \\&\quad \rightarrow \frac{16\pi ^4Q_1^6-\alpha B_1^2+4\gamma \pi ^2Q_1^3B_1}{-36Q_1},\nonumber \\ R_2= & {} \frac{16\pi ^4Q_2^6-\alpha B_2^2+4\gamma \pi ^2Q_2^3B_2}{-36Q_2}+o\left( \mathscr {J}_1, \mathscr {J}_2\right) \nonumber \\&\quad \rightarrow \frac{16\pi ^4Q_2^6-\alpha B_2^2+4\gamma \pi ^2Q_2^3B_2}{-36Q_2}, \end{aligned}$$
(61)

\(\text {when} \left( \mathscr {J}_1, \mathscr {J}_2\right) \rightarrow 0\), and assuming that

$$\begin{aligned}&u_0=0,~Q_1=\frac{k_1+k_2}{2i\pi },~Q_2=\frac{k_3+k_4}{2i\pi },\nonumber \\&B_1=\frac{5k_1^3+5k_2^3}{2\gamma i\pi },~B_2=\frac{5k_3^3+5k_4^3}{2\gamma i\pi },\nonumber \\&\epsilon _1=\frac{-i\pi \lambda _{11}+\ln A_1}{2i\pi },~\epsilon _2=\frac{-i\pi \lambda _{22}+\ln A_2}{2i\pi },\nonumber \\&\lambda _{12}=\frac{\ln A_{12}}{2i\pi },~\alpha =\frac{\gamma ^2}{5}, \end{aligned}$$
(62)

where \(k_1, k_2, k_3, k_4, A_1, A_2, A_{12}, \alpha \) and \(\gamma \) are determined by Eq. (20). We can rewrite Eq. (46) as

$$\begin{aligned}&\vartheta (\zeta _1, \zeta _2, \lambda )=1+\left( e^{2\pi i\zeta _1}+e^{-2\pi i\zeta _1}\right) e^{i\pi \lambda _{11}}\nonumber \\&\quad +\,\left( e^{2\pi i\zeta _2}+e^{-2\pi i\zeta _2}\right) e^{i\pi \lambda _{22}}\nonumber \\&\quad +\left[ e^{2\pi i\left( \zeta _1+\zeta _2\right) }+e^{-2\pi i\left( \zeta _1+\zeta _2\right) }\right] \nonumber \\&\qquad e^{i\pi \left( \lambda _{11}+2\lambda _{12}+\lambda _{22}\right) }+\cdots \nonumber \\&=1+A_1e^{\xi _{1}+\xi _{2}}+A_2e^{\xi _{3}+\xi _{4}}\nonumber \\&\quad +A_{12}e^{\xi _{1}+\xi _{2}+\xi _{3}+\xi _{4}},\nonumber \\&\text {when} \left( \mathscr {J}_1, \mathscr {J}_2\right) \rightarrow 0. \end{aligned}$$
(63)

Thus, we notice that Two-Periodic Wave Solutions (55) approach to Two-Soliton Solutions (20) under the limiting conditions \(\left( \mathscr {J}_1, \mathscr {J}_2\right) \rightarrow 0\) [\(\mathscr {J}_1\) and \(\mathscr {J}_2\) are defined in (53)].

4 Conclusions

Fluid mechanics has the applications in a wide range of disciplines, such as oceanography, astrophysics, meteorology, and biomedical engineering. In this paper, we have investigated the (\(2+1\))-dimensional gCDGKS equation, i.e., Eq. (1), in fluid mechanics. Based on the Pfaffian technique and Constraint (11) on the real constant \(\alpha \), the Nth-Order Pfaffian Solutions (16) have been obtained. One- and two-soliton solutions, i.e., Solutions (19) and (20), have been derived via the Nth-Order Pfaffian Solutions (16). One- and two-periodic-wave solutions, i.e., Solutions (36) and (55), have been constructed via the Hirota–Riemann method. Results can be summarized as follows:

  1. 1.

    Amplitude of the one soliton is irrelevant to the real constant \(\gamma \), the velocity along the x direction of the one soliton is independent of \(\gamma \), while the velocity along the y direction of the one soliton is proportional to \(\gamma \);

  2. 2.

    We show the propagation of the one soliton in Fig. 1 and the interaction between the two solitons in Fig. 2, and found that the one soliton keeps its amplitude and velocity invariant during the propagation and total amplitude of the two solitons in the interaction region is lower than that of any soliton;

  3. 3.

    One-periodic wave has been viewed as a superposition of the overlapping solitary waves, placed one period apart, as shown in Fig. 3;

  4. 4.

    Periodic behaviors for the two-periodic wave have existed along the x and y directions, respectively, as depicted in Fig. 4;

  5. 5.

    With the asymptotic behaviors of One-Periodic-Wave Solutions (36) and Two-Periodic-Wave Solutions (55), we have noticed that One-Periodic-Wave Solutions (36) approach to One-Soliton Solutions (19) under the limiting condition with respect to \(\varDelta \) in (33), i.e., \(\varDelta \rightarrow 0\), that Two-Periodic-Wave Solutions (55) approach to Two-Soliton Solutions (20) under the limiting conditions with respect to \(\mathscr {J}_1\) and \(\mathscr {J}_2\) in (53), i.e., \(\left( \mathscr {J}_1, \mathscr {J}_2\right) \rightarrow 0\).