1 Introduction

With the development of nonlinear science, an increasing number of scholars dedicate to soliton theory, and a series of methods are applied to study soliton solutions of nonlinear equation including \(\bar{\partial }\) method [1,2,3,4,5,6,7], Bäcklund transformation [8, 9], Hirota bilinear method [10, 11], inverse scatter transformation [12,13,14] and so on [15,16,17,18,19,20]. These methods give a lot of meaningful solutions which are extremely useful in investigating nonlinear evolution equations.

Recently, a (3 + 1)-dimension Hirota bilinear equation

$$\begin{aligned} u_{yt} - u_{xxxy} - 3(u_x u_y)_x- 3u_{xx} + 3u_{zz} = 0, \end{aligned}$$
(1)

is proposed by using a multivariate polynomial and two types of resonant multiple wave solutions are found by applying the linear superposition principle in [21]. As an extension of the KdV equation, Eq. (1) admits the similar physical meaning as KdV equation, it can be used to describe the nonlinear waves in fluid dynamics, plasma physics, and weakly dispersive media [22]. For special case \(z=y\) and \(z=t\), two classes of lump solutions to the dimensionally reduced are derived respectively in [23].

In order to investigate nonlinear dynamical phenomena in shallow water, plasma and nonlinear optics, a generalized (2 + 1)-dimensional Hirota bilinear equation

$$\begin{aligned} u_{yt} + c_1\left[ u_{xxxy} + 3(2u_x u_y + uu_{xy}) + 3u_{xx}\int ^x_{-\infty }u_y dx\right] +c_2u_{yy}=0, \end{aligned}$$
(2)

is given, and two types of interaction solutions including lump-kink and lump-soliton are derived in [24]. Equation (2) is extended to a more generalized form [25]

$$\begin{aligned} u_{yt} + c_1\left[ u_{xxxy} + 3(2u_x u_y + uu_{xy}) + 3u_{xx}\int ^x_{-\infty }u_y dx\right] +c_2u_{yy}+c_3u_{xx}=0, \end{aligned}$$
(3)

which reflexes richer physical meaning in nonlinear optics, fluid mechanics, and plasma physics. Equation (3) can be considered as a (2 + 1)-dimension generalized nonlinear wave equation with \(c_1\), \(c_2\) and \(c_3\) as real constants. And Eq. (3) is studied by Hirota bilinear method, on the basis of which, M-lump, high-order breather wave and interaction solutions are constructed and their dynamical behaviors are also discussed in [25]. Moreover, Eq. (3) admits Lax pair based on bilinear Bäcklund transformation, the mixed rogue-solitary wave solutions and mixed rogue-periodic wave solutions are studied in [26].

Riemann–Hilbert (RH) approach, as a powerful tool in solving nonlinear evolution equations and asymptotic analysis, is used in numerous equations like Fokas–Lenells equation [27], Wadati–Konno–Ichikawa equation [28], Sasa–Satsuma equation [29] and so on [30]. But it is not universal in solving nonlinear evolution equation, the first application of \(\bar{\partial }\)-dressing method in solving Kadomtsev–Petviashvili II (KP II equation indicates above fact. Ablowitz et al. proof that inverse spectral problem of KP II equation can be solved via \(\bar{\partial }\)-dressing method [31]. As a generalization of RH problem, \(\bar{\partial }\)-problem is firstly proposed by Zakharov and Shabat [32]. The main purpose of this method is using Cauchy–Green formula to establish connection between potential functions and solutions of \(\bar{\partial }\)-problem. Until now, this method has solved some equations successfully including KP II equation [31], cNLS-MB equation [1], Sasa–Satsuma equation [2], mixed Chen–Lee–Liu derivative nonlinear Schödinger equation [4] and so on.

To the best of our knowledge, the \(\bar{\partial }\)-dressing method has not been applied to Eq. (3) yet. In this paper, we will consider Eq. (3) with inverse spectral transformation and \(\bar{\partial }\)-problem. The paper is organized as follows. In Sect. 2, characteristic functions and Green’s functions of spatial spectral problem are given on basis of Fourier transformation and Fourier inverse transformation. And we rewritten Green’s function as single integral form by residue theorem and Jordan theorem. In Sect. 3, we obtain \(\bar{\partial }\)-problem by calculate \(\bar{\partial }\) derivative of characteristic function, and simplify the \(\bar{\partial }\)-problem by virtue of symmetry relation of Green’s function. In Sect. 4, we determine time evolution of the scattering data \(F(z_1,z_2,t)\), then, the form solution of Eq. (3) is expressed. Finally, a conclusion of this paper is given.

2 Characteristic Function and Green’s Function

In this section, we use inverse spectral transformation according to characteristic function and Green’s function of Lax pair to consider Eq. (3). The equation is the compatibility condition of the following two linear equations:

$$\begin{aligned} (3c_1 \partial _{xy} + 3c_1 v_y + c_3 - \beta \partial _x)\psi = 0, \end{aligned}$$
(4)
$$\begin{aligned} (\partial _t + c_1\partial ^3_x + 3c_1 v_x \partial _x + c_2 \partial _y - \alpha )\psi = 0, \end{aligned}$$
(5)

when \(u=v_x\) with \(\alpha\) and \(\beta\) as the constants. In our analysis, we always assume that u decays to zero sufficiently fast as \(x, y\rightarrow \pm \infty\), then, a Jost solution of spectral equation (4) is derived as

$$\begin{aligned} \psi \sim e^{izx + \frac{ic_3 + z\beta }{3c_1 z} y}, \quad |x|,|y| \rightarrow \infty . \end{aligned}$$
(6)

Inserting the Jost solution into (5), we can solve \(\alpha =-ic_1z^3 + \frac{c_2(ic_3+z\beta )}{3c_1z}\). To introduce the Lax pair with z as spectral parameter, we make the following transformation

$$\begin{aligned} \phi (x,y,z) =\psi (x,y)e^{-izx - \frac{ic_3 + z\beta }{3c_1 z} y}, \end{aligned}$$
(7)

then, we can see that

$$\begin{aligned} \phi (x,y,z) \sim 1, \quad |x|,|y| \rightarrow \infty . \end{aligned}$$
(8)

Based on the transformation (7), we find that the Lax pair (4)-(5) becomes

$$\begin{aligned}{} & {} 3c_1 \phi _{xy} + \frac{ic_3}{z}\phi _{x} + 3ic_1z\phi _y = -3c_1v_y\phi , \end{aligned}$$
(9)
$$\begin{aligned}{} & {} \phi _t + c_1\phi _{xxx} + 3c_1z\phi _{xx} - 3c_1z^2\phi _x + 3c_1v_x\phi _x - ic_1z\phi + 3ic_1zv_x\phi \nonumber \\{} & {} \quad + \frac{ic_2c_3 + c_2z\beta }{3c_1z}\phi - \alpha \phi = 0. \end{aligned}$$
(10)

To get the function \(\phi (x,y,z)\) that is bounded in the plane xy from the above system (9), (10), we consider the Green’s function of Eq. (9)

$$\begin{aligned} 3c_1 G_{xy} + \frac{ic_3}{z}G_{x} + 3ic_1zG_y = \delta (x)\delta (y). \end{aligned}$$
(11)

Making Fourier transformation of x and y on both sides of Eq. (11), and using the properties of multivariate \(\delta\) function, we have

$$\begin{aligned} \hat{G}(\xi ,\eta ,z) \equiv F[G] = \frac{1}{2\pi } \frac{1}{-3c_1 \xi \eta - \frac{c_3}{z}\xi - 3c_1z \eta }. \end{aligned}$$
(12)

Then making Fourier inverse transformation in Eq. (12), we can solve Green’s function with expression

$$\begin{aligned} G(x,y,z) = \frac{1}{4\pi ^2} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } \frac{e^{i(\xi x + \eta y)}}{-3c_1 \xi \eta - \frac{c_3}{z}\xi - 3c_1z \eta } d\xi d\eta , \end{aligned}$$
(13)

thus, the general solution of Eq. (9) can be written as the convolution of Green’s function G(x, y, z) and \(-3c_1v_y(x,y)\phi (x,y,z)\), that is

$$\begin{aligned} \phi (x,y,z)= & {} G(x,y,z)*[-3c_1v_y(x,y) \phi (x,y,z)] \nonumber \\= & {} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } G(x-x', y-y',z)[-3c_1v_{y'}(x',y') \phi (x',y',z)]dx'dy'. \end{aligned}$$
(14)

For zero potential solution \(v(x,y) = 0\), we choose two characteristic function solutions of spectral problem Eq. (9) which are linearly independent \(M_0 = 1\), \(N_0 = e^{-2iz_1[x-\frac{c_3}{3c_1(z_1^2+z_2^2)}y]}\), where \(z=z_1 + iz_2\) is complex. Hence we can express two characteristic functions corresponding to general potential of Eq. (9) as

$$\begin{aligned}{} & {} M = 1+ \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } G(x-x', y-y',z)[-3c_1v_{y'}(x',y') M(x',y',z)]dx'dy', \end{aligned}$$
(15)
$$\begin{aligned}{} & {} N = e^{-2iz_1[x+\frac{c_3}{3c_1(z_1^2+z_2^2)}y]} +\int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } G(x-x', y-y',z)[-3c_1v_{y'}(x',y') N(x',y',z)]dx'dy', \end{aligned}$$
(16)

where Green’s function is written as

$$\begin{aligned} G(x-x', y-y',z) = \frac{1}{4\pi ^2} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } \frac{e^{i[\xi (x-x') + \eta (y-y')]}}{-3c_1 \xi \eta - \frac{c_3}{z}\xi - 3c_1z \eta } d\xi d\eta . \end{aligned}$$
(17)

We can decompose (17) into two integrals

$$\begin{aligned} G(x-x', y-y',z) = \frac{1}{2\pi } \int ^{\infty }_{-\infty } g(\xi , y-y', z) e^{i\xi (x-x')} d\xi , \end{aligned}$$
(18)

where

$$\begin{aligned} g(\xi , y-y', z) = \frac{1}{2\pi } \int ^{\infty }_{-\infty } \frac{e^{i\eta (y-y')}}{-3c_1 \xi \eta - \frac{c_3}{z}\xi - 3c_1z \eta } d\eta . \end{aligned}$$
(19)

From the integral (19), we can see that integrand has first-order singular point

$$\begin{aligned} \eta _1 = -\frac{c_3 \xi }{3c_1 (z^2 + z\xi )}. \end{aligned}$$

When \(y-y'\), we make a sufficiently large semicircle in the upper half plane \(z\in \mathbb {C}_+\), \(C_R: \eta = Re^{i\theta }\), \(R> |\eta _1|\), \(0\le \theta \le \pi\), then \([-R, R]\cup C_R\) can make up a closed curve, whose direction is counterclockwise. Thus, if \(\eta _1\) locates in the upper half plane, \(\eta _1\) is surrounded by curve, if \(\eta _1\) locates in the lower half plane, the integrand is analytically continued inside the curve. Using the residue theorem, we obtain

$$\begin{aligned}{} & {} \frac{1}{2\pi i} \int ^R_{-R} \frac{e^{i\eta (y-y')}}{3ic_1 \xi \eta + \frac{ic_3}{z}\xi + 3ic_1z \eta } d\eta + \frac{1}{2\pi i} \int _{C_R} \frac{e^{i\eta (y-y')}}{3ic_1 \xi \eta + \frac{ic_3}{z}\xi + 3ic_1z \eta } d\eta \\{} & {} \quad = \underset{\eta = \eta _1}{\text {Res}}\left[ \frac{e^{i\eta (y-y')}}{3ic_1 \xi \eta + \frac{ic_3}{z}\xi +3ic_1z \eta }\right] . \end{aligned}$$

By virtue of Jordan theorem, we can see that the second integral above is equal to zero as \(R\rightarrow \infty\), then we calculate that

$$\begin{aligned} g(\xi , y-y', z)= & {} \underset{\eta = \eta _1}{\text{ Res }}\left[ \frac{e^{i\eta (y-y')}}{3ic_1 \xi \eta + \frac{ic_3}{z}\xi + 3ic_1z \eta }\right] = \left\{ \begin{array}{lll} &{} e^{\frac{-ic_3\xi }{3c_1(z^2 + z\xi )}(y-y')}, &{}{\text {Im}}\eta _1 >0 \\ &{} 0,&{}{\text {Im}}\eta _1 <0 \end{array}\right. \nonumber \\= & {} H[\text {Im}\eta _1(y-y')]e^{\frac{-ic_1\xi }{3c_1(z^2 + z\xi )}(y-y')}. \end{aligned}$$
(20)

Then by techniques similar to those used above, we can show that when \(y-y'<0\)

$$\begin{aligned} g(\xi , y-y', z)= & {} -\underset{\eta = \eta _1}{\text {Res}} \left[ \frac{e^{i\eta (y-y')}}{3ic_1 \xi \eta + \frac{ic_3}{z}\xi + 3ic_1z \eta }\right] = \left\{ \begin{array}{lll} &{} -e^{\frac{-ic_3\xi }{3c_1(z^2 + z\xi )}(y-y')}, &{}{\text{ Im }}\eta _1 <0 \\ &{} 0, &{}{\text {Im}}\eta _1 >0 \end{array}\right. \nonumber \\= & {} -H[\text {Im}\eta _1(y-y')]e^{\frac{-ic_1\xi }{3c_1(z^2 + z\xi )}(y-y')}. \end{aligned}$$
(21)

Together with formula (20)-(21), we can rewritten \(g(\xi , y-y', z)\) as

$$\begin{aligned} g(\xi , y-y', z) = sgn(y-y')H[\text {Im}\eta _1(y-y')]e^{\frac{-ic_3\xi }{3c_1(z^2 + z\xi )}(y-y')}, \end{aligned}$$
(22)

where \(H(\cdot )\) is Heaviside function. Inserting (22) into (18), hence the Green’s function \(G(x-x', y-y', z)\) is expressed as

$$\begin{aligned} G(x-x', y-y', z) = \frac{sgn(y-y')}{2\pi } \int ^{\infty }_{-\infty } H[\text {Im}\eta _1(y-y')]e^{i\xi (x-x') + \frac{-ic_3\xi }{3c_1(z^2 + z\xi )}(y-y')} d\xi . \end{aligned}$$

The Green’s function G(x, y, z) has no jump points along the real axis, and it is analytic in the complex z-plane, however, characteristic functions M and N are not analytic in the whole \(\mathbb {C}\) plane. This make it impossible to use Riemann–Hilbert method solving equation, nevertheless, \(\bar{\partial }\) method becomes a efficient option.

3 Scattering Equation and \(\bar{\partial }\)-Problem

The primary goal of this section is to establish connection between characteristic equations and \(\bar{\partial }\)-problem, that is, we need to represented Eqs. (15) and (16) as \(\bar{\partial }\)-problem. To do this, we calculate \(\bar{\partial }\) derivative of Eq. (15)

$$\begin{aligned} \bar{\partial }M(x,y,z)= & {} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } [\bar{\partial }G(x-x', y-y', z)][-3c_1v_{y'}(x',y') M(x',y',z)]dx'dy' \nonumber \\{} & {} + \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } G(x-x', y-y', z)[-3c_1v_{y'}(x',y') \bar{\partial }M(x',y',z)]dx'dy'. \end{aligned}$$
(23)

From (17), we can calculate that

$$\begin{aligned}{} & {} \bar{\partial }G(x-x', y-y', z) = \frac{1}{4\pi ^2} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } e^{i[\xi (x-x') + \eta (y-y')]} \bar{\partial }\frac{1}{-3c_1 \xi \eta - \frac{c_3}{z}\xi - 3c_1z \eta } d\xi d\eta \nonumber \\{} & {} \quad = \frac{1}{4\pi } \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } \frac{z}{\eta } e^{i[\xi (x-x') + \eta (y-y')]} \nonumber \\{} & {} \qquad \times \delta (-3c_1z_1^2 + 3c_1z_2^2 - 3c_1z_1\xi -c_3\frac{\xi }{\eta }) \delta (-3c_1z_2\xi - 6c_1z_1z_2) d\xi d\eta \nonumber \\{} & {} \quad = \frac{1}{4\pi } \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } \frac{z |\eta | e^{i[\xi (x-x') + \eta (y-y')]}}{\eta |3c_1z_2| |3c_1z_1^2 - 3c_1z_2^2 + 3c_1z_1\xi |} \nonumber \\{} & {} \qquad \times \delta (\eta + \frac{c_3\xi }{3c_1z_1^2 - 3c_1z_2^2 + 3c_1z_1\xi }) \delta (\xi + 2z_1) d\xi d\eta \nonumber \\{} & {} \quad = \frac{1}{4\pi } \frac{z |\eta | e^{i[\xi (x-x') + \eta (y-y')]}}{\eta |3c_1z_2| |3c_1z_1^2 - 3c_1z_2^2 + 3c_1z_1\xi |} \bigg |_{\xi = -2z_1, \eta = -\frac{2c_3z_1}{3c_1(z_1^2 + z_2^2)}} \nonumber \\{} & {} \quad = \frac{1}{4\pi }sgn(-c_1c_3z_1)\frac{z_1 + iz_2}{|3c_1z_2| |3c_1z_1^2 - 3c_1z_2^2 + 3c_1z_1\xi |} e^{-2iz_1[(x-x') + \frac{c_3}{3c_1(z_1^2 + z_2^2)}(y-y')]}. \end{aligned}$$
(24)

Inserting (24) into Eq. (23), we have

$$\begin{aligned} \bar{\partial }M(x,y,z)= & {} F(z_1, z_2)e^{-2iz_1[x + \frac{c_3}{3c_1(z_1^2 + z_2^2)}y]} \nonumber \\{} & {} + \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } G(x-x',y-y',z) [-3c_1v_{y'}(x',y') \bar{\partial }M(x',y',z)]dx'dy', \end{aligned}$$
(25)

where spectral data

$$\begin{aligned}{} & {} F(z_1,z_2) = \frac{1}{4\pi } \frac{sgn(-c_1c_3z_1)(z_1 + iz_2)}{|3c_1z_2| |3c_1z_1^2 - 3c_1z_2^2 + 3c_1z_1\xi |} \nonumber \\{} & {} \quad \times \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } [-3c_1v_{y'}(x',y') M(x',y',z)] e^{2iz_1[x' + \frac{c_3}{3c_1(z_1^2 + z_2^2)}y']} dx'dy'. \end{aligned}$$
(26)

Multiplying both sides of Eq. (16) by \(F(z_1,z_2)\), then subtracting Eq. (25), and we assume that the corresponding homogeneous integral equation has only zero solution, the scatting equation with the form of \(\bar{\partial }\) problem can be derived as

$$\begin{aligned} \bar{\partial }M(x,y,z) = F(z_1,z_2)N(x,y,z). \end{aligned}$$
(27)

If we can write N(x, y, z) in terms of M(x, y, z), the \(\bar{\partial }\)-problem (27) will be greatly simplified. To do this, calculate the symmetry relation of Green’s function (17)

$$\begin{aligned}{} & {} G(x-x',y-y',-\bar{z}) = \frac{1}{4\pi ^2} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } \frac{e^{i[\xi (x-x') + \eta (y-y')]}}{-3c_1 \xi \eta + \frac{c_3}{\bar{z}}\xi + 3c_1\bar{z} \eta } d\xi d\eta \nonumber \\{} & {} \quad = \frac{1}{4\pi ^2} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } \frac{e^{i[\xi (x-x') + \eta (y-y')]}}{-3c_1(\xi -2z_1)(\eta -\frac{2c_3z_1}{3c_1z_1^2+3c_1z_2^2})-\frac{c_3}{z}(\xi -2z_1)-3c_1z(\eta -\frac{2c_3z_1}{3c_1z_1^2+3c_1z_2^2})} d\xi d\eta , \end{aligned}$$
(28)

where \(\bar{z}\) is the complex conjugate of z. Making transformation \(\xi -2z_1\rightarrow \xi\), \(\eta -\frac{2c_3z_1}{3c_1z_1^2+3c_1z_2^2} \rightarrow \eta\), then the symmetry of Green’s function is obtained

$$\begin{aligned} G(x-x',y-y',-\bar{z}) = G(x-x',y-y',z)e^{2iz_1[(x-x') + \frac{c_3}{3c_1(z_1^2 + z_2^2)}(y-y')]}. \end{aligned}$$
(29)

Setting \(z\rightarrow -\bar{z}\) in Eq. (15) and using (29), we have

$$\begin{aligned}{} & {} M(x,y,-\bar{z})e^{-2iz_1[x + \frac{c_3}{3c_1(z_1^2 + z_2^2)}y]} \nonumber \\{} & {} \quad = e^{-2iz_1[x + \frac{c_3}{3c_1(z_1^2 + z_2^2)}y]}\nonumber \\{} & {} \qquad + \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } G(x-x', y-y',z)[-3c_1v_{y'}(x',y') M(x',y',-\bar{z})] e^{-2iz_1[x' + \frac{c_3}{3c_1(z_1^2 + z_2^2)}y']} dx'dy', \end{aligned}$$
(30)

then comparing Eq. (30) with Eq. (16), we can rewritten N(x, y, z) as

$$\begin{aligned} N(x,y,z) = M(x,y,-\bar{z}) e^{-2iz_1[x + \frac{c_3}{3c_1(z_1^2 + z_2^2)}y]}, \end{aligned}$$

In view of what has been discussed above, the \(\bar{\partial }\)-problem (27) is transformed into

$$\begin{aligned} \bar{\partial }M(x,y,z) = F(z_1,z_2) M(x,y,-\bar{z})e^{-2iz_1[x + \frac{c_3}{3c_1(z_1^2 + z_2^2)}y]}. \end{aligned}$$
(31)

Until now, we have connected characteristic function M(x, y, z) and \(\bar{\partial }\)-problem, what we need do next is to express M(x, y, z) by solving \(\bar{\partial }\)-problem (31).

4 Inverse Spectral Problem

The solution of inverse spectral problem (31) can be shown by Cauchy–Green formula as

$$\begin{aligned} M(x,y,z) = 1 + \frac{1}{2\pi i} \iint \frac{d\zeta \wedge d\bar{\zeta }}{\zeta - z} F(\zeta _1, \zeta _2)M(x,y,-\bar{\zeta }) e^{-2i\zeta _1(x+\frac{c_3}{3c_1 \zeta _1^2 + 3c_1\zeta _2^2}y)}, \end{aligned}$$
(32)

where \(\zeta = \zeta _1 + i\zeta _2\). Together with (15) and (32), two representations with Green’s function and \(\bar{\partial }\) of \(M-1\) are obtained

$$\begin{aligned} M-1 = {\left\{ \begin{array}{ll} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } G(x-x', y-y',z)[-3c_1v_{y'}(x',y') M(x',y',z)]dx'dy', \\ \frac{1}{2\pi i} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } \frac{d\zeta \wedge d\bar{\zeta }}{\zeta - z} F(\zeta _1, \zeta _2)M(x,y,-\bar{\zeta }) e^{-2i\zeta _1(x+\frac{c_3}{3c_1 \zeta _1^2 + 3c_1\zeta _2^2}y)}. \end{array}\right. } \end{aligned}$$
(33)

By comparing terms of the same power in \(z^{-1}\), reconstruction formula can be derived. Calculating integral (13), we have

$$\begin{aligned} G(x,y,z)= & {} \frac{1}{4\pi ^2} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } \frac{e^{i(\xi x + \eta y)}}{-3c_1 \xi \eta - \frac{c_3}{z}\xi - 3c_1z \eta } d\xi d\eta \nonumber \\= & {} -\frac{1}{12c_1z\pi ^2} \int ^{\infty }_{-\infty } \frac{e^{i\eta y}}{\eta } d\eta \int ^{\infty }_{-\infty } e^{i\xi x} d\xi + O(z^{-2})\nonumber \\= & {} -\frac{1}{6c_1z\pi } \int ^{\infty }_{-\infty } \frac{e^{i\eta y}}{\eta } d\eta \int ^{\infty }_{-\infty } \frac{e^{i\xi x}}{2\pi } d\xi + O(z^{-2}) \nonumber \\= & {} -\frac{i}{6c_1z} \delta (x)sgn(y) + O(z^{-2}). \end{aligned}$$
(34)

Inserting (34) into the first equation of (33) and take notice of \(M = 1 + O(z^{-1})\), we get

$$\begin{aligned} M - 1= & {} -\frac{i}{6c_1z} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } \delta (x-x')sgn(y-y')[-3c_1v_{y'}(x',y') M(x',y',z)]dx'dy' + O(z^{-2}) \nonumber \\= & {} -\frac{i}{6c_1z} \int ^{\infty }_{-\infty } sgn(y-y')[-3c_1v_{y'}(x,y') M(x,y',z)]dy' + O(z^{-2}) \nonumber \\= & {} -\frac{i}{6z}\left( \int ^y_{-\infty } -3v_{y'}(x,y')dy' - \int ^{\infty }_y -3v_{y'}(x,y')dy' \right) + O(z^{-2})\nonumber \\= & {} \frac{i}{z}v(x,y) + O(z^{-2}). \end{aligned}$$
(35)

We expand \(1/(\zeta - z)\) at large z in the second integral of (33) as

$$\begin{aligned} M - 1 = \frac{i}{2\pi z} \int ^{\infty }_{-\infty } \int ^{\infty }_{-\infty } d\zeta \wedge d\bar{\zeta } F(\zeta _1, \zeta _2)M(x,y,-\bar{\zeta }) e^{-2i\zeta _1(x+\frac{c_3}{3c_1 \zeta _1^2 + 3c_1\zeta _2^2}y)} + O(z^{-2}), \end{aligned}$$
(36)

then compare (35) and (36), we get reconstruction formula

$$\begin{aligned} v(x,y) = \frac{1}{2\pi } \iint d\zeta \wedge d\bar{\zeta } F(\zeta _1, \zeta _2)M(x,y,-\bar{\zeta }) e^{-2i\zeta _1(x+\frac{c_3}{3c_1 \zeta _1^2 + 3c_1\zeta _2^2}y)}. \end{aligned}$$
(37)

Further, we need to determine the time evolution of the scattering data \(F(z_1, z_2, t)\). Substituting (7) into Eq. (31), we find that

$$\begin{aligned} \bar{\partial }\psi (x,y,z,t) = F(z_1, z_2, t)\psi (x,y,t,-\bar{z}), \end{aligned}$$
(38)

then, by differentiating (38) with respect to t, we get

$$\begin{aligned}{}[\bar{\partial } \psi (z)]_t = F_t(z_1, z_2, t)\psi (-\bar{z}) + F(z_1,z_2,t) \psi _t(-\bar{z}). \end{aligned}$$
(39)

From (5), we see that \(\phi (z)\) satisfies

$$\begin{aligned}{} & {} [\bar{\partial } \psi (z)]_t = -(c_1\partial ^3_x + 3c_1 v_x \partial _x + c_2 \partial _y - \alpha ) \bar{\partial } \psi (z), \nonumber \\{} & {} \psi _t(-\bar{z}) = -(c_1\partial ^3_x + 3c_1 v_x \partial _x + c_2 \partial _y - \alpha ) \psi (-\bar{z}), \end{aligned}$$
(40)

inserting (40) into (39), we can solve \(F(z_1,z_2,t)\) with expression

$$\begin{aligned} F(z_1,z_2,t) = F(z_1,z_2)e^{-ic_1(z^3+\bar{z}^3)t + c_2 \left[ \frac{ic_3(\bar{z}-z)+ 2|z|^2\beta }{3c_1 |z|^2} \right] t}. \end{aligned}$$
(41)

Replacing \(F(\zeta _1,\zeta _2)\) with \(F(\zeta _1,\zeta _2,t)\) in (37), the potential v(x, y) is shown as

$$\begin{aligned} v(x,y) = \frac{1}{2\pi } \iint d\zeta \wedge d\bar{\zeta } F(\zeta _1, \zeta _2)M(x,y,-\bar{\zeta }) e^\vartheta , \end{aligned}$$
(42)

where \(\vartheta = \exp \left[ {-2i\zeta _1(x+\frac{c_3}{3c_1 \zeta _1^2 + 3c_1\zeta _2^2}y)} - ic_1(z^3+\bar{z}^3)t + c_2 (\frac{ic_3(\bar{z}-z)+ 2|z|^2\beta }{3c_1 |z|^2})t \right]\). Differentiating (42) with respect to x, the formal solution of (3) is obtained

$$\begin{aligned} u(x,y) = \frac{1}{2\pi } \partial _x \iint d\zeta \wedge d\bar{\zeta } F(\zeta _1, \zeta _2)M(x,y,-\bar{\zeta }) e^\vartheta . \end{aligned}$$

5 Conclusion

In this paper, we investigate generalized (2 + 1)-dimensional nonlinear wave equation based on the \(\bar{\partial }\)-problem. By using Fourier transformation and Fourier inverse transformation, we give the representation of Green’s function and characteristic functions, and further transform one of the characteristic functions into \(\bar{\partial }\)-problem by calculating \(\bar{\partial }\) derivative. After determining the time evolution of spectral data, we obtain the form solution of Eq. (3).