Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions

Abstract

A.S. Fokas has obtained integrable nonlinear partial differential equations (PDEs) in 4 + 2 dimensions by complexifying the independent variables. In this work, the complexification of the independent variables of the 2 + 1-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation yields the 4 + 2 integrable extension of the CDGKS equation. Then, by transforming two temporal variables, the CDGKS equation in three dimensions is reduced, and the Lax pairs of the corresponding equations are given. Finally, the solutions of Cauchy problems for the CDGKS equation in three spatial and two temporal dimensions are constructed by introducing a novel nonlocal d-bar formalism, in which several new long derivative operators, $D_x$, $D_y$, and $D_t$, are constructed for the study of the initial value problem for the CDGKS equation. Some significant propositions and results are presented in this paper.

1. Introduction

The well-known Korteweg–de Vries (KdV) equation is a typical integrable nonlinear evolution equation with one spatial variable, which has two integrable versions in two spatial dimensions, i.e., the Kadomtsev–Petviashvili I (KPI) and II (KPII) equations [1]. The inverse scattering transform method for solving 1 + 1-dimensional integrable evolution equations can be considered a nonlinear Fourier transform method [2,3]. In [4], the spectral analysis of the time-independent part of the Lax pair through the formulation of the Riemann–Hilbert problem allows the construction of the classical Fourier transform pair, as well as the nonlinear generalization of this pair required for the realization of the inverse scattering transform method. It has been found that the initial value problem for a large class of nonlinear equations can be solved by the inverse scattering transform method [5,6,7]. The modified Korteweg–de Vries equation [8], Schrödinger equation [9], and Camassa–Holm equation [10] have been investigated. The role of the Riemann–Hilbert problem is now referred to as the $\overline{\partial }$ (d-bar)-problem [11,12,13]. In [14,15], a more powerful $\overline{\partial }$-dressing method was proposed by Beals and Coifman for solving the $\overline{\partial }$-problem. The $\overline{\partial }$-method was first used by Ablowitz and A.S. Fokas to solve the inverse scattering problem of KPII and DSII [16]. Making use of the $\overline{\partial }$-method, a number of integrable 2 + 1-dimensional integrable nonlinear evolution equations have been studied [17,18,19]. Fokas, Mark Ablowitz, and his collaborators provided a formal method for solving the Cauchy problem of equations for 2 + 1 dimensions, i.e., two spatial and one temporal dimension. This method was later applied in several publications [20,21,22].

It has been found that the multi-dimensional equations obtained by generalization have a wide range of applications in mathematical physics, including neural network algorithms and various architectures, science and engineering, fiber optics, and fluid mechanics [23,24,25]. The derivation of multi-dimensional equations has become a focus of research in recent years. Many scholars have investigated multi-dimensional equations, especially equations with more than two spatial variables. Fokas, using complexifications of the independent variables, was able to obtain integrable nonlinear PDEs in 4 + 2 dimensions. For example, 4 + 2 generalizations of Kadomtsev–Petviashvili (KP) and Davey–Stewartson-type (DS) systems are obtained in [26]. The solutions of the Cauchy problem for 4 + 2-dimensional KP and DS equations are constructed in [27,28] by a nonlocal d-bar formalism, respectively. The solutions of the Cauchy problems for the 4 + 2 generalization of three-wave interaction equations are given in [29]. The solutions of these Cauchy problems are constructed by a novel nonlocal d-bar formalism, where the nonlocality takes the form of a two-dimensional integral: the d-bar problem involves the two-dimensional integration of $k_R$ and $k_I$, where k is a complex variable. Thus, the nonlinear Fourier transform depends on four real parameters, $l_R$, $l_I$, $k_R$, and $k_I$, where $l_R$ and $l_I$ are the real and imaginary parts of the spectral parameter l, respectively, which is consistent with the fact that the potential function u depends on four spatial variables. The construction of nonlinear evolution equations in three spatial dimensions has always been an important open problem in the field of integrability research. More importantly, recently, a breakthrough has been achieved: finally, integrable evolution equations in 3 + 1 dimensions have been constructed, and the associated initial value problem has been solved [30], which may have given the author a way to construct three-dimensional integrable equations. For example, Fokas, a well-known expert on mathematics and physics, expanded some (2 + 1)-dimensional integrable equations into the corresponding multi-dimensional integrable ones. Based on his works, we provide some rich mathematical structures by deducing the 4 + 2 integrable extension of the CDGKS equation, such as structures possessing nonlocal commutators different from (2 + 1) dimensions, extending the application scope of the d-bar method and Fourier transformations, and so on.

In this paper, we study the solutions of Cauchy problems for the CDGKS equation in three spatial and two temporal dimensions. These two extended three-dimensional integrable equations are given below, which are constructed in Section 2. The first equation,

$$\begin{array}{cc}\hfill u_t+& \alpha u_{xxxxx}+\beta u_x{u}_{xx}+\gamma u{u}_{xxx}+\delta u^2{u}_x+{\alpha }^{\prime }(u_{xxy}+i{u}_{xxz})+{\beta }^{\prime }u(u_y+i{u}_z)\hfill \\ \hfill & +{\gamma }^{\prime }{u}_x{\partial }_x^{-1}(u_y+i{u}_z)+{\delta }^{\prime }{\partial }_x^{-1}(u_{yy}-u_{zz}+2i{u}_{yz})=0,\hfill \end{array}$$

(1)

is the extended CDGKS equation in 3 + 1 dimensions, where $u(x,y,z,t)$ depends on the three spatial variables x, y, and z and on the one temporal variable t, and $\alpha$, $\beta$, $\gamma$, $\delta$, ${\alpha }^{\prime }$, ${\beta }^{\prime }$, ${\gamma }^{\prime }$, and ${\delta }^{\prime }$ are arbitrary real or complex constants. In the particular case where ${\partial }_z=0$ and some coefficients take special values, Equation (1) reduces to the (2 + 1)-dimensional CDGKS equation, which has an important role in fluid dynamics as well as in fields such as plasma physics. In [31], symmetry reductions and group-invariant solutions are studied.

The second equation,

$$\begin{array}{cc}\hfill u_t+& i{u}_{\rho }+\alpha u_{xxxxx}+\beta u_x{u}_{xx}+\gamma u{u}_{xxx}+\delta u^2{u}_x+{\alpha }^{\prime }(u_{xxy}+i{u}_{xxz})+{\beta }^{\prime }u(u_y+i{u}_z)\hfill \\ \hfill & +{\gamma }^{\prime }{u}_x{\partial }_x^{-1}(u_y+i{u}_z)+{\beta }^{\prime }{\partial }_x^{-1}(u_{yy}-u_{zz}+2i{u}_{yz})=0,\hfill \end{array}$$

(2)

is the integrable generalization of the CDGKS equation in 3 + 2 dimensions, where $u(x,y,z,t,\rho )$ depends on the three spatial variables x, y, and z and on the two temporal variables t and $\rho$. It is noted that $\rho$ in Equation (2) is defined as the second time-like variable. When ${\partial }_z={\partial }_{\rho }=0$, Equation (2) also reduces to the (2 + 1)-dimensional CDGKS equation. Since the (2 + 1)-dimensional CDGKS equation has some physical significance, Equations (1) and (2) have, of course, some potential applications. In addition, Equations (1) and (2) possess more general Lax pairs than the (2 + 1)-dimensional CDGKS equation, which can be used to obtain a richer mathematical structure. Equations (1) and (2) have some rich mathematical structures (nonlocal commutators, d-bar equations for eigenfunction $\psi$, the Green’s function G, Fourier transformations, and so on), as presented in this paper.

In this paper, by setting the time dependence of the nonlinear Fourier transform to $exp(i\omega t+i\upsilon \rho )$ and $exp(i\tilde{\omega }t+i\tilde{\upsilon }\rho )$ (where $\omega$, $\upsilon$, $\tilde{\omega }$, and $\tilde{\upsilon }$ are specific real functions of the spectral variables $l_R$, $l_I$, $\xi$, and $\rho$, defined as the second time-like variable), we study the solutions of Cauchy problems for the CDGKS equation [31] in 3 + 2 dimensions, which is obtained by reducing the 4 + 2 generalization of the CDGKS equation. The results obtained in this paper are based on some new long derivative operators, $D_x$, $D_y$, and $D_t$, which are introduced to reduce these 4 + 2-dimensional equations and study the solutions of the initial value problems for the equation in three spatial and two time dimensions.

2. Three-Dimensional Extensions of the CDGKS Equation

In reference [31], by complexifying the independent variables x, y, and t of the CDGKS equation, the following 4 + 2 integrable extension of the CDGKS equation is obtained:

$$\begin{array}{cc}\hfill 36u_{\overline{t}}=& -u_{\overline{x}\overline{x}\overline{x}\overline{x}\overline{x}}-15u_{\overline{x}}{u}_{\overline{x}\overline{x}}-15u{u}_{\overline{x}\overline{x}\overline{x}}-45u^2{u}_{\overline{x}}+5u_{\overline{x}\overline{x}\overline{y}}\hfill \\ \hfill & +15u{u}_{\overline{y}}+15u_{\overline{x}}{\partial }_{\overline{x}}^{-1}{u}_{\overline{y}}+5{\partial }_{\overline{x}}^{-1}{u}_{\overline{y}\overline{y}}=0,\hfill \end{array}$$

(3)

where

$$x=x_1+i{x}_2,y=y_1+i{y}_2,t=t_1+i{t}_2,x_1,x_2,y_1,y_2,t_1,t_2\in \mathbb{R}.$$

(4)

Equation (4) satisfy

$${\partial }_{\overline{x}}={\displaystyle \frac{1}{2}}(x_1+i{x}_2),{\partial }_{\overline{y}}={\displaystyle \frac{1}{2}}(y_1+i{y}_2),{\partial }_{\overline{t}}={\displaystyle \frac{1}{2}}(t_1+i{t}_2),$$

(5)

and

$$({\partial }_{\overline{x}}^{-1})f(x_1,x_2)={\displaystyle \frac{1}{\pi }}\int {\int }_{R^2}{\displaystyle \frac{x_1^{\prime }{x}_2^{\prime }}{x-x^{\prime }}}d{x}_1^{\prime }d{x}_2^{\prime }.$$

(6)

It is noted that the dependent variable $u(x_1,x_2,y_1,y_2,t_1,t_2)$ depends on the four spatial variables $x_1$, $x_2$, $y_1$, and $y_2$ and on the two temporal variables $t_1$ and $t_2$, and $\overline{x}$, $\overline{y}$, and $\overline{t}$ are the complex conjugates of x, y, and t, respectively.

Letting

$$\xi =a{x}_1+b{x}_2,\tau =\tilde{a}{t}_1+\tilde{b}{t}_2,$$

(7)

where $a,b,\tilde{a},\tilde{b}$ are real constants, we can obtain

$${\partial }_{\overline{x}}=A{\displaystyle \frac{\partial }{{\partial }_{\xi }}},{\partial }_{\overline{t}}=\tilde{A}{\displaystyle \frac{\partial }{{\partial }_{\tau }}},A={\displaystyle \frac{a+ib}{2}},\tilde{A}={\displaystyle \frac{\tilde{a}+i\tilde{b}}{2}}.$$

(8)

Replacing u with $Uu$ and y with $y/\overline{Y}$, where U and Y are complex constants, and using the above equation, we can see that Equation (3) has the following form:

$$\begin{array}{cc}\hfill u_{\tau }+& \alpha u_{\xi \xi \xi \xi \xi }+\beta u_{\xi }{u}_{\xi \xi }+\gamma u{u}_{\xi \xi \xi }+\delta u^2{u}_{\xi }+{\alpha }^{\prime }{u}_{\xi \xi \overline{y}}\hfill \\ \hfill & +{\beta }^{\prime }u{u}_{\overline{y}}+{\gamma }^{\prime }{u}_{\xi }{\partial }_{\xi }^{-1}{u}_{\overline{y}}+{\delta }^{\prime }{\partial }_{\xi }^{-1}{u}_{\overline{y}\overline{y}}=0,\hfill \end{array}$$

(9)

where

$$\begin{array}{cc}\hfill & \alpha ={\displaystyle \frac{A^5}{36\tilde{A}}},\beta ={\displaystyle \frac{5A^3}{12\tilde{A}}},\gamma ={\displaystyle \frac{5A^{3}U}{12\tilde{A}}},\delta ={\displaystyle \frac{5A{U}^2}{4\tilde{A}}},{\alpha }^{\prime }=-{\displaystyle \frac{5A^{2}Y}{\tilde{A}}},\hfill \\ \hfill & {\beta }^{\prime }=-{\displaystyle \frac{3UY}{\tilde{A}}},{\gamma }^{\prime }=-{\displaystyle \frac{15Y}{\tilde{A}}},{\delta }^{\prime }=-{\displaystyle \frac{Y^2}{5A\tilde{A}}}.\hfill \end{array}$$

(10)

Renaming $\tau$, $\xi$, $y_1$, $y_2$, ${\displaystyle \frac{{\beta }^{\prime }}{2}}$, and ${\displaystyle \frac{{\gamma }^{\prime }}{2}}$ to t, x, y, z, $\beta$, and $\gamma$, Equation (9) with $\tilde{a}{t}_1=2t$, $t_2=0$ becomes 3 + 1-dimensional Equation (1). And, Equation (9) with $\tilde{a}{t}_1=2t$, $t_2=2\rho$ becomes 3 + 2-dimensional Equation (2).

Equation (3) possesses the following Lax pair:

$$\begin{array}{cc}\hfill & [D_{\overline{y}}-D_{\overline{x}}^3-u{D}_{\overline{x}}]\psi =0,\hfill \\ \hfill & [12D_{\overline{t}}-3D_{\overline{x}}^5-15u{D}_{\overline{x}}^3-15u_{\overline{x}}{D}_{\overline{x}}^2-(5{\partial }_{\overline{x}}^{-1}{u}_{\overline{y}}+10u_{\overline{x}\overline{x}}+15u^2)D_{\overline{x}}]\psi =0,\hfill \end{array}$$

(11)

where $D_{\overline{x}}={\partial }_{\overline{x}}+l$, $D_{\overline{y}}={\partial }_{\overline{y}}+l^3$, $D_{\overline{t}}={\partial }_{\overline{t}}+l^5$. Through the series of changes of variables mentioned above, Equation (3) becomes

$$\begin{array}{c}\hfill [{\displaystyle \frac{Y}{A^3}}{\partial }_{\overline{y}}-{\partial }_{\xi }-3l{\partial }_{\xi }^2-3l^2{\partial }_{\xi }-{\displaystyle \frac{U}{A^2}}u({\partial }_{\xi }+l)]\psi =0,\end{array}$$

(12)

and by replacing l with $Al$, we obtain Equation (9) as follows:

$$\begin{array}{c}\hfill [-{\displaystyle \frac{36{\beta }^{\prime }}{5\alpha }}{\partial }_{\overline{y}}-{\partial }_{\xi }-3l{\partial }_{\xi }^2-3l^2{\partial }_{\xi }-{({\displaystyle \frac{{\alpha }^{\prime }}{45\alpha }})}^{{\displaystyle \frac{1}{2}}}u({\partial }_{\xi }+l)]\psi =0.\end{array}$$

(13)

Then, the t-dependent part of the Lax pair of Equation (3) becomes

$$\begin{array}{c}\hfill [{\partial }_{\tau }-9\alpha {({\partial }_{\xi }+l)}^5+3\gamma u{({\partial }_{\xi }+l)}^3-3\gamma u_{\xi }{({\partial }_{\xi }+l)}^2-({\displaystyle \frac{2{\gamma }^2}{\beta }}{u}_{\xi \xi }-{\displaystyle \frac{108{\beta }^{\prime }\gamma }{\alpha }}{\partial }_{\xi }^{-1}{u}_{\overline{y}}+{\displaystyle \frac{{\gamma }^2}{5\alpha }}{u}^2)({\partial }_{\xi }+l)]\psi =0.\end{array}$$

(14)

Renaming $\tau$, $\xi$, $y_1$, and $y_2$ to t, x, y, and z and renaming ${\displaystyle \frac{{\beta }^{\prime }}{2}}$, ${\displaystyle \frac{{\gamma }^{\prime }}{2}}$ to $\beta$, $\gamma$, we can obtain the Lax pair of integrable 3 + 1 Equation (1):

$$\begin{array}{c}\hfill [-{\displaystyle \frac{36{\beta }^{\prime }}{5\alpha }}({\partial }_y+i{\partial }_z)-{\partial }_x-3l{\partial }_x^2-3l^2{\partial }_x-{({\displaystyle \frac{{\alpha }^{\prime }}{45\alpha }})}^{{\displaystyle \frac{1}{2}}}u({\partial }_x+l)]\psi =0\end{array}$$

(15)

and

$$\begin{array}{cc}\hfill [{\partial }_t& -9\alpha {({\partial }_x+l)}^5+3\gamma u{({\partial }_x+l)}^3-3\gamma u_x{({\partial }_x+l)}^2\hfill \\ \hfill & -({\displaystyle \frac{2{\gamma }^2}{\beta }}{u}_{xx}-{\displaystyle \frac{108{\beta }^{\prime }\gamma }{\alpha }}{\partial }_x^{-1}(u_y+i{u}_z)+{\displaystyle \frac{{\gamma }^2}{5\alpha }}{u}^2)({\partial }_x+l)]\psi =0.\hfill \end{array}$$

(16)

Similarly, the Lax pair of integrable 3 + 2 Equation (2) is constructed as follows:

$$\begin{array}{c}\hfill [-{\displaystyle \frac{36{\beta }^{\prime }}{5\alpha }}({\partial }_y+i{\partial }_z)-{\partial }_x-3l{\partial }_x^2-3l^2{\partial }_x-{({\displaystyle \frac{{\alpha }^{\prime }}{45\alpha }})}^{{\displaystyle \frac{1}{2}}}u({\partial }_x+l)]\psi =0\end{array}$$

(17)

and

$$\begin{array}{cc}\hfill [{\partial }_t& +i{\partial }_{\rho }-9\alpha {({\partial }_x+l)}^5+3\gamma u{({\partial }_x+l)}^3-3\gamma u_x{({\partial }_x+l)}^2\hfill \\ \hfill & -({\displaystyle \frac{2{\gamma }^2}{\beta }}{u}_{xx}-{\displaystyle \frac{108{\beta }^{\prime }\gamma }{\alpha }}{\partial }_x^{-1}(u_y+i{u}_z)+{\displaystyle \frac{{\gamma }^2}{5\alpha }}{u}^2)({\partial }_x+l)]\psi =0.\hfill \end{array}$$

(18)

3. Spectral Analysis of the $\mathit{t}$-Independent Part of the Lax Pair

Next, we focus on the spectral analysis of the following equation:

$$\begin{array}{c}\hfill [{\partial }_y+i{\partial }_z-{\partial }_x^3-3l{\partial }_x^2-3l^2{\partial }_x-u({\partial }_x+l)]\psi =0,\end{array}$$

(19)

where $x,y,z\in \mathbb{R}$, and $k\in \mathbb{C}$. Then, Equation (19) can be generated from Equation (15) by replacing $-{\displaystyle \frac{36{\beta }^{\prime }}{5\alpha }}y$, $-{\displaystyle \frac{36{\beta }^{\prime }}{5\alpha }}z$, and ${({\displaystyle \frac{{\alpha }^{\prime }}{45\alpha }})}^{{\displaystyle \frac{1}{2}}}u$ with y, z, and u, respectively.

Here, we assume that $u(x,y,z)\to 0$ as $\left|x\right|$, $\left|y\right|$, $\left|z\right|$$\to \infty$. The general solution of Equation (19) is the convolution of $G(x,y,z,l)$ and $u(x,y,z)({\partial }_x+l)\psi (x,y,z,l)$, i.e.,

$$\psi (x,y,z,l)=1+{\int }_{R^3}G(x-x^{\prime },y-y^{\prime },z-z^{\prime },l)u(x^{\prime },y^{\prime },z^{\prime })({\partial }_{x^{\prime }}+l)\psi (x^{\prime },y^{\prime },z^{\prime },l)d{x}^{\prime }d{y}^{\prime }d{z}^{\prime }.$$

(20)

By calculating the $\overline{\partial }$-derivative of (20), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \psi }{\partial \overline{l}}}(x,y,z,l)& ={\int }_{R^3}[\overline{\partial }G(x-x^{\prime },y-y^{\prime },z-z^{\prime },l)]u(x^{\prime },y^{\prime },z^{\prime })({\partial }_{x^{\prime }}+l)\psi (x^{\prime },y^{\prime },z^{\prime },l)d{x}^{\prime }d{y}^{\prime }d{z}^{\prime }\hfill \\ \hfill & +{\int }_{R^3}G(x-x^{\prime },y-y^{\prime },z-z^{\prime },l)u(x^{\prime },y^{\prime },z^{\prime })\overline{\partial }({\partial }_{x^{\prime }}+l)\psi (x^{\prime },y^{\prime },z^{\prime },l)d{x}^{\prime }d{y}^{\prime }d{z}^{\prime }\hfill \\ \hfill & =H(x,y,z,l)\hfill \\ \hfill & +{\int }_{R^3}G(x-x^{\prime },y-y^{\prime },z-z^{\prime },l)u(x^{\prime },y^{\prime },z^{\prime })\overline{\partial }({\partial }_{x^{\prime }}+l)\psi (x^{\prime },y^{\prime },z^{\prime },l)d{x}^{\prime }d{y}^{\prime }d{z}^{\prime },\hfill \end{array}$$

(21)

where $H(x,y,z,l)$ reads

$$H(x,y,z,l)={\int }_{R^3}[\overline{\partial }G(x-x^{\prime },y-y^{\prime },z-z^{\prime },l)]u(x^{\prime },y^{\prime },z^{\prime })({\partial }_{x^{\prime }}+l)\psi (x^{\prime },y^{\prime },z^{\prime },l)d{x}^{\prime }d{y}^{\prime }d{z}^{\prime }.$$

(22)

Then, we have that

$$G_y+i{G}_z-G_{xxx}-3l{G}_{xx}-3l^2{G}_x=\delta (x)\delta (y)\delta (z),$$

(23)

$$G(x,y,z,l)={\displaystyle \frac{1}{{(2\pi )}^3}}{\int }_{R^3}{\displaystyle \frac{e^{i(\xi x+\eta y+\zeta z)}}{i{\xi }^3+3l{\xi }^2-3il\xi +i\eta -\zeta }}d\xi d\eta d\zeta ,$$

(24)

$$\widehat{G}(x,y,z,l)={\displaystyle \frac{1}{{(2\pi )}^{\frac{3}{2}}}}{\displaystyle \frac{1}{i{\xi }^3+3l{\xi }^2-3il\xi +i\eta -\zeta }},$$

(25)

using the identity of multivariate $\delta$ functions

$$\delta (x)\delta (y)\delta (z)={\displaystyle \frac{1}{{(2\pi )}^3}}{\int }_{R^3}{e}^{i(\xi x+\eta y+\zeta z)}d\xi d\eta d\zeta ,$$

(26)

where l is an arbitrary complex number $l=l_1+i{l}_2$.

To compute $\partial G/\partial \overline{l}$, we find

$$i{\xi }^3+3l{\xi }^2-3il\xi +i\eta -\zeta =-3i\xi [l^2+il\xi -{\xi }^2/3-(i\zeta +\eta )/3\xi ].$$

(27)

This equation with the identity

$${\displaystyle \frac{\partial }{\partial \overline{l}}}({\displaystyle \frac{1}{l-\tilde{l}}})=\pi \delta (l_R-{\tilde{l}}_R)\delta (l_I-{\tilde{l}}_I),l=l_R+i{l}_I,\tilde{l}={\tilde{l}}_R+i{\tilde{l}}_I,$$

(28)

yields the formula

$${\displaystyle \frac{\partial G}{\partial \overline{l}}}(x,y,z,l)=-{\displaystyle \frac{3}{8i{\pi }^2}}{\int }_{R^3}\xi e^{i[\xi x+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(3l_1{\xi }^2+6\xi l_1{l}_2)z]}d\xi .$$

(29)

We obtain $H(x,y,z,l)$ as

$$H(x,y,z,l)={\int }_R\xi \widehat{u}(\xi ,l)e^{i[\xi x+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(3l_1{\xi }^2+6\xi l_1{l}_2)z]}d\xi ,$$

(30)

where $\widehat{u}(\xi ,l)$ satisfies

$$\begin{array}{cc}\hfill \widehat{u}(\xi ,l)=& -{\displaystyle \frac{3}{8i{\pi }^2}}{\int }_{R^3}\xi e^{-i[-\xi x^{\prime }+({\xi }^3+3{\xi }^2{l}_2+3\xi l_2^2-3\xi l_1^2)y^{\prime }-(3l_1{\xi }^2+6\xi l_1{l}_2)z^{\prime }]}\hfill \\ \hfill & \times u(x^{\prime },y^{\prime },z^{\prime })({\partial }_{x^{\prime }}+l)\psi (x^{\prime },y^{\prime },z^{\prime },l)d{x}^{\prime }d{y}^{\prime }d{z}^{\prime }.\hfill \end{array}$$

(31)

In order to define a classical d-bar problem, we try to find the relationship between $\psi$ and $\partial \psi /\partial \overline{l}$. In Equation (20), replacing $\psi (x,y,z,l)$ with $\psi (x,y,z,l+i\xi )$ and multiplying by $e^{i[\xi x+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(3{\xi }^2{l}_1+6\xi l_1{l}_2)z]}$ obtains

$$\begin{array}{cc}\hfill & e^{i[\xi x+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(3{\xi }^2{l}_1+6\xi l_1{l}_2)z]}\psi (x,y,z,l+i\xi )\hfill \\ \hfill & =e^{i[\xi x+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(3l_1{\xi }^2+6\xi l_1{l}_2)z]}\hfill \\ \hfill & +{\int }_{R^3}\breve{G}(x-x^{\prime },y-y^{\prime },z-z^{\prime },l,\xi )u(x^{\prime },y^{\prime },z^{\prime })({\partial }_{x^{\prime }}+l)\psi (x^{\prime },y^{\prime },z^{\prime },l+i\xi )\hfill \\ \hfill & \times e^{-i[\xi x^{\prime }+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y^{\prime }+(3l_1{\xi }^2+6\xi l_1{l}_2)z^{\prime }]}d{x}^{\prime }d{y}^{\prime }d{z}^{\prime },\hfill \end{array}$$

(32)

where

$$\breve{G}(x,y,z,l,\xi )={\displaystyle \frac{1}{{(2\pi )}^3}}{\int }_{R^3}{\displaystyle \frac{e^{i[(\tilde{\xi }+\xi )x+(\tilde{\eta }-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(\tilde{\zeta }+l_1{\xi }^2+6\xi l_1{l}_2)z]}}{i{\tilde{\xi }}^3+3l{\tilde{\xi }}^2+3i\xi {\tilde{\xi }}^2+3i{\xi }^2\tilde{\xi }-3i{l}^2\tilde{\xi }+6l\xi \tilde{\xi }+i\tilde{\eta }-\tilde{\zeta }}}d\tilde{\xi }d\tilde{\eta }d\tilde{\zeta }.$$

(33)

By the transformation

$$\breve{\xi }=\tilde{\xi }+\xi ,\breve{\eta }=\tilde{\eta }-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2,\breve{\zeta }=\tilde{\zeta }+l_1{\xi }^2+6\xi l_1{l}_2,$$

(34)

we further obtain

$$\breve{G}(x,y,z,l,\xi )={\displaystyle \frac{1}{{(2\pi )}^3}}{\int }_{R^3}{\displaystyle \frac{e^{i(\breve{\xi }x+\breve{\eta }y+\breve{\zeta }z)}}{i{\breve{\xi }}^3+3l{\breve{\xi }}^2-3i{l}^2\breve{\xi }+i\breve{\eta }-\breve{\zeta }}}d\breve{\xi }d\breve{\eta }d\breve{\zeta }=G(x,y,z,l).$$

(35)

For Equation (32), by replacing $\breve{G}$ with G, multiplying the resulting equation by $\xi \widehat{u}(\xi ,l)$, and integrating over $\xi$, the relationship between the functions $\psi$ and $\partial \psi /\partial \overline{l}$ can be constructed:

$${\displaystyle \frac{\partial \psi }{\partial \overline{l}}}(x,y,z,l)={\int }_R\widehat{u}(\xi ,l)e^{i[\xi x+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(3l_1{\xi }^2+6\xi l_1{l}_2)z]}\psi (x,y,z,l+i\xi )d\xi ,$$

(36)

where $\psi (x,y,z,l)\to 1$ as $l\to \infty$. The solution of this d-bar problem (36) is presented as follows:

$$\psi (x,y,z,l)=1-{\displaystyle \frac{1}{\pi }}{\int }_{R^3}{\displaystyle \frac{d\xi d{k}_{R}d{k}_I}{k-l}}\xi e^{i[\xi x+(-{\xi }^3-3{\xi }^2{k}_2-3\xi k_2^2+3\xi k_1^2)y+(3k_1{\xi }^2+6\xi k_1{k}_2)z]}\widehat{u}(\xi ,k)\psi (x,y,z,k+i\xi ),$$

(37)

and $l\in \mathbb{C}$, $k=k_1+i{k}_2$.

We can expand $\psi (x,y,z,l)$ into the following series:

$$\psi (x,y,z,l)=1+{\displaystyle \frac{{\psi }_1(x,y,z)}{l}}+O(l^{-2}),l\to \infty ,$$

(38)

where

$${\psi }_1(x,y,z)={\displaystyle \frac{1}{\pi }}{\int }_{R^3}\xi e^{i[\xi x+(-{\xi }^3-3{\xi }^2{k}_2-3\xi k_2^2+3\xi k_1^2)y+(3k_1{\xi }^2+6\xi k_1{k}_2)z]}\widehat{u}(\xi ,k)\psi (x,y,z,k+i\xi )d\xi d{k}_{R}d{k}_I.$$

(39)

Putting Equation (38) into (19), we further have

$$\begin{array}{cc}\hfill u(x,y,z)& =-3{({\psi }_1(x,y,z))}_x.\hfill \end{array}$$

(40)

Therefore,

$$u(x,y,z)=-{\displaystyle \frac{3}{\pi }}{\partial }_x{\int }_{R^3}\xi e^{i[\xi x+(-{\xi }^3-3{\xi }^2{k}_2-3\xi k_2^2+3\xi k_1^2)y+(3k_1{\xi }^2+6\xi k_1{k}_2)z]}\widehat{u}(\xi ,k)\psi (x,y,z,k+i\xi )d\xi d{k}_{R}d{k}_I.$$

(41)

In conclusion, the nonlinear Fourier transform pairs (both direct and inverse transforms) are constructed by spectral analysis of Equation (19), where the direct transform is given by Equation (31), with $\psi$ defined in terms of u by Equation (20), and the inverse transform is given by Equation (41), with $\psi$ defined in terms of $\widehat{u}$ by Equation (37).

In the linear limit of u, $\psi \to 1$, the above nonlinear Fourier transform pair becomes

$$\widehat{u}(\xi ,l)=-{\displaystyle \frac{3}{8i{\pi }^2}}{\int }_{R^3}\xi e^{i[\xi x+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(3l_1{\xi }^2+6\xi l_1{l}_2)z]}u(x,y,z)({\partial }_x+l)dxdydz,$$

(42)

$$\begin{array}{c}\hfill u(x,y,z)=-{\displaystyle \frac{3}{\pi }}{\partial }_x{\int }_{R^3}{e}^{i[\xi x+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(3l_1{\xi }^2+6\xi l_1{l}_2)z]}\widehat{u}(\xi ,l)d\xi d{l}_{R}d{l}_I.\end{array}$$

(43)

By the variable transformations $l_1=\xi$, $l_2=-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2$, and $l_3=3l_1{\xi }^2+6\xi l_1{l}_2$, the above transform pair maps to the usual three-dimensional Fourier transform pair.

$$\widehat{u}(\xi ,l)=-{\displaystyle \frac{3}{8i{\pi }^2}}{\int }_{R^3}\xi e^{i[l_{1}x+l_{2}y+l_{3}z]}u(x,y,z)({\partial }_x+l)dxdydz,$$

(44)

$$\begin{array}{c}\hfill u(x,y,z)=-{\displaystyle \frac{3}{\pi }}{\partial }_x{\int }_{R^3}{e}^{i[l_{1}x+l_{2}y+l_{3}z]}\widehat{u}(\xi ,l)d\xi d{l}_{R}d{l}_I.\end{array}$$

(45)

4. The Solution of the Cauchy Problem for Equation (48)

In this section, we study the solution of the Cauchy problem for the 3 + 2-dimensional CDGKS equation. First, some propositions and remarks are given.

Proposition 1. 

Given the complex-valued function $u(x,y,z)\in S({\mathbb{R}}^3)$, whose $L_1$ and $L_2$ norms are sufficiently small, let ${\psi }_0(x,y,z,l)$ denote the solution of Equation (20), where u is replaced by $u_0$. Let ${\widehat{u}}_0(\xi ,l)$ be defined by the right-hand side of Equation (31), with u and ψ replaced by $u_0$ and ${\psi }_0$, respectively. Let $\psi (x,y,z,t,\rho ,l)$ be the solution of Equation (37), with $\widehat{u}(\xi ,l)$ replaced by

$$\widehat{u}(\xi ,l,t,\rho )={\widehat{u}}_0(\xi ,l)e^{i(\omega t+\upsilon \rho )}$$

(46)

and

$$\begin{array}{cc}\hfill & \omega ={\displaystyle \frac{1}{28}}[{\xi }^5+5l_2{\xi }^4-10(l_1^2-l_2^2-2){\xi }^3+10(l_2^3-3l_1^2{l}_2+3l_2){\xi }^2+5(l_1^4+l_2^4-6l_1^2{l}_2^2+30l_2^2-30l_1^2)\xi ],\hfill \\ \hfill & \upsilon ={\displaystyle \frac{5}{9}}[-l_1{\xi }^4-4l_1{l}_2{\xi }^3+2(l_2^3-3l_1{l}_2^2+3l_1){\xi }^2+4(l_1^3{l}_2-l_1{l}_2^2+3l_1{l}_2)\xi ].\hfill \end{array}$$

(47)

Let $u(x,y,z,t,\rho )$ be defined by the right-hand side of Equation (41), with $\widehat{u}(\xi ,l)$ and $\psi (x,y,z,l)$ replaced by $\widehat{u}(\xi ,l,t,\rho )$ and $\psi (x,y,z,t,\rho ,l)$, respectively. Then, $u(x,y,z,t,\rho )$ satisfies

$$\begin{array}{cc}\hfill 36u_t& +i{u}_{\rho }+u_{xxxxx}+15u_x{u}_{xx}+15u{u}_{xxx}+45u^2{u}_x-5(u_{xxy}+i{u}_{xxz})\hfill \\ \hfill & -15u(u_y+i{u}_z)-15u_x{\partial }_x^{-1}(u_y+i{u}_z)-5{\partial }_x^{-1}(u_{yy}-u_{zz}+2i{u}_{yz})=0,\hfill \end{array}$$

(48)

with

$$u(x,y,z,0,0)=u_0(x,y,z).$$

(49)

Proof. 

The equation satisfied by u is equivalent to the d-bar problem:

$${\displaystyle \frac{\partial \psi }{\partial \overline{l}}}(l)={\int }_R\xi \mathrm{\Gamma }(l,\xi )\psi (l+i\xi ){\widehat{u}}_0(\xi ,l)d\xi ,$$

(50)

and

$$\psi (l)=1+{\displaystyle \frac{{\psi }_1}{l}}+O(l^{-2}),l\to \infty ,$$

(51)

where $\mathrm{\Gamma }(l,\xi )$ is defined by

$$\mathrm{\Gamma }(l,\xi )=e^{i[\xi x+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(3l_1{\xi }^2+6\xi l_1{l}_2)z+\omega t+\upsilon \rho ]}.$$

(52)

For convenience of notation, we have suppressed the x, y, z, t, and $\rho$ dependence of $\psi$ and $\mathrm{\Gamma }$.

In what follows, we will use the dressing method to prove that if $\psi$ satisfies (50) and u is defined in terms of $\psi$ by

$$u(x,y,z,t,\rho )=-3({\psi }_1{(x,y,z,t,\rho )}_x,$$

(53)

or equivalently via Equation (41), with $\widehat{u}(\xi ,l)$ and $\psi (x,y,z,l)$ replaced by $\widehat{u}(\xi ,l,t,\rho )$ and $\psi (x,y,z,t,\rho ,l)$, respectively, then u solves Equation (48).

Next, some long derivative operators are introduced:

$$D_x={\partial }_x+l,D_y={\partial }_y+i{\partial }_z+l^3,D_t={\partial }_t+i{\partial }_{\rho }+l^5.$$

(54)

We find that these operators are interchangeable. For example,

$${\displaystyle \frac{\partial }{\partial \overline{l}}}(D_t\psi (l))={\int }_R\mathrm{\Gamma }(l,\xi )D_t\psi (l+i\xi )d\chi ,d\chi =\xi {\widehat{u}}_0(\xi ,l)d\xi ,$$

(55)

which satisfies the following identity:

$$i\omega -\upsilon +l^5={(l+i\xi )}^5.$$

(56)

In order to verify Equation (55), we have

$${\displaystyle \frac{\partial }{\partial \overline{l}}}({\psi }_t(l))={\int }_R\mathrm{\Gamma }(l,\xi )[i\omega \psi (l+i\xi )+{\psi }_t(l+i\xi )]d\chi ,$$

(57)

$${\displaystyle \frac{\partial }{\partial \overline{l}}}(i{\psi }_{\rho }(l))={\int }_R\mathrm{\Gamma }(l,\xi )[-\upsilon \psi (l+i\xi )+i{\psi }_{\rho }(l+i\xi )]d\chi ,$$

(58)

$${\displaystyle \frac{\partial }{\partial \overline{l}}}(l^5\psi (l))={\int }_R\mathrm{\Gamma }(l,\xi )l^5\psi (l+i\xi )d\chi .$$

(59)

Adding these three equations yields Equation (55). Similarly, we further obtain

$${\displaystyle \frac{\partial }{\partial \overline{l}}}(D_x\psi (l))={\int }_R\mathrm{\Gamma }(l,\xi )D_x\psi (l+i\xi )d\chi ,$$

(60)

and

$${\displaystyle \frac{\partial }{\partial \overline{l}}}(D_y\psi (l))={\int }_R\mathrm{\Gamma }(l,\xi )D_y\psi (l+i\xi )d\chi .$$

(61)

In order to ensure the validity of the property in (61), we obtain that

$$i\omega -\upsilon +l^3={(l+i\xi )}^3.$$

(62)

Therefore, the function ${\mathrm{\Theta }}_1(l)$ is defined by

$$(D_y-D_x^3-u{D}_x)\psi =0,$$

(63)

which forms the $(t,\rho )$-independent part of the Lax pair for Equation (48). We have suppressed the x, y, z, t, and $\rho$ dependence of ${\mathrm{\Theta }}_1(l)$, satisfying Equation (50). When ${\mathrm{\Theta }}_1(l)=O(1/l)$, $\left|l\right|\to \infty$, that is, ${\mathrm{\Theta }}_1(l)\to 0$. If u from Equation (53), then this is indeed the case.

By defining the $D_t$ operator in (54), the $(t,\rho )$-dependent part ${\mathrm{\Theta }}_2(l)$ of the Lax pair for Equation (48) is constructed:

$$(3D_t-3D_x^5+U_1{D}_x^3+U_2{D}_x^2+U_3{D}_x)\psi =0,$$

(64)

where $U_i(i=1,2,3)$ are functions of x, y, z, t, and $\rho$. The terms of $O(l^{-1})$ and $O(l^0)$ of the equation ${\mathrm{\Theta }}_1(l)$ present

$$u{\psi }_1+3{\psi }_{1xx}+3{\psi }_{2x}=0,$$

(65)

$${\psi }_{1y}+i{\psi }_{1z}-{\psi }_{1xxx}-3{\psi }_{2xx}-3{\psi }_{3x}-u{\psi }_{1x}-u{\psi }_2=0.$$

(66)

Combining Equations (65) and (66), we obtain that

$${\psi }_{3x}=-{\displaystyle \frac{1}{9}}{\partial }_x^{-1}{u}_y-{\displaystyle \frac{i}{9}}{\partial }_x^{-1}{u}_z-{\displaystyle \frac{2}{9}}{u}_{xx}-{\displaystyle \frac{1}{9}}{u}^2-{\displaystyle \frac{1}{9}}{u}_x{\partial }_x^{-1}u-{\displaystyle \frac{1}{27}}u{\partial }_x^{-1}(u{\partial }_x^{-1}u).$$

(67)

The terms of $O(l^1)$, $O(l^2)$, and $O(l^3)$ of the equation ${\mathrm{\Theta }}_2(l)$ satisfy

$$\begin{array}{cc}\hfill & U_1-15{\psi }_{1x}=0,\hfill \\ \hfill & U_2-30{\psi }_{1xx}-15{\psi }_{2x}+U_1{\psi }_1=0,\hfill \\ \hfill & U_3-30{\psi }_{1xxx}-3U_1{\psi }_{1x}-30{\psi }_{2xx}-15{\psi }_{3x}+U_2{\psi }_1+U_1{\psi }_2.\hfill \end{array}$$

(68)

By reorganizing Equation (68), we can obtain

$$\begin{array}{cc}\hfill & U_1=-5u,\hfill \\ \hfill & U_2=-5u_x\hfill \\ \hfill & U_3=-{\displaystyle \frac{5}{3}}{\partial }_x^{-1}{u}_y-{\displaystyle \frac{5}{3}}i{\partial }_x^{-1}{u}_z-{\displaystyle \frac{10}{3}}{u}_{xx}-{\displaystyle \frac{5}{3}}{u}^2.\hfill \end{array}$$

(69)

□

Equation (48) contains the $(t,\rho )$-dependent part and $(t,\rho )$-independent part of the Lax pair to be determined, and the compatibility of the Lax pair yields Equation (48).

Remark 1. 

An important point is that the exponential $exp[i\xi x+i(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+i(3l_1{\xi }^2+6\xi l_1{l}_2)+i\omega t+i\upsilon \rho ]$ meets the linear form of Equation (48); this equation can be solved numerically by letting $u=ϵ+i\lambda$. By solving this equation and making the real and imaginary parts of the equation zero, two equations can be derived; thus, ω and υ (47) are defined. The main roles of the long derivative operators $D_x$, $D_y$, and $D_t$ lie in conveniently obtaining the Lax pairs of the (3+1)-dimensional equation form, where the d-bar equations for eigenfunction ψ and the Green’s function G can be expressed by the variable l containing in the long derivatives.

Remark 2. 

If the initial condition $u_0$ satisfies the relation

$$\overline{u_0(x,y,z)}=u_0(x,y,-z),$$

(70)

then the solution of Equation (48) satisfies the following relation:

$$\overline{u(x,y,z,t,\rho )}=u(x,y,-z,t,-\rho ).$$

(71)

Proof. 

Substituting $\xi \to -\xi$, $\eta \to -\eta$ into (24) results in

$$\begin{array}{cc}\hfill G(x,y,-z,\overline{l})& ={\displaystyle \frac{1}{{(2\pi )}^3}}{\int }_{R^3}{\displaystyle \frac{e^{i(-\xi x-\eta y-\zeta z)}}{-i{\xi }^3-3\overline{l}{\xi }^2+3i\overline{l}\xi -i\eta -\zeta }}d\xi d\eta d\zeta =\overline{G(x,y,z,l)}.\hfill \end{array}$$

(72)

Letting $z\to -z$ NS $l\to \overline{l}$ in Equation (20) and utilizing (70) and (72), Equation (20) together with the assumption on $u_0$ satisfy

$$\begin{array}{cc}\hfill \psi (x,y,-z,\overline{l})& =1+{\int }_{R^3}G(x-x^{\prime },y-y^{\prime },z^{\prime }-z,\overline{l})\psi (x^{\prime },y^{\prime },-z^{\prime },\overline{l})\hfill \\ \hfill & \times [u_0(x^{\prime },y^{\prime },-z^{\prime })({\partial }_{x^{\prime }}+\overline{l})]d{x}^{\prime }d{y}^{\prime }d{z}^{\prime }\hfill \\ \hfill & =1+{\int }_{R^3}\overline{G(x-x^{\prime },y-y^{\prime },z-z^{\prime },l)}\psi (x^{\prime },y^{\prime },-z^{\prime },\overline{l})\hfill \\ \hfill & \times [\overline{u_0(x^{\prime },y^{\prime },-z^{\prime })}({\partial }_{x^{\prime }}+l)]d{x}^{\prime }d{y}^{\prime }d{z}^{\prime }\hfill \\ \hfill & =\overline{\psi (x,y,z,l)}.\hfill \end{array}$$

(73)

Then, putting $\xi \to -\xi$, $z\to -z$, and $l\to \overline{l}$ into Equation (31) implies

$$\begin{array}{cc}\hfill -\widehat{u}(-\xi ,\overline{l})& ={\displaystyle \frac{3}{8i{\pi }^2}}{\int }_{R^3}\xi e^{-i[\xi x^{\prime }+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y^{\prime }+(3l_1{\xi }^2+6\xi l_1{l}_2)z^{\prime }]}\hfill \\ \hfill & \times u(x^{\prime },y^{\prime },-z^{\prime })({\partial }_{x^{\prime }}+\overline{l})\psi (x^{\prime },y^{\prime },-z^{\prime },\overline{l})d{x}^{\prime }d{y}^{\prime }d{z}^{\prime }=\overline{\widehat{u}(\xi ,l)}.\hfill \end{array}$$

(74)

Substituting $\xi \to -\xi$, $z\to -z$, $l\to \overline{l}$, and $\rho \to -\rho$ into Equation (50), we then obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \psi (x,y,-z,t,-\rho ,\overline{l})}{\partial \overline{l}}}& ={\int }_R(-\xi ){\widehat{u}}_0(-\xi ,\overline{l})e^{i[-\xi x+({\xi }^3-3{\xi }^2{l}_2+3\xi l_2^2-3\xi l_1^2)y-(3l_1{\xi }^2+6\xi l_1{l}_2)z-\omega t-\upsilon \rho ]}\hfill \\ \hfill & \times \psi (x,y,-z,\overline{l}-i\xi )d\xi .\hfill \end{array}$$

(75)

The conjugate of Equation (50) satisfies

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \overline{\psi (x,y,z,t,\rho ,l)}}{\partial \overline{l}}}& ={\int }_R\xi \overline{{\widehat{u}}_0(\xi ,l)}{e}^{-i[\xi x+(-{\xi }^3-3{\xi }^2{l}_2-3\xi l_2^2+3\xi l_1^2)y+(3l_1{\xi }^2+6\xi l_1{l}_2)z+\omega t+\upsilon \rho ]}\hfill \\ \hfill & \times \overline{\psi (x,y,z,l+i\xi )}d\xi .\hfill \end{array}$$

(76)

Combining (75) and (76) and utilizing Equation (74), the following symmetry relation is obtained:

$$\begin{array}{c}\hfill \overline{\psi (x,y,z,t,\rho ,l)}=\psi (x,y,-z,t,-\rho ,\overline{l}).\end{array}$$

(77)

Therefore, the definition of u implies that u satisfies Equation (71). □

5. Conclusions and Discussion

In this paper, we generalized the 2 + 1-dimensional CDGKS equation to three spatial dimensions, i.e., 3 + 1 and 3 + 2 dimensions, and constructed the Lax pairs for the equation in three spatial dimensions. The solutions of Cauchy initial value problems for the CDGKS equation in 3 + 2 dimensions were investigated by a novel nonlocal d-bar formalism different from the classical d-bar method. Several propositions given in this paper yield the main result, which provides a nonlocal d-bar formalism for solving the initial value problems for these high-dimensional equations.

In the future, other solutions, such as multi-solitons and multi-breathers associated with the 3 + 1 and 3 + 2 nonlinear equations introduced in this paper, will be studied. At the same time, we will discuss initial value problems for integrable extensions of other types of high-dimensional equations [32,33,34,35,36,37], which play an important role in many areas of mathematical physics. The possibility of studying the long-time asymptotics of high-dimensional equations according to the method obtained by the authors in [38] is one of the planned future research problems. The methods and propositions presented in this paper may provide inspiration for dealing with similar high-dimensional nonlinear equations.