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The Dbar-dressing method for the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

  • Shifei Sun ,
  • Biao Li *
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  • School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, China

Author to whom any correspondence should be addressed.

Received date: 2023-09-25

  Revised date: 2023-12-01

  Accepted date: 2023-12-07

  Online published: 2024-01-19

Copyright

© 2024 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

In this work, the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation is studied by means of the $\bar{{\boldsymbol{\partial }}}$-dressing method. A new $\bar{{\boldsymbol{\partial }}}$ problem has been constructed by analyzing the characteristic function and the Green's function of its Lax representation. Based on solving the $\bar{\partial }$ equation and choosing the proper spectral transformation, the solution of the DJKM equation is constructed. Furthermore, the more general solution of the DJKM equation can be also obtained by ensuring the evolution of the time spectral data.

Cite this article

Shifei Sun , Biao Li . The Dbar-dressing method for the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation[J]. Communications in Theoretical Physics, 2024 , 76(1) : 015003 . DOI: 10.1088/1572-9494/ad1324

1. Introduction

Nonlinear evolution equations (NLEEs) have been widely studied as integrable models that can describe some nonlinear phenomena and analyze various properties in different fields, such as plasma physics, fluid, solid-state materials and nonlinear optics. It is always an important research topic when finding a method to solve integrable models extensively.
With the in-depth study of nonlinear systems, there have been many mature methods to study the important properties of nonlinear systems. such as Darboux transformation [1, 2], Hirota bilinear method [35], Bäcklund transformation [6, 7], Riemann–Hilbert method [813], Painlevé analysis [14, 15] and algebra-geometric method [16, 17]. These methods have made great contributions to the study of nonlinear system solutions, integrability, symmetry, scattering problems and long time asymptotic behavior, etc.
However, we still need a more straightforward, efficient and unified approach. Among them, the Riemann–Hilbert method and the
$\bar{\partial }$
-dressing method have wide applicability in soliton decomposition and the study of new solutions to equations. But sometimes in the study of partial differential equations (PDEs), by analyzing the analytic properties of Green's function and eigenfunction, we find that when the analytic properties of Green's function and eigenfunction of the equation in the same plane are inconsistent, it is difficult to obtain the solution of the equation by local Riemann–Hilbert method. This is where the
$\bar{\partial }$
-dressing method shows its superiority. The
$\bar{\partial }$
-dressing method proposed by Zakharov and Shabat [18] is a remarkable method for investigating spectral problems and soliton solutions of NLEEs. Although other methods can also be used for that aim, the
$\bar{\partial }$
-dressing method is the most succinct and clear way to the final results. In recent years, the
$\bar{\partial }$
-dressing method has been applied to different types of equations successfully such as Kadomtsev-Petviashvili (KP) equation with integrable boundary [19], coupled Gerdjikov-Ivanon (GI) equation [20], (2+1)-dimensional Sawada–Kotera (SK) equation [21] etc. In addition, the $\bar{\partial }$-dressing method also has excellent performance in studying the long-time asymptotic behavior of solutions, and many scholars have made a lot of achievements recently. Borghese et al obtained asymptotic expansion in any fixed space-time cone for the focusing nonlinear Schrödinger (NLS) equation in the solitonic region [22]; Jenkins et al studied the soliton resolution of  the derivative NLS equation for generic initial data in a weighted Soblev space [23]; Wang and Fan studied the defocusing NLS equation with nonzero background and analyzed the large-time asymptotic in a solitonless region and two transition regions, respectively [24, 25].
In the present paper, we mainly consider the following (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation [26]:
$\begin{eqnarray}\begin{array}{l}{u}_{{xxxxy}}+4{u}_{{xxy}}{u}_{x}+2{u}_{{xxx}}{u}_{y}\\ \quad +\,6{u}_{{xy}}{u}_{{xx}}+{u}_{{yyy}}-2{u}_{{xxt}}=0.\end{array}\end{eqnarray}$
This equation can be constructed via the bilinear equation of the KP hierarchy [27] and has been proved integrable. The Lax pair and the infinite conservation laws of equation (1.1) have been studied. The integrability and multi-shock wave solutions of the DJKM equation are studied utilizing the Bell polynomials scheme, the Hirota bilinear method, and symbolic computation [28]. However, to our knowledge, there is still no research work on the DJKM equation using the
$\bar{\partial }$
-dressing method.
This paper is organized as follows: In section 2, we derived the Green's function and the characteristic function of the (2+1)-dimensional DJKM equation, and the corresponding spectral problem is obtained. In section 3, we obtained the scattering equation and the
$\bar{\partial }$
problem of the DJKM equation by using the inverse scattering transformation. In section 4, the inverse spectral problem, the characteristic function of time development, and the solution of equation (1.1) are obtained, respectively. Finally, the conclusions and discussions have been given in section 5.

2. Characteristic function and Green' function

Firstly, we will introduce the $\bar{\partial }$-dressing method and its necessary notations briefly. Consider a matrix $\bar{\partial }$ problem
$\begin{eqnarray}\begin{array}{l}\bar{\partial }\psi (x,t,z)=\psi (x,t,z)R(x,t,z),\\ \psi (x,t,z)\to I,z\to \infty ,\end{array}\end{eqnarray}$
where I is the unit matrix, ψ(x, t, z) and R(x, t, z) is 2 × 2 matrix, R(x, t, z) is the spectral transform matrix.
Then the equation (2.1) admits a solution
$\begin{eqnarray}\begin{array}{l}\psi (x,t,k)=I+\displaystyle \frac{1}{2\pi {\rm{i}}}\\ \,\times \,\displaystyle \int \displaystyle \int \displaystyle \frac{\psi (\zeta )R(\zeta )}{\zeta -z}{\rm{d}}\zeta \wedge {\rm{d}}\bar{\zeta }\equiv I+\psi {{RC}}_{z},\end{array}\end{eqnarray}$
where Ck denotes the Cauchy–Green integral operator acting on the left.
For functions of two variables defined on ${{\mathbb{C}}}^{2}$, we consider the special nonlocal $\bar{\partial }$ problem as follows
$\begin{eqnarray}\begin{array}{l}\bar{\partial }\psi (x,t,z)=\displaystyle \int \displaystyle \int \psi (x,t,\zeta )R(x,t,z,\zeta ){\rm{d}}\zeta \wedge {\rm{d}}\bar{\zeta }\\ \quad \equiv \,\psi (x,t,z){R}_{z}F,\quad z,\zeta \in {\mathbb{C}},\\ \psi (x,t,z)=I+O({z}^{-1}),\quad z\to \infty ,\end{array}\end{eqnarray}$
which R(z, ζ) is a function of two variables defined on ${{\mathbb{C}}}^{2}$, similarly, the equation (2.3) also admit a solution
$\begin{eqnarray}\begin{array}{l}\psi (x,t,z)=I+\displaystyle \frac{1}{2\pi {\rm{i}}}\displaystyle \int \displaystyle \int \displaystyle \frac{{\rm{d}}\zeta \wedge {\rm{d}}\bar{\zeta }}{\zeta -z}\\ \quad \displaystyle \times \int \displaystyle \int \psi (x,t,m)R(x,t,\zeta ,m){\rm{d}}m\wedge {\rm{d}}\bar{m}\\ \quad \equiv \,I+\psi (x,t,z){R}_{z}{{FC}}_{z}.\end{array}\end{eqnarray}$
By studying equations (2.3) and (2.4), we can obtain the spectral problem and evolutionary equation hierarchy of (2+1)-dimensional integrable system, but in this paper, we will not discuss this. In this section, through the Fourier transformation and inverse spectral transformation, we will consider the DJKM equation's characteristic function and Green's function of the Lax representation. As we know, the Lax pair of equation (1.1) can be rewritten as the following linear system
$\begin{eqnarray}{\psi }_{{xx}}+{u}_{x}\psi +{\psi }_{y}+\lambda \psi =0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\psi }_{t}-{u}_{y}{\psi }_{x}+{\psi }_{{yy}}+\displaystyle \frac{1}{2}\psi {\partial }_{x}^{-1}{u}_{{yy}}\\ \quad +\displaystyle \frac{1}{2}{u}_{{xy}}\psi +2\lambda {\psi }_{y}=0.\end{array}\end{eqnarray}$
where λ is an arbitrary parameter, and the inverse integrable operator is defined as
$\begin{eqnarray*}{\partial }_{x}^{-1}(u)=\displaystyle \frac{1}{2}\left({\int }_{-\infty }^{x}-{\int }_{x}^{\infty }u(x^{\prime} ){\rm{d}}x^{\prime} \right).\end{eqnarray*}$
Supposing that the potential function u(x, y)  rapidly tends to 0 when (x, y) → ± ∞ , an asymptotic solution can be derived as
$\begin{eqnarray*}\psi \sim {{\rm{e}}}^{{\rm{i}}{zx}+({z}^{2}-\lambda )y},\quad x,y\to \pm \infty ,\end{eqnarray*}$
which is called the Jost solution and parameter z is an arbitrary. In order to introduce the Lax pair with z as the spectral parameter, we can make a transformation
$\begin{eqnarray}\phi (x,y,z)=\psi (x,y){{\rm{e}}}^{-{\rm{i}}{zx}-({z}^{2}-\lambda )y},\end{eqnarray}$
then we can get an asymptotic condition of the new parameter φ(x, y, z)
$\begin{eqnarray*}\phi \sim 1,\quad x,y\to \pm \infty .\end{eqnarray*}$
Substituting the above equation into equations (2.5) and (2.6), we can get the Lax pair of φ(x, y, z)
$\begin{eqnarray}{\phi }_{{xx}}+{\phi }_{y}+2{\rm{i}}z{\phi }_{x}=-{u}_{x}\phi ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\phi }_{t}-{u}_{y}{\phi }_{x}+{\phi }_{{yy}}+2{z}^{2}{\phi }_{y}\\ \quad +\left[-{\rm{i}}{{zu}}_{y}+({z}^{4}-{\lambda }^{2})+\displaystyle \frac{1}{2}{\partial }_{x}^{-1}{u}_{{yy}}\right.\\ \quad +\,\left.\displaystyle \frac{1}{2}{u}_{{xy}}\right]\phi =0.\end{array}\end{eqnarray}$
In order to obtain the bounded function φ(x, y, z) defined on the xy plane, we considered the Green's function of equation (2.8) as follows:
$\begin{eqnarray}{G}_{{xx}}+{G}_{y}+2{\rm{i}}{{zG}}_{x}=\delta (x)\delta (y).\end{eqnarray}$
By virtue of the Fourier transformation, Fourier inverse transformation and the properties of δ function, we can obtain
$\begin{eqnarray}{\mathscr{F}}(G)(\xi ,\eta ,z)=\displaystyle \frac{1}{2\pi }\displaystyle \frac{1}{-{\xi }^{2}+{\rm{i}}\eta -2z\xi },\end{eqnarray}$
and we can get the Green's function of equation (2.10)
$\begin{eqnarray}G(x,y,z)=\displaystyle \frac{1}{4{\pi }^{2}}\int \int \displaystyle \frac{{{\rm{e}}}^{{\rm{i}}(\xi x+\eta y)}}{-{\xi }^{2}+{\rm{i}}\eta -2z\xi }{\rm{d}}\xi {\rm{d}}\eta .\end{eqnarray}$
So the general solution of equation (2.8) is the convolution of the Green's function G(x, y, z) and −ux(x, y)φ(x, y), it can be written as follows
$\begin{eqnarray}\begin{array}{l}\phi (x,y,z)=G(x,y,z)\ast (-{u}_{x}\phi )\\ \quad ={\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }G(x-x^{\prime} ,y-y^{\prime} ,z)\\ \quad \times [-{u}_{x^{\prime} }(x^{\prime} ,y^{\prime} )\phi (x^{\prime} ,y^{\prime} ,z)]{\rm{d}}x^{\prime} {\rm{d}}y^{\prime} .\end{array}\end{eqnarray}$
For the zero potential solution u(x, y) = 0, we can choose two linearly independent solutions M0 = 1, ${N}_{0}={e}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}$ of the characteristic function, where z = z1 + z2 is complex. Then the two characteristic functions corresponding to the general potential can be expressed as
$\begin{eqnarray}\begin{array}{l}M(x,y,z)=1+{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }G\left(x-{x}^{{\prime} },y-{y}^{{\prime} },z\right)\\ \quad \times \,\left[-{u}_{x^{\prime} }\left({x}^{{\prime} },{y}^{{\prime} }\right)M\left({x}^{{\prime} },{y}^{{\prime} },z\right)\right]{\rm{d}}{x}^{{\prime} }{\rm{d}}{y}^{{\prime} },\end{array}\end{eqnarray}$
and
$\begin{eqnarray}\begin{array}{l}N(x,y,z)={{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}+{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }G\left(x-{x}^{{\prime} },y-{y}^{{\prime} },z\right)\\ \quad \times \,\left[-{u}_{x^{\prime} }\left({x}^{{\prime} },{y}^{{\prime} }\right)N\left({x}^{{\prime} },{y}^{{\prime} },z\right)\right]{\rm{d}}{x}^{{\prime} }{\rm{d}}{y}^{{\prime} },\end{array}\end{eqnarray}$
where the Green's function
$\begin{eqnarray}\begin{array}{l}G\left(x-{x}^{{\prime} },y-{y}^{{\prime} },z\right)\\ \quad =\,\tfrac{1}{4{\pi }^{2}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\tfrac{{{\rm{e}}}^{{\rm{i}}\left(\xi \left(x-{x}^{{\prime} }\right)+\eta \left(y-{y}^{{\prime} }\right)\right)}}{-{\xi }^{2}+{\rm{i}}\eta -2z\xi }{\rm{d}}\xi {\rm{d}}\eta \\ \quad =\,\tfrac{1}{2\pi }{\displaystyle \int }_{-\infty }^{\infty }g\left(\xi ,y-{y}^{{\prime} },z\right){{\rm{e}}}^{{\rm{i}}\xi \left(x-{x}^{{\prime} }\right)}{\rm{d}}\xi ,\end{array}\end{eqnarray}$
with
$\begin{eqnarray}g\left(\xi ,y-{y}^{{\prime} },z\right)=\displaystyle \frac{1}{2\pi }{\int }_{-\infty }^{\infty }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}\eta \left(y-{y}^{{\prime} }\right)}}{-{\xi }^{2}+{\rm{i}}\eta -2z\xi }{\rm{d}}\eta .\end{eqnarray}$
We can see that the first-order singularity of the above formula is
$\begin{eqnarray*}{\eta }_{1}=-{\rm{i}}\xi (\xi +2z),\quad \mathrm{Im}\eta =\xi (\xi +2{z}_{1}).\end{eqnarray*}$
When $y-y^{\prime} \gt 0$, we make a sufficiently large semicircle in the upper half plane CR η = Reiθ, R > ∣η1∣, 0 ≤ θπ, then [−R, R] ∪ CR forms a closed curve with the counterclockwise direction. If η1 is in the upper half plane, the η1 is inside the curve, and if η1 is in the lower half plane, the integrand is analytic in the curve, then by making use of residue theorem, we can obtain
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{2\pi {\rm{i}}}{\displaystyle \int }_{-R}^{R}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}\eta \left(y-{y}^{{\prime} }\right)}}{-{\xi }^{2}+{\rm{i}}\eta -2z\xi }{\rm{d}}\eta \\ \quad +\,\displaystyle \frac{1}{2\pi {\rm{i}}}{\displaystyle \int }_{{C}_{R}}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}\eta \left(y-{y}^{{\prime} }\right)}}{-{\xi }^{2}+{\rm{i}}\eta -2z\xi }{\rm{d}}\eta \\ \quad =\mathop{\mathrm{Res}}\limits_{\eta ={\eta }_{1},\mathrm{Im}{\eta }_{1}\gt 0}\left[\displaystyle \frac{1}{2\pi {\rm{i}}}{\displaystyle \int }_{-R}^{R}\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}\eta \left(y-{y}^{{\prime} }\right)}}{-{\xi }^{2}+{\rm{i}}\eta -2z\xi }{\rm{d}}\eta \right].\end{array}\end{eqnarray}$
Let R → ∞ , by the Jordan theorem, we can find the limitation of the second formula of the previous equation is 0, then we have
$\begin{eqnarray}\begin{array}{l}g\left(\xi ,y-{y}^{{\prime} },z\right)=\mathop{\mathrm{Res}}\limits_{\eta ={\eta }_{1},\mathrm{Im}{\eta }_{1}\gt 0}\left[\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}\eta \left(y-{y}^{{\prime} }\right)}}{\eta +{\rm{i}}\xi (\xi +2z)}\right]\\ \quad =\,\left\{\begin{array}{cc}{{\rm{e}}}^{\xi (\xi +2z)\left(y-{y}^{{\prime} }\right)}, & \xi \left(\xi +2{z}_{1}\right)\gt 0,\\ 0, & \xi \left(\xi +2{z}_{1}\right)\lt 0;\end{array}\right.\\ \quad =\,H\left[\xi \left(\xi +2{z}_{1}\right)\left(y-{y}^{{\prime} }\right)\right]{{\rm{e}}}^{\xi (\xi +2z)\left(y-{y}^{{\prime} }\right)},\end{array}\end{eqnarray}$
where H( · ) is the Heaviside function. Similarly, when $y-y^{\prime} \lt 0$, we can also write the following expression of $g(\xi ,y-y^{\prime} ,z)$
$\begin{eqnarray}\begin{array}{l}g\left(\xi ,y-{y}^{{\prime} },z\right)=\mathop{\mathrm{Res}}\limits_{\eta ={\eta }_{1},\mathrm{Im}{\eta }_{1}\lt 0}\left[\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}\eta \left(y-{y}^{{\prime} }\right)}}{\eta +{\rm{i}}\xi (\xi +2z)}\right]\\ \quad =\,\left\{\begin{array}{cc}-{{\rm{e}}}^{\xi (\xi +2z)\left(y-{y}^{{\prime} }\right)}, & \xi \left(\xi +2{z}_{1}\right)\lt 0,\\ 0, & \xi \left(\xi +2{z}_{1}\right)\gt 0;\end{array}\right.\\ \quad =\,-H\left[\xi \left(\xi +2{z}_{1}\right)\left(y-{y}^{{\prime} }\right)\right]{{\rm{e}}}^{\xi (\xi +2z)\left(y-{y}^{{\prime} }\right)}.\end{array}\end{eqnarray}$
In short, we can combine the above two equations
$\begin{eqnarray}\begin{array}{l}g(\xi ,y-y^{\prime} ,z)=\mathrm{sgn}(y-y^{\prime} )\\ \quad \times \,H[\xi (\xi +2{z}_{1})(y-y^{\prime} )]{{\rm{e}}}^{\xi (\xi +2z)(y-y^{\prime} )},\end{array}\end{eqnarray}$
therefore
$\begin{eqnarray}\begin{array}{l}G(x-x^{\prime} ,y-y^{\prime} ,z)=\displaystyle \frac{\mathrm{sgn}(y-y^{\prime} )}{2\pi }\\ \quad \times \,{\displaystyle \int }_{-\infty }^{\infty }H[\xi (\xi +2{z}_{1})(y-y^{\prime} )]{{\rm{e}}}^{{\rm{i}}\xi (x-x^{\prime} )+\xi (\xi +2z)(y-y^{\prime} )}.\end{array}\end{eqnarray}$
Obviously, the above Green's function is analytic in the z − plane and has no jump across the real axis, but the functions M and N are not analytic in the same plane, so we need to use the $\bar{\partial }$-dressing method instead of the local RH method to solve the inverse scattering problem of the DJKM equation.

3. Scattering equation and $\bar{\partial }$-problem

Firstly, we consider the $\bar{\partial }$ derivative of the characteristic function equation (2.14) as follows:
$\begin{eqnarray}\begin{array}{l}\bar{\partial }M={\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }[\bar{\partial }G(x-x^{\prime} ,y-y^{\prime} ,z)][-{u}_{x^{\prime} }(x^{\prime} ,y^{\prime} )M(x^{\prime} ,y^{\prime} ,z)]{\rm{d}}x^{\prime} {\rm{d}}y^{\prime} \\ \quad +\,{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }G(x-x^{\prime} ,y-y^{\prime} ,z)[-{u}_{x^{\prime} }(x^{\prime} ,y^{\prime} )\bar{\partial }M(x^{\prime} ,y^{\prime} ,z)]{\rm{d}}x^{\prime} {\rm{d}}y^{\prime} ,\end{array}\end{eqnarray}$
then we need to calculate the $\bar{\partial }G(x-x^{\prime} ,y-y^{\prime} ,z)$ at first
$\begin{eqnarray}\begin{array}{l}\bar{\partial }G(x-x^{\prime} ,y-y^{\prime} ,z)\\ \quad =\,\displaystyle \frac{1}{4{\pi }^{2}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }{{\rm{e}}}^{{\rm{i}}[\xi (x-x^{\prime} )+\eta (y-y^{\prime} )]}\bar{\partial }\\ \quad \times \,\left(\displaystyle \frac{1}{-{\xi }^{2}+{\rm{i}}\eta -2z\xi }\right){\rm{d}}\xi {\rm{d}}\eta \\ \quad =\,\displaystyle \frac{1}{4{\pi }^{2}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }-\displaystyle \frac{1}{2\xi }{{\rm{e}}}^{{\rm{i}}[\xi (x-x^{\prime} )+\eta (y-y^{\prime} )]}\\ \quad \times \,\delta \left({z}_{1}+\displaystyle \frac{\xi }{2}\right)\delta \left({z}_{2}-\displaystyle \frac{\eta }{2\xi }\right){\rm{d}}\xi {\rm{d}}\eta \\ \quad =\,\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }-\displaystyle \frac{| \xi | }{\xi }{{\rm{e}}}^{{\rm{i}}[\xi (x-x^{\prime} )+\eta (y-y^{\prime} )]}\\ \quad \times \,\delta (2{z}_{1}+\xi )\delta (2\xi {z}_{2}-\eta ){\rm{d}}\xi {\rm{d}}\eta \\ \quad =\,\left\{\begin{array}{cc}\tfrac{1}{2\pi }{{\rm{e}}}^{{\rm{i}}[\xi (x-x^{\prime} )+\eta (y-y^{\prime} )]}{| }_{\xi =-2{z}_{1},\eta =2\xi {z}_{2}}, & {z}_{1}\gt 0,\\ -\tfrac{1}{2\pi }{{\rm{e}}}^{{\rm{i}}[\xi (x-x^{\prime} )+\eta (y-y^{\prime} )]}{| }_{\xi =-2{z}_{1},\eta =2\xi {z}_{2}}, & {z}_{1}\lt 0;\end{array}\right.\\ \quad =\,\displaystyle \frac{1}{2\pi }\mathrm{sgn}({z}_{1}){{\rm{e}}}^{-2{\rm{i}}{z}_{1}[(x-x^{\prime} )+2{z}_{2}(y-y^{\prime} )]}.\end{array}\end{eqnarray}$
Naturally, substituting equation (3.2) into equation (3.1), the following equation could be derived as
$\begin{eqnarray}\begin{array}{l}\bar{\partial }M(x,y,z)=F({z}_{1},{z}_{2}){{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}\\ \quad +{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }G(x-x^{\prime} ,y-y^{\prime} ,z)\\ \quad \times [-{u}_{x^{\prime} }(x^{\prime} ,y^{\prime} )\bar{\partial }M(x^{\prime} ,y^{\prime} ,z)]{\rm{d}}x^{\prime} {\rm{d}}y^{\prime} ,\end{array}\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{l}F({z}_{1},{z}_{2})=\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\mathrm{sgn}({z}_{1}){{\rm{e}}}^{2{\rm{i}}{z}_{1}(x^{\prime} +2{z}_{2}y^{\prime} )}\\ \quad \times \,[-{u}_{x^{\prime} }(x^{\prime} ,y^{\prime} )M(x^{\prime} ,y^{\prime} ,z)]{\rm{d}}x^{\prime} {\rm{d}}y^{\prime} .\end{array}\end{eqnarray}$
We multiply both sides of equation (2.15) by F(z1, z2) and subtract the above equation (3.3), we can get
$\begin{eqnarray}\begin{array}{l}F({z}_{1},{z}_{2})N(x,y,z)-\bar{\partial }M(x,y,z)\\ \quad =\,-{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }G(x-x^{\prime} ,y-y^{\prime} ,z){u}_{x^{\prime} }(x^{\prime} ,y^{\prime} )\\ \quad \times \,[F({z}_{1},{z}_{2})N(x^{\prime} ,y^{\prime} ,z)-\bar{\partial }M(x^{\prime} ,y^{\prime} ,z)]{\rm{d}}x^{\prime} {\rm{d}}y^{\prime} .\end{array}\end{eqnarray}$
Assuming that the corresponding homogeneous integral equation has only a trivial solution, we can obtain the scattering equation in the following form of the linear $\bar{\partial }$ problem
$\begin{eqnarray}\bar{\partial }M(x,y,z)=F({z}_{1},{z}_{2})N(x,y,z).\end{eqnarray}$
In order to get a standard formula for N with M, which enable us to search for the symmetry of the Green's function
$\begin{eqnarray}\begin{array}{l}G(x-x^{\prime} ,y-y^{\prime} ,-\bar{z})=\displaystyle \frac{1}{4{\pi }^{2}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}[\xi (x-x^{\prime} )+\eta (y-y^{\prime} )]}}{-{\xi }^{2}+{\rm{i}}\eta +2\bar{z}\xi }{\rm{d}}\xi {\rm{d}}\eta \\ \quad =\,\displaystyle \frac{1}{4{\pi }^{2}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}[\xi (x-x^{\prime} )+\eta (y-y^{\prime} )]}}{-{\left(\xi -2{z}_{1}\right)}^{2}+{\rm{i}}(\eta -4{z}_{1}{z}_{2})-2({z}_{1}+{\rm{i}}{z}_{2})(\xi -2{z}_{1})}{\rm{d}}\xi {\rm{d}}\eta ,\end{array}\end{eqnarray}$
where z = z1 + iz2, $\bar{z}={z}_{1}-{\rm{i}}{z}_{2}$. Making a pair of transformation ξ − 2z1s, η − 4z1z2m, we can get the symmetry of Green's function
$\begin{eqnarray}\begin{array}{l}G(x-x^{\prime} ,y-y^{\prime} ,-\bar{z})\\ =\,\displaystyle \frac{1}{4{\pi }^{2}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}[(s+2{z}_{1})(x-x^{\prime} )+(m+4{z}_{1}{z}_{2})(y-y^{\prime} )]}}{-{s}^{2}+{\rm{i}}m-2{zs}}{\rm{d}}s{\rm{d}}m\\ \quad =\,\displaystyle \frac{1}{4{\pi }^{2}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}[s(x-x^{\prime} )+m(y-y^{\prime} )]}}{-{s}^{2}+{\rm{i}}m-2{zs}}{\rm{d}}s{\rm{d}}m\\ \quad \times \,{{\rm{e}}}^{2{\rm{i}}{z}_{1}(x-x^{\prime} )+4{\rm{i}}{z}_{1}{z}_{2}(y-y^{\prime} )}\\ \quad =\,G(x-x^{\prime} ,y-y^{\prime} ,z){{\rm{e}}}^{2{\rm{i}}{z}_{1}(x-x^{\prime} )+4{\rm{i}}{z}_{1}{z}_{2}(y-y^{\prime} )}.\end{array}\end{eqnarray}$
Replace z in equation (2.14) with $-\bar{z}$ and multiply equation (2.14) with ${{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}$, we can find
$\begin{eqnarray}\begin{array}{l}M(x,y,-\bar{z}){{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}\\ \quad =\,{{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}+{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }G(-\bar{z})\\ \quad \times \,[-{u}_{x^{\prime} }(x^{\prime} ,y^{\prime} )M(x^{\prime} ,y^{\prime} ,-\bar{z})]{\rm{d}}x^{\prime} {\rm{d}}y^{\prime} {{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}\\ \quad =\,{{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}+{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }G(z)\\ \quad \times \,[-{u}_{x^{\prime} }(x^{\prime} ,y^{\prime} )M(x^{\prime} ,y^{\prime} ,-\bar{z}){{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x^{\prime} +2{z}_{2}y^{\prime} )}]{\rm{d}}x^{\prime} {\rm{d}}y^{\prime} .\end{array}\end{eqnarray}$
Combine with equation (2.15), the relation of N and M can be expressed as
$\begin{eqnarray}N(x,y,z)=M(x,y,-\bar{z}){{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}.\end{eqnarray}$
Therefore, the $\bar{\partial }$ problem for the characteristic function M  turns out to be
$\begin{eqnarray}\bar{\partial }M(x,y,z)=F({z}_{1},{z}_{2})M(x,y,-\bar{z}){{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}.\end{eqnarray}$

4. Inverse spectral problem

For the inverse problem equation (3.11), we can obtain a solution as follows by virtue of the Cauchy–Green formula
$\begin{eqnarray}\begin{array}{l}M(x,y,z)=1+\displaystyle \frac{1}{2\pi {\rm{i}}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{\rm{d}}\zeta \wedge {\rm{d}}\bar{\zeta }}{\zeta -z}F({\zeta }_{1},{\zeta }_{2})M(x,y,-\bar{\zeta }){{\rm{e}}}^{-2{\rm{i}}{\zeta }_{1}(x+2{\zeta }_{2}y)},\end{array}\end{eqnarray}$
where ζ = ζ1 + iζ2. From equation (2.14) and equation (4.1) we have two representations for M − 1:
$\begin{eqnarray}\begin{array}{l}M(x,y,z)-1\,=\,\left\{\begin{array}{l}\displaystyle \frac{1}{2\pi {\rm{i}}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{\rm{d}}\zeta \wedge {\rm{d}}\bar{\zeta }}{\zeta -z}F({\zeta }_{1},{\zeta }_{2})M(x,y,-\bar{\zeta }){{\rm{e}}}^{-2{\rm{i}}{\zeta }_{1}(x+2{\zeta }_{2}y)},\\ {\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }G\left(x-{x}^{{\prime} },y-{y}^{{\prime} },z\right)\left[-{u}_{x^{\prime} }\left({x}^{{\prime} },{y}^{{\prime} }\right)M\left({x}^{{\prime} },{y}^{{\prime} },z\right)\right]{\rm{d}}{x}^{{\prime} }{\rm{d}}{y}^{{\prime} }.\end{array}\right.\end{array}\end{eqnarray}$
By comparing the order of O(z−1) and using the properties of the Dirac delta function, the Fourier transformation and the Fourier inverse transformation, we can derive equation (2.16) as follows
$\begin{eqnarray}\begin{array}{l}G(x,y,z,\bar{z})=\displaystyle \frac{1}{4{\pi }^{2}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}(\xi x+\eta y)}}{-{\xi }^{2}+{\rm{i}}\eta -2z\xi }{\rm{d}}\xi {\rm{d}}\eta \\ \quad =\,-\displaystyle \frac{{\rm{i}}}{4\pi z}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{1}{2\pi }{{\rm{e}}}^{{\rm{i}}\eta y}{\rm{d}}\eta {\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}\xi x}}{\xi }{\rm{d}}\xi +O({z}^{-2})\\ \quad =\,-\displaystyle \frac{{\rm{i}}}{4z}\delta (y)\mathrm{sgn}(x)+O({z}^{-2}).\end{array}\end{eqnarray}$
Besides, M(x, y, z) = 1 + O(z−1). Hence, the Green's function representation of M − 1 is
$\begin{eqnarray}\begin{array}{l}M-1=\displaystyle \frac{{\rm{i}}}{4z}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\mathrm{sgn}\left(x-{x}^{{\prime} }\right)\delta \left(y-{y}^{{\prime} }\right)\\ \quad \times \,{u}_{x^{\prime} }\,\left({x}^{{\prime} },{y}^{{\prime} }\right){\rm{d}}{x}^{{\prime} }{\rm{d}}{y}^{{\prime} }+O\left({z}^{-2}\right)\\ \quad =\,\displaystyle \frac{{\rm{i}}}{4z}{\displaystyle \int }_{-\infty }^{\infty }\mathrm{sgn}\left(x-{x}^{{\prime} }\right){u}_{x^{\prime} }\left({x}^{{\prime} },y\right){\rm{d}}{x}^{{\prime} }+O\left({z}^{-2}\right)\\ \quad =\,\displaystyle \frac{{\rm{i}}}{4z}\left({\displaystyle \int }_{-\infty }^{x}{u}_{x^{\prime} }\left({x}^{{\prime} },y\right){\rm{d}}{x}^{{\prime} }\right.\\ \quad \left.-\,{\displaystyle \int }_{x}^{\infty }{u}_{x^{\prime} }\left({x}^{{\prime} },y\right){\rm{d}}{x}^{{\prime} }\right)+O\left({z}^{-2}\right)\\ \quad =\,\displaystyle \frac{{\rm{i}}}{2z}u(x,y)+O({z}^{-2}).\end{array}\end{eqnarray}$
From the second equation of the representation (4.2), we can also obtain
$\begin{eqnarray}\begin{array}{l}M-1=\displaystyle \frac{{\rm{i}}}{2\pi z}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}\zeta \wedge {\rm{d}}\bar{\zeta }F\left({\zeta }_{1},{\zeta }_{2}\right)\\ \quad \times \,M(x,y,-\bar{\zeta }){{\rm{e}}}^{-2{\rm{i}}{\zeta }_{1}\left(x+2{\zeta }_{2}y\right)}+O\left({z}^{-2}\right).\end{array}\end{eqnarray}$
By comparing equation (4.4) and equation (4.5), we can get the reconstruction formula
$\begin{eqnarray}\begin{array}{l}u(x,y)=\displaystyle \frac{1}{\pi }{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}\zeta \wedge {\rm{d}}\bar{\zeta }F\left({\zeta }_{1},{\zeta }_{2}\right)\\ \quad \times \,M(x,y,-\bar{\zeta }){{\rm{e}}}^{-2{\rm{i}}{\zeta }_{1}\left(x+2{\zeta }_{2}y\right)}.\end{array}\end{eqnarray}$
In order to determine the temporal evolution of scattering data F(z1, z2, t), substituting equation (2.7) into equation (3.11), it is intuitive to find that
$\begin{eqnarray}\begin{array}{l}\bar{\partial }\psi (x,y,t,z)=F({z}_{1},{z}_{2},t)\psi (x,y,t,-\bar{z})\\ \,\times \,{{\rm{e}}}^{{\rm{i}}\bar{z}x-({\bar{z}}^{2}-\lambda )y}{{\rm{e}}}^{-2{\rm{i}}{z}_{1}(x+2{z}_{2}y)}{{\rm{e}}}^{{\rm{i}}{zx}+({z}^{2}-\lambda )y}\\ \quad =\,F({z}_{1},{z}_{2},t)\psi (x,y,t,-\bar{z}){{\rm{e}}}^{-{\rm{i}}{zx}-({z}^{2}-\lambda )y}\\ \,\times \,{{\rm{e}}}^{{\rm{i}}{zx}+({z}^{2}-\lambda )y}\\ \quad =\,F({z}_{1},{z}_{2},t)\psi (x,y,t,-\bar{z}).\end{array}\end{eqnarray}$
By differentiating both sides of the above equation in regard of t, we can get
$\begin{eqnarray}\begin{array}{l}{\left(\bar{\partial }\psi (z)\right)}_{t}={F}_{t}({z}_{1},{z}_{2},t)\psi (-\bar{z})\\ \quad +\,F({z}_{1},{z}_{2},t){\psi }_{t}(-\bar{z}).\end{array}\end{eqnarray}$
By equation (2.6) and the above formula, we can get two equations for ψ(z) as the following form
$\begin{eqnarray}\begin{array}{l}\bar{\partial }{\psi }_{t}(z)=\bar{\partial }({u}_{y}{\partial }_{x}-{\partial }_{{yy}}-\displaystyle \frac{1}{2}{\partial }_{x}^{-1}{u}_{{yy}}\\ \quad -\displaystyle \frac{1}{2}{u}_{{xy}}-2\lambda {\partial }_{y})\psi (z),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\psi }_{t}(-\bar{z})=\left({u}_{y}{\partial }_{x}-{\partial }_{{yy}}-\displaystyle \frac{1}{2}{\partial }_{x}^{-1}{u}_{{yy}}\right.\\ \quad -\left.\displaystyle \frac{1}{2}{u}_{{xy}}-2\lambda {\partial }_{y}\right)\psi (-\bar{z}).\end{array}\end{eqnarray}$
Then by choosing λ = iz3 and substituting equations (4.9) and (4.10) in equation (4.8), we can receive the following result
$\begin{eqnarray}F({z}_{1},{z}_{2},t)=F({z}_{1},{z}_{2}){{\rm{e}}}^{-2{\rm{i}}({z}^{3}+{\bar{z}}^{3})t+y},\end{eqnarray}$
where we define the F(z1, z2, 0) = F(z1, z2).
Finally, we can obtain the characteristic function with time evolution
$\begin{eqnarray}\begin{array}{l}M(x,y,z)=1+\displaystyle \frac{1}{2\pi {\rm{i}}}{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{\rm{d}}\zeta \wedge {\rm{d}}\bar{\zeta }}{\zeta -z}F({\zeta }_{1},{\zeta }_{2})\\ \quad \times \,M(x,y,-\bar{\zeta }){{\rm{e}}}^{-2{\rm{i}}{\zeta }_{1}(x+2{\zeta }_{2}y)-2{\rm{i}}({z}^{3}+{\bar{z}}^{3})t+y},\end{array}\end{eqnarray}$
and the formal solution of the DJKM equation can be expressed as
$\begin{eqnarray}\begin{array}{l}u(x,y)=\displaystyle \frac{1}{\pi }{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}\zeta \wedge {\rm{d}}\bar{\zeta }F\left({\zeta }_{1},{\zeta }_{2}\right)M(x,y,-\bar{\zeta })\\ \times \,{{\rm{e}}}^{-2{\rm{i}}{\zeta }_{1}\left(x+2{\zeta }_{2}y\right)-2{\rm{i}}({z}^{3}+{\bar{z}}^{3})t+y},\end{array}\end{eqnarray}$
where the F(z1, z2) has been shown as equation (3.4).

5. Conclusion

In this paper, we proposed a new systematical solution procedure of the (2+1)-dimensional DJKM equation. Via the $\bar{\partial }$-dressing method, we first introduced the characteristic function and Green's function of the DJKM equation, respectively. Secondly, the $\bar{\partial }$ problem, scattering equation and inverse spectral problem of the DJKM equation have been presented. The $\bar{\partial }$ -dressing method can solve the problem that the local RH method has difficulty solving when the analytic properties of Green's function and eigenfunction are inconsistent. After determining the time evolution, we can apply the $\bar{\partial }$-dressing method to formalize the construction of the new solution of the (2+1)-dimensional DJKM equation.
When the $\bar{\partial }$ -dressing method is used to study the solution of the equations, we find that for the (1+1)-dimensional integrable systems, the $\bar{\partial }$ -dressing method can get the n-soliton solution of the system easily, but for most (2+1)-dimensional integrable systems like the one in this paper, it is worth mentioning that the solution proposed in this paper is the formal solution of the equation. At present, we cannot reduce this kind of solution to special solutions such as soliton solutions. We will consider this possibility in future research work.

Declarations

Author contributions

SFS studied the conceptualization and wrote (reviewed and edited) the manuscript and BL funded the acquisition and project administration. All authors approved the final manuscript.

Funding

This work is supported by National Natural Science Foundation of China under Grant Nos. 12 175 111 and 11 975 131, and K C Wong Magna Fund in Ningbo University.

Conflict of interest

The authors declare that they have no competing interests.

Ethics approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

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