Lie symmetry analysis, optimal system and conservation laws of a new (2【-逻*辑*与-】plus;1)-dimensional

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Lie symmetry analysis, optimal system and conservation laws of a new (2【-逻辑与-】plus;1)-dimensional

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Mengmeng Wang¹, Shoufeng Shen¹, Lizhen Wang^,²^,^∗¹Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China
²Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi’an 710127, China

First author contact: ^∗Author to whom any correspondence should be addressed.
Received:2021-02-19Revised:2021-04-23Accepted:2021-04-29Online:2021-06-25

Abstract
In this paper, Lie point symmetries of a new (2+1)-dimensional KdV system are constructed by using the symbolic computation software Maple. Then, the one-dimensional optimal system, associated with corresponding Lie algebra, is obtained. Moreover, the reduction equations and some explicit solutions based on the optimal system are presented. Finally, the nonlinear self-adjointness is provided and conservation laws of this KdV system are constructed.
Keywords： Lie point symmetry;optimal system;symmetry reduction;exact solution;conservation law

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Mengmeng Wang, Shoufeng Shen, Lizhen Wang. Lie symmetry analysis, optimal system and conservation laws of a new (2+1)-dimensional KdV system. Communications in Theoretical Physics, 2021, 73(8): 085004- doi:10.1088/1572-9494/abfcb8

1. Introduction

Integrable systems are widely used to describe the complex natural phenomena in fluid physics, quantum physics, condensed matter physics, biology and mathematical physics etc. Some important models such as the KdV equation, KP equation, nonlinear Schrödinger equation, NNV equation and Sine-Godon equation have been discovered and then investigated extensively by many researchers. By observing the fact that all the known (2+1)-dimensional integrable systems have the centerless Virasoro type subaglebra $[\sigma ({f}_{1}),\sigma ({f}_{2})]=\sigma ({\dot{f}}_{1}{f}_{2}-{\dot{f}}_{2}{f}_{1})$, where ${f}_{i},i=1,2$ are two arbitrary functions with one independent variable and $\sigma ({f}_{i}),i\,=\,1,2$ are the symmetries of the given system and there are no known nonintegrable systems possessing such Virasoro symmetry algebra. Lou and Hu introduced an idea that if an f-independent model owns the Virasoro symmetry algebra, the model is Virasoro integrable. Utilizing this theory, some new (2+1)-dimensional and (3+1)-dimensional Virasoro integrable systems [1–4] have been constructed. In this paper, we are interested in the new (2+1)-dimensional KdV system(1)$\begin{eqnarray}{\rm{\Delta }}=\left\{\begin{array}{l}{v}_{t}+{{uv}}_{x}+{u}_{x}=0\\ {u}_{{yt}}+{u}_{x}{u}_{y}+{{uu}}_{{xy}}+{v}_{{xxx}}=0\end{array}\right.,\end{eqnarray}$which has the same centerless Virasoro type subalgebra as the coupled KdV system.

An important task of studying integrable systems is to find their exact solutions. Many powerful methods also have been found, such as the inverse scattering transformation (IST), Darboux transformation, Hirota’s bilinear method, classical and non-classical Lie symmetry analysis methods and Clarkson-Kruskal’s direct method etc. Lie symmetry analysis, an effective algebra method playing an essential role in constructing exact solutions, were originally derived by Sophus Lie. It is also used to solve nonlinear partial differential equations with variable coefficients [18–20] and to obtain the numerical solutions [21]. Classification of all group invariant solutions is an important step in Lie group analysis. To classify group invariant solutions, one should classify the subalgebra of the Lie algebra generated by Lie point symmetries, so the definition of one-dimensional optimal system was came up with by Ovsiannikov [5, 9, 10] using a global matrix for the adjoint transformation, and then Olver [6, 16] introduced one method, which was used invariants and adjoint operators to simplify a general element in Lie algebra as much as possible. In 2010, Ibragimov ([11–13] and references there in) put forward a different method based on ${C}_{\beta \gamma }^{\alpha }$ , the structural constants of Lie algebra. In 2015, a direct algorithm of one-dimensional optimal system was presented by Hu et al [14, 15, 17] who obtained new invariant never addressed by Killing form in the example of heat equation. Moreover, another important task of studying nonlinear evolution equations is to analyze its integrability, which always can be implied by the existence of conservation laws. Noether’s theorem presents a connection between symmetries and conservation laws, but it is so strict that it is not applicable to evolution equations, to different equations of odd order, etc. In order to overcome its restrictions, Ibragimov generalized Noether’s theorem in [7, 8].

The plan of this article is as follows: section 2 performs Lie point symmetry method on the (2+1)-dimensional KdV system, and obtains the one-dimensional optimal system combining the Ovsiannikov’s global matrix method and Ibragimov’s method. Section 3 is devoted to getting reduction equations by reducing the associated Lagrange characteristic equations of one-dimensional optimal operators. Section 4 proves the nonlinear self-adjointness of (2+1)-dimensional KdV system and constructs the conservation laws. In section 5 we make some brief comments.

2. Lie Symmetry and Optimal System of Subalgebras

Let us begin with a one-parameter Lie group of point transformation:$\begin{eqnarray*}\begin{array}{rcl}{x}^{* } & = & x+\varepsilon \xi (x,y,t,u,v)+o({\varepsilon }^{2}),\\ {y}^{* } & = & y+\varepsilon \eta (x,y,t,u,v)+o({\varepsilon }^{2}),\\ {t}^{* } & = & t+\varepsilon \tau (x,y,t,u,v)+o({\varepsilon }^{2}),\\ {u}^{* } & = & u+\varepsilon \phi (x,y,t,u,v)+o({\varepsilon }^{2}),\\ {v}^{* } & = & v+\varepsilon \psi (x,y,t,u,v)+o({\varepsilon }^{2}).\end{array}\end{eqnarray*}$The corresponding generator of Lie algebra has the form:$\begin{eqnarray*}V=\xi \displaystyle \frac{\partial }{\partial x}+\eta \displaystyle \frac{\partial }{\partial y}+\tau \displaystyle \frac{\partial }{\partial t}+\phi \displaystyle \frac{\partial }{\partial u}+\psi \displaystyle \frac{\partial }{\partial v}.\end{eqnarray*}$

If ${{pr}}^{(3)}V$ is the third order prolongation, the invariant condition$\begin{eqnarray*}{{pr}}^{(3)}V({\rm{\Delta }}){| }_{{\rm{\Delta }}=0}=0,\end{eqnarray*}$yields the following determining equations:$\begin{eqnarray*}\begin{array}{l}{\xi }_{y}={\xi }_{u}={\xi }_{v}=0,\\ {\eta }_{x}={\eta }_{t}={\eta }_{u}={\eta }_{v}=0,\\ {\tau }_{x}={\tau }_{y}={\tau }_{u}={\tau }_{v}=0,\\ {\phi }_{y}={\phi }_{v}={\phi }_{{uu}}={\phi }_{{yu}}=0,\\ {\psi }_{x}={\psi }_{u}={\psi }_{{vv}}=0,\\ {\psi }_{{xv}}-{\xi }_{{xx}}=0,\\ {\psi }_{v}-{\phi }_{u}+{\eta }_{y}+{\tau }_{t}-3{\xi }_{x}=0,\\ \phi -{\xi }_{t}+u{\tau }_{t}-u{\xi }_{x}=0,\\ {\phi }_{u}+{\tau }_{t}-{\xi }_{x}=0,\\ {\phi }_{{tu}}+{\phi }_{x}+u{\phi }_{{xu}}=0,\\ -{\psi }_{v}+{\phi }_{u}+{\tau }_{t}-{\xi }_{x}=0,\\ {\phi }_{x}+{\psi }_{t}=0,\\ {\phi }_{{yt}}+{\psi }_{{xxx}}=0.\end{array}\end{eqnarray*}$

By solving these equations, we obtain the following Lie point symmetry generators:$\begin{eqnarray*}\begin{array}{rcl}{V}_{1} & = & -x\displaystyle \frac{\partial }{\partial x}-2t\displaystyle \frac{\partial }{\partial t}+u\displaystyle \frac{\partial }{\partial u},\\ {V}_{2} & = & \displaystyle \frac{x}{2}\displaystyle \frac{\partial }{\partial x}+y\displaystyle \frac{\partial }{\partial y}+\displaystyle \frac{t}{2}\displaystyle \frac{\partial }{\partial t},\\ {V}_{3} & = & {f}_{3}^{{\prime} }(t)x\displaystyle \frac{\partial }{\partial x}+\displaystyle \frac{\partial }{\partial y}+2{f}_{3}(t)\displaystyle \frac{\partial }{\partial t}+({f}_{3}^{{\prime} ^{\prime} }(t)x\\ & & +{f}_{3}^{{\prime} }(t)u)\displaystyle \frac{\partial }{\partial u}-{f}_{3}^{{\prime} }(t)\displaystyle \frac{\partial }{\partial v},\\ {V}_{4} & = & \displaystyle \frac{\partial }{\partial t},\\ {V}_{5} & = & ({f}_{5}(t)+1)\displaystyle \frac{\partial }{\partial x}+{f}_{5}^{{\prime} }(t)\displaystyle \frac{\partial }{\partial u},\\ {V}_{6} & = & {f}_{6}(y)\displaystyle \frac{\partial }{\partial v}.\end{array}\end{eqnarray*}$Because of the arbitrariness of the functions ${f}_{3},{f}_{5},{f}_{6}$, we can obtain an infinite-dimensional Lie algebra expanded by ${V}_{1},{V}_{2},\cdots ,{V}_{6}$. Let ${f}_{3}(t)=t,{f}_{5}(t)=1,{f}_{6}(y)=1$, the following six Lie point symmetries operators are obtained:$\begin{eqnarray*}\begin{array}{rcl}{V}_{1} & = & -x\displaystyle \frac{\partial }{\partial x}-2t\displaystyle \frac{\partial }{\partial t}+u\displaystyle \frac{\partial }{\partial u},\\ {V}_{2} & = & \displaystyle \frac{x}{2}\displaystyle \frac{\partial }{\partial x}+y\displaystyle \frac{\partial }{\partial y}+\displaystyle \frac{t}{2}\displaystyle \frac{\partial }{\partial t},\\ {V}_{3} & = & x\displaystyle \frac{\partial }{\partial x}+\displaystyle \frac{\partial }{\partial y}+2t\displaystyle \frac{\partial }{\partial t}-u\displaystyle \frac{\partial }{\partial u}-\displaystyle \frac{\partial }{\partial v},\\ {V}_{4} & = & \displaystyle \frac{\partial }{\partial t},\\ {V}_{5} & = & 2\displaystyle \frac{\partial }{\partial x},\\ {V}_{6} & = & \displaystyle \frac{\partial }{\partial v}.\end{array}\end{eqnarray*}$By means of linear transformation, this six Lie point symmetries can be rewritten in a simpler form:(2)$\begin{eqnarray}\begin{array}{rcl}{V}_{1} & = & \displaystyle \frac{\partial }{\partial x},\\ {V}_{2} & = & \displaystyle \frac{\partial }{\partial y},\\ {V}_{3} & = & x\displaystyle \frac{\partial }{\partial x}+4y\displaystyle \frac{\partial }{\partial y}+u\displaystyle \frac{\partial }{\partial u},\end{array}\end{eqnarray}$

(3)$\begin{eqnarray}\begin{array}{rcl}{V}_{4} & = & \displaystyle \frac{\partial }{\partial v},\\ {V}_{5} & = & x\displaystyle \frac{\partial }{\partial x}+2t\displaystyle \frac{\partial }{\partial t}-u\displaystyle \frac{\partial }{\partial u},\\ {V}_{6} & = & \displaystyle \frac{\partial }{\partial t}.\end{array}\end{eqnarray}$

The corresponding one-parameter Lie transformation group can be obtained by solving the initial problem for the system of ordinary differential equations$\begin{eqnarray*}\begin{array}{rcl}\frac{{\rm{d}}{x}^{* }}{{\rm{d}}\varepsilon } & = & {\xi }^{}({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* }),{x}^{* }{| }_{\varepsilon =0}=x,\\ \frac{{\rm{d}}{y}^{* }}{{\rm{d}}\varepsilon } & = & \eta ({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* }),{y}^{* }{| }_{\varepsilon =0}=y,\\ \frac{{\rm{d}}{t}^{* }}{{\rm{d}}\varepsilon } & = & \tau ({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* }),{t}^{* }{| }_{\varepsilon =0}=t,\\ \frac{{\rm{d}}{u}^{* }}{{\rm{d}}\varepsilon } & = & \phi ({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* }),{u}^{* }{| }_{\varepsilon =0}=u,\\ \frac{{\rm{d}}{v}^{* }}{{\rm{d}}\varepsilon } & = & {\rm{\Psi }}({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* }),{v}^{* }{| }_{\varepsilon =0}=v.\end{array}\end{eqnarray*}$

Hence, we obtain the following six one-parameter groups H₆ of symmetries:$\begin{eqnarray*}\begin{array}{l}{g}_{1}:({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* })\to (x+\varepsilon ,y,t,u,v),\\ {g}_{2}:({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* })\to (x,y+\varepsilon ,t,u,v),\\ {g}_{3}:({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* })\to (x{{\rm{e}}}^{\varepsilon },y{{\rm{e}}}^{4\varepsilon },t,u{{\rm{e}}}^{\varepsilon },v),\\ {g}_{4}:({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* })\to (x,y,t,u,v+\varepsilon ),\\ {g}_{5}:({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* })\to (x{{\rm{e}}}^{\varepsilon },y,t{{\rm{e}}}^{2\varepsilon },u{{\rm{e}}}^{-\varepsilon },v),\\ {g}_{6}:({x}^{* },{y}^{* },{t}^{* },{u}^{* },{v}^{* })\to (x,y,t+\varepsilon ,u,v).\end{array}\end{eqnarray*}$

Since any linear combination of infinitesimal generator is also an infinitesimal generator, there are infinitely many different symmetry subgroups for the differential equation. Therefore it is necessary and significant to find a way to determine which subgroups can produce the essentially different types of group invariant solutions. To this end, utilizing Ovsiannikov’s global matrix method [5, 9, 10] combining the Ibragimov’s method [11–13], one can construct the optimal system of one-dimensional subalgebras. An arbitrary operator of Lie algebra L₆ is written as$\begin{eqnarray*}V={a}_{1}{V}_{1}+{a}_{2}{V}_{2}+{a}_{3}{V}_{3}+{a}_{4}{V}_{4}+{a}_{5}{V}_{5}+{a}_{6}{V}_{6}.\end{eqnarray*}$Based on six Lie point symmetries, we can compute the nonzero commutation relations which are listed as follows(4)$\begin{eqnarray}\begin{array}{rcl}[{V}_{1},{V}_{3}] & = & {V}_{1},[{V}_{2},{V}_{3}]=4{V}_{2},\\ [{V}_{1},{V}_{5}] & = & {V}_{1},[{V}_{5},{V}_{6}]=-2{V}_{6}.\end{array}\end{eqnarray}$Using equation (4) and following formula for the adjoint representation$\begin{eqnarray*}\begin{array}{rcl}\mathrm{Ad}(\exp (\varepsilon {V}_{i})){V}_{j} & = & {V}_{j}-\varepsilon [{V}_{i},{V}_{j}]\\ & & +\displaystyle \frac{{\varepsilon }^{2}}{2}[{V}_{i},[{V}_{i},{V}_{j}]]-\cdots ,\end{array}\end{eqnarray*}$the adjoint action of the Lie group H₆ on the Lie algebra L₆ has been obtained and listed in table 1.

Table 1.
Table 1.The Adjoint Action of the Lie Group H₆ on the Lie Algebra L₆.

Ad	V₁	V₂	V₃	V₄	V₅	V₆
V₁	V₁	V₂	${V}_{3}-\varepsilon {V}_{1}$	V₄	${V}_{5}-\varepsilon {V}_{1}$	V₆
V₂	V₁	V₂	${V}_{3}-4\varepsilon {V}_{2}$	V₄	V₅	V₆
V₃	${V}_{1}{{\rm{e}}}^{\varepsilon }$	${V}_{2}{{\rm{e}}}^{4\varepsilon }$	V₃	V₄	V₅	V₆
V₄	V₁	V₂	V₃	V₄	V₅	V₆
V₅	${V}_{1}{{\rm{e}}}^{\varepsilon }$	V₂	V₃	V₄	V₅	${V}_{6}{{\rm{e}}}^{-2\varepsilon }$
V₆	V₁	V₂	V₃	V₄	${V}_{5}-2\varepsilon {V}_{6}$	V₆

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Applying the adjoint action of V₁ to $V\,={a}_{1}{V}_{1}\,+{a}_{2}{V}_{2}+{a}_{3}{V}_{3}+\cdots +{a}_{6}{V}_{6}$ and with the help of adjoint representation table, we have$\begin{eqnarray*}\begin{array}{l}\mathrm{Ad}(\exp ({\varepsilon }_{1}{V}_{i}))({a}_{1}{V}_{1}+{a}_{2}{V}_{2}+{a}_{3}{V}_{3}\,+\cdots +\,{a}_{6}{V}_{6})\\ =\,{a}_{1}\mathrm{Ad}(\exp ({\varepsilon }_{1}{V}_{1})){V}_{1}+{a}_{2}\mathrm{Ad}(\exp ({\varepsilon }_{1}{V}_{1})){V}_{2}\,+\cdots +\,{a}_{6}\mathrm{Ad}(\exp ({\varepsilon }_{1}{V}_{1})){V}_{6}\\ =\,({V}_{1},{V}_{2}\cdots ,{V}_{6}){A}_{1}{\left({a}_{1},{a}_{2},\cdots ,{a}_{6}\right)}^{{\rm{T}}},\end{array}\end{eqnarray*}$with(5)$\begin{eqnarray}{A}_{1}=\left(\begin{array}{cccccc}1 & 0 & -{\varepsilon }_{1} & 0 & -{\varepsilon }_{1} & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\end{array}\right).\end{eqnarray}$Similarly,(6)$\begin{eqnarray}\begin{array}{rcl}{A}_{2} & = & \left(\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & -4{\varepsilon }_{2} & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\end{array}\right);\\ {A}_{3} & = & \left(\begin{array}{cccccc}{{\rm{e}}}^{{\varepsilon }_{3}} & 0 & 0 & 0 & 0 & 0\\ 0 & {{\rm{e}}}^{4{\varepsilon }_{3}} & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\end{array}\right).\end{array}\end{eqnarray}$

$\begin{eqnarray*}{A}_{4}=I\end{eqnarray*}$

(7)$\begin{eqnarray}\begin{array}{rcl}{A}_{5} & = & \left(\begin{array}{cccccc}{{\rm{e}}}^{{\varepsilon }_{5}} & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & {{\rm{e}}}^{-2{\varepsilon }_{5}}\end{array}\right);\\ {A}_{6} & = & \left(\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & -2{\varepsilon }_{6} & 1\end{array}\right).\end{array}\end{eqnarray}$

Then, the general adjoint transformation matrix A is constructed by$\begin{eqnarray*}\begin{array}{l}A={A}_{1}{A}_{2}{A}_{3}{A}_{4}{A}_{5}{A}_{6}\\ \quad =\left(\begin{array}{cccccc}{{\rm{e}}}^{{\varepsilon }_{5}}{{\rm{e}}}^{{\varepsilon }_{3}} & 0 & -{\varepsilon }_{1} & 0 & -{\varepsilon }_{1} & 0\\ 0 & {{\rm{e}}}^{4{\varepsilon }_{3}} & -4{\varepsilon }_{2} & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & -2{{\rm{e}}}^{-2{\varepsilon }_{5}}{\varepsilon }_{6} & {{\rm{e}}}^{-2{\varepsilon }_{5}}\end{array}\right).\\ \end{array}\end{eqnarray*}$

Adjoint transformation equation for the (2+1)-dimensional KdV system is$\begin{eqnarray*}A{\left({a}_{1},{a}_{2},\cdots ,{a}_{6}\right)}^{{\rm{T}}}={\left({b}_{1},{b}_{2},\cdots ,{b}_{6}\right)}^{{\rm{T}}},\end{eqnarray*}$namely(8)$\begin{eqnarray}\left(\begin{array}{c}{{\rm{e}}}^{{\varepsilon }_{3}+{\varepsilon }_{5}}{a}_{1}-{\varepsilon }_{1}({a}_{3}+{a}_{5})\\ {{\rm{e}}}^{4{\varepsilon }_{3}}{a}_{2}-4{\varepsilon }_{2}{a}_{3}\\ {a}_{3}\\ {a}_{4}\\ {a}_{5}\\ -2{{\rm{e}}}^{-2{\varepsilon }_{5}}{\varepsilon }_{6}{a}_{5}+{a}_{6}{{\rm{e}}}^{-2{\varepsilon }_{5}}\end{array}\right)=\left(\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\\ {b}_{4}\\ {b}_{5}\\ {b}_{6}\end{array}\right).\end{eqnarray}$

The transformation A maps the vector V to the vector$\begin{eqnarray*}\begin{array}{rcl}{V}^{* } & = & {b}_{1}{V}_{1}^{* }+{b}_{2}{V}_{2}^{* }+{b}_{3}{V}_{3}^{* }+{b}_{4}{V}_{4}^{* }\\ & & +{b}_{5}{V}_{5}^{* }+{b}_{6}{V}_{6}^{* }.\end{array}\end{eqnarray*}$V is equivalent to V^* if and only if equation (8) has the nonzero solution.

Theorem 1 The following operators provide a one-parameter optimal system to the Lie algebra L₆.

$\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Omega }}}_{1} & = & {V}_{1}{{\rm{\Omega }}}_{7}={V}_{3}\pm {V}_{5}\\ {{\rm{\Omega }}}_{13} & = & {V}_{4}\pm {V}_{6}{{\rm{\Omega }}}_{19}={V}_{2}\pm {V}_{4}\pm {V}_{5}\\ {{\rm{\Omega }}}_{2} & = & {V}_{2}{{\rm{\Omega }}}_{8}={V}_{2}\pm {V}_{5}\\ {{\rm{\Omega }}}_{14} & = & {V}_{4}\pm {V}_{2}{{\rm{\Omega }}}_{20}={V}_{3}\pm {V}_{4}\pm {V}_{6}\\ {{\rm{\Omega }}}_{3} & = & {V}_{3}{{\rm{\Omega }}}_{9}={V}_{4}\pm {V}_{5}\\ {{\rm{\Omega }}}_{15} & = & {V}_{1}\pm {V}_{2}{{\rm{\Omega }}}_{21}={V}_{2}\pm {V}_{4}\pm {V}_{6}\\ {{\rm{\Omega }}}_{4} & = & {V}_{4}{{\rm{\Omega }}}_{10}={V}_{3}\pm {V}_{4}\\ {{\rm{\Omega }}}_{16} & = & {V}_{1}\pm {V}_{4}{{\rm{\Omega }}}_{22}={V}_{1}\pm {V}_{2}\pm {V}_{4}\\ {{\rm{\Omega }}}_{5} & = & {V}_{5}{{\rm{\Omega }}}_{11}={V}_{3}\pm {V}_{6}\\ {{\rm{\Omega }}}_{17} & = & {V}_{1}\pm {V}_{6}{{\rm{\Omega }}}_{23}={V}_{1}\pm {V}_{2}\pm {V}_{6}\\ {{\rm{\Omega }}}_{6} & = & {V}_{6}{{\rm{\Omega }}}_{12}={V}_{2}\pm {V}_{6}\\ {{\rm{\Omega }}}_{18} & = & {V}_{3}\pm {V}_{4}\pm {V}_{5}{{\rm{\Omega }}}_{24}={V}_{1}\pm {V}_{4}\pm {V}_{6}\\ {{\rm{\Omega }}}_{25} & = & {V}_{1}\pm {V}_{2}\pm {V}_{4}\pm {V}_{6}\end{array}\end{eqnarray*}$

Proof.the construction of the optimal system of one-dimensional subalgebras will be divided into four cases.

Case 1. ${a}_{5}\ne 0,{a}_{3}\ne 0$.We take ${\varepsilon }_{1}=\tfrac{{{\rm{e}}}^{{\varepsilon }_{3}+{\varepsilon }_{5}}{a}_{1}}{{a}_{3}+{a}_{5}}$ in the transformation (8) and reduce ${b}_{1}=0$. By taking ${\varepsilon }_{2}=\tfrac{{{\rm{e}}}^{4{\varepsilon }_{3}}{a}_{2}}{4{a}_{3}}$, we can make ${b}_{2}=0$. By taking ${\varepsilon }_{6}=\tfrac{{a}_{6}}{2{a}_{5}}$, we can make ${b}_{6}=0$. Thus, the vector $b={({b}_{1},{b}_{2},\cdots ,{b}_{6})}^{{\rm{T}}}$ is reduced to the form$\begin{eqnarray*}b={\left(\mathrm{0,0},{b}_{3},{b}_{4},{b}_{5},0\right)}^{{\rm{T}}}.\end{eqnarray*}$The corresponding representatives for the optimal system are obtained, namely$\begin{eqnarray*}{V}_{3}\pm {V}_{5},{V}_{3}\pm {V}_{4}\pm {V}_{5}.\end{eqnarray*}$

Case 2.${a}_{5}\ne 0,{a}_{3}=0$. We take ${\varepsilon }_{1}=\tfrac{{{\rm{e}}}^{{\varepsilon }_{3}+{\varepsilon }_{5}}{a}_{1}}{{a}_{5}}$ in the transformation (8) to reduce ${b}_{1}=0$. By taking ${\varepsilon }_{6}=\tfrac{{a}_{6}}{2{a}_{5}}$, we can make ${b}_{6}=0$. Then the vector $b={({b}_{1},{b}_{2},\cdots ,{b}_{6})}^{{\rm{T}}}$ is hence reduced to the form$\begin{eqnarray*}b={\left(0,{b}_{2},0,{b}_{4},{b}_{5},0\right)}^{{\rm{T}}}.\end{eqnarray*}$The corresponding representatives for the optimal system are obtained, namely$\begin{eqnarray*}{V}_{5},{V}_{2}\pm {V}_{5},{V}_{4}\pm {V}_{5},{V}_{2}\pm {V}_{4}\pm {V}_{5}.\end{eqnarray*}$

Case 3.${a}_{5}=0,{a}_{3}\ne 0$. We take ${\varepsilon }_{1}=\tfrac{{{\rm{e}}}^{{\varepsilon }_{3}+{\varepsilon }_{5}}{a}_{1}}{{a}_{3}}$ in the transformation (8) and reduce ${b}_{1}=0$. By taking ${\varepsilon }_{2}=\tfrac{{{\rm{e}}}^{4{\varepsilon }_{3}}{a}_{2}}{4{a}_{3}}$, we can make ${b}_{2}=0$. The vector $b={({b}_{1},{b}_{2},\cdots ,{b}_{6})}^{{\rm{T}}}$ is hence reduced to the form$\begin{eqnarray*}b={\left(\mathrm{0,0},{b}_{3},{b}_{4},0,{b}_{6}\right)}^{{\rm{T}}}.\end{eqnarray*}$The corresponding representatives for the optimal system are obtained, namely$\begin{eqnarray*}{V}_{3},{V}_{3}\pm {V}_{4},{V}_{3}\pm {V}_{6},{V}_{3}\pm {V}_{4}\pm {V}_{6}.\end{eqnarray*}$

Case 4.${a}_{5}=0,{a}_{3}=0$. The vector $b={({b}_{1},{b}_{2},\cdots ,{b}_{6})}^{{\rm{T}}}$ is reduced to the form$\begin{eqnarray*}b={\left({b}_{1},{b}_{2},0,{b}_{4},0,{b}_{6}\right)}^{{\rm{T}}}.\end{eqnarray*}$The corresponding representatives for the optimal system are obtained, namely$\begin{eqnarray*}\begin{array}{l}{V}_{1},{V}_{2},{V}_{4},{V}_{6},{V}_{1}\pm {V}_{2},{V}_{1}\pm {V}_{4},{V}_{1}\pm {V}_{6},{V}_{2}\pm {V}_{4},{V}_{2}\pm {V}_{6},{V}_{4}\pm {V}_{6},{V}_{1}\,\pm \\ {V}_{2}\pm {V}_{4},{V}_{1}\pm {V}_{2}\pm {V}_{6},{V}_{1}\pm {V}_{4}\pm {V}_{6},{V}_{2}\pm {V}_{4}\pm {V}_{6},{V}_{1}\pm {V}_{2}\pm {V}_{4}\pm {V}_{6}.\end{array}\end{eqnarray*}$

3. Similarity Reductions and the Invariant Solutions

In this section, we mainly lay emphasis on the similarity reductions of the (2+1)-dimensional KdV equation (1) on the basis of the optimal system of the one-dimensional subalgebras. Meanwhile, we give four examples of general invariant solutions that can be obtained directly by solving the reduced equations. In addition, table 2 has listed the other reduced equations, which are nonlinear partial differential equations (NLPDEs). To obtain the invariant solutions of this reduced NLPDEs, we need use the Lie point symmetry method again.

Table 2.
Table 2.Reductions of the (2+1)-dimensional KdV system.

Case	Similarity variables	invariant solutions
	$\xi =\tfrac{x}{{t}^{\tfrac{1}{2}}},\eta =y$
${{\rm{\Omega }}}_{9}{V}_{4}+{V}_{5}$	$u=\tfrac{f(\xi ,\eta )}{x}$	${\xi }^{3}{g}_{\xi }+4f+2\xi {g}_{\xi }f-2\xi {f}_{\xi }=0$
	$v={ln}(x)+g(\xi ,\eta )$	${\xi }^{3}{f}_{\xi \eta }-2\xi {f}_{\eta }{f}_{\xi }+4{f}_{\eta }f-2\xi {{ff}}_{\xi \eta }-4-2{\xi }^{3}{g}_{\xi \xi \xi }=0$
	$\xi =x,\eta =y-t$
${{\rm{\Omega }}}_{12}{V}_{2}+{V}_{6}$	$u=f(\xi ,\eta )$	$-{g}_{\eta }-{g}_{\xi }f+{f}_{\xi }=0$
	$v=g(\xi ,\eta )$	$-{f}_{\eta \eta }+{f}_{\eta }{f}_{\xi }+{f}_{\xi \eta }f+{g}_{\xi \xi \xi }=0$
	$\xi =x,\eta =y$
${{\rm{\Omega }}}_{13}{V}_{4}+{V}_{6}$	$u=f(\xi ,\eta )$	$1-{g}_{\xi }f+{f}_{\xi }=0$
	$v=t+g(\xi ,\eta )$	${f}_{\eta }{f}_{\xi }+{f}_{\xi \eta }f+{g}_{\xi \xi \xi }=0$
	$\xi =x-y,\eta =t$
${{\rm{\Omega }}}_{15}{V}_{1}+{V}_{2}$	$u=f(\xi ,\eta )$	${g}_{\eta }-{g}_{\xi }f+{f}_{\xi }=0$
	$v=g(\xi ,\eta )$	$-{f}_{\xi \eta }-{f}_{\xi }^{2}-{{ff}}_{\xi \eta }+{g}_{\xi \xi \xi }=0$
	$\xi =y,\eta =x-t$
${{\rm{\Omega }}}_{17}{V}_{1}+{V}_{6}$	$u=f(\xi ,\eta )$	$-{g}_{\eta }-{g}_{\eta }f+{f}_{\eta }=0$
	$v=g(\xi ,\eta )$	$-{f}_{\xi \eta }+{f}_{\xi }{f}_{\eta }+{{ff}}_{\xi \eta }+{g}_{\eta \eta \eta }=0$
	$\xi =\tfrac{x}{{y}^{\tfrac{1}{2}}},\eta =\tfrac{t}{{y}^{\tfrac{1}{2}}}$
${{\rm{\Omega }}}_{18}{V}_{3}+{V}_{4}+{V}_{5}$	$u=f(\xi ,\eta )$	$\tfrac{1}{2}+\eta {g}_{\eta }-\eta {g}_{\xi }f+\eta {f}_{\xi }=0$
	$v=\tfrac{1}{2}{ln}(t)+g(\xi ,\eta )$	$\xi {f}_{\xi \eta }+\eta {f}_{\eta \eta }+{f}_{\eta }+\xi {f}_{\xi }^{2}+\eta {f}_{\xi }{f}_{\eta }+\xi {{ff}}_{\xi \xi }+\eta {{ff}}_{\xi \eta }+{{ff}}_{\xi }-2{g}_{\eta \eta \eta }=0$
	$\xi =x,\eta =y-t$
${{\rm{\Omega }}}_{21}{V}_{2}+{V}_{4}+{V}_{6}$	$u=f(\xi ,\eta )$	$1-{g}_{\eta }-{g}_{\xi }f+{f}_{\xi }=0$
	$v=t+g(\xi ,\eta )$	$-{f}_{\eta \eta }+{f}_{\xi }{f}_{\eta }+{{ff}}_{\xi \eta }+{g}_{\xi \xi \xi }=0$
	$\xi =x-y,\eta =t$
${{\rm{\Omega }}}_{22}{V}_{1}+{V}_{2}+{V}_{4}$	$u=f(\xi ,\eta )$	${g}_{\eta }-f-{g}_{\xi }f+{f}_{\xi }=0$
	$v=x+g(\xi ,\eta )$	$-{f}_{\xi \eta }-{f}_{\xi }^{2}-{{ff}}_{\xi \xi }+{g}_{\xi \xi \xi }=0$
	$\xi =x-y,\eta =y-t$
${{\rm{\Omega }}}_{23}{V}_{1}+{V}_{2}+{V}_{6}$	$u=f(\xi ,\eta )$	$-{g}_{\eta }-{{fg}}_{\xi }+{f}_{\xi }=0$
	$v=g(\xi ,\eta )$	${f}_{\xi \eta }-{f}_{\eta \eta }-{f}_{\xi }^{2}+{f}_{\xi }{f}_{\eta }-{{ff}}_{\xi \xi }+{{ff}}_{\xi \eta }+{g}_{\xi \xi \xi }=0$
	$\xi =y,\eta =x-t$
${{\rm{\Omega }}}_{24}{V}_{1}+{V}_{4}+{V}_{6}$	$u=f(\xi ,\eta )$	$-{g}_{\eta }-{{fg}}_{\eta }+{f}_{\eta }=0$
	$v=x+g(\xi ,\eta )$	$-{f}_{\xi \eta }+{f}_{\xi }{f}_{\eta }+{{ff}}_{\xi \eta }+{g}_{\eta \eta \eta }=0$
	$\xi =x-y,\eta =y-t$
${{\rm{\Omega }}}_{25}{V}_{1}+{V}_{2}+{V}_{4}+{V}_{6}$	$u=f(\xi ,\eta )$	$-{g}_{\eta }-f-{{fg}}_{\xi }+{f}_{\xi }=0$
	$v=x+g(\xi ,\eta )$	${f}_{\xi \eta }-{f}_{\eta \eta }-{f}_{\xi }^{2}+{f}_{\xi }{f}_{\eta }-{{ff}}_{\xi \xi }+{{ff}}_{\xi \eta }+{g}_{\xi \xi \xi }=0$

New window|CSV

Example 1 ${{\rm{\Omega }}}_{1}$ The characteristic equation of ${{\rm{\Omega }}}_{1}\,={V}_{1}=\tfrac{\partial }{\partial x}$ is(9)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}x}{1}=\displaystyle \frac{{\rm{d}}y}{0}=\displaystyle \frac{{\rm{d}}t}{0}=\displaystyle \frac{{\rm{d}}u}{0}=\displaystyle \frac{{\rm{d}}v}{0}.\end{eqnarray}$Solving equation (9) leads to the following group invariants(10)$\begin{eqnarray}\xi =y,\eta =t,u=f(\xi ,\eta ),v=g(\xi ,\eta ).\end{eqnarray}$Substituting (10) into (1) we obtain reduced (1+1)-dimensional PDEs as follows.(11)$\begin{eqnarray}\left\{\begin{array}{l}{g}_{\eta }=0,\\ {f}_{\xi \eta }=0.\end{array}\right.\end{eqnarray}$Then, one group invariant solution for system (1) is given by(12)$\begin{eqnarray}\begin{array}{l}u(x,y,t)={F}_{1}(t)+{F}_{2}(y),\\ v(x,y,t)={F}_{3}(y),\end{array}\end{eqnarray}$where ${F}_{1}(t)$ is an arbitrary function with respect to t and ${F}_{2}(y),{F}_{3}(y)$ are arbitrary functions with respect to y.

Example 2 ${{\rm{\Omega }}}_{2}$ The characteristic equation of ${{\rm{\Omega }}}_{2}\,={V}_{2}=\tfrac{\partial }{\partial y}$ is(13)$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}x}{0} & = & \displaystyle \frac{{\rm{d}}y}{1}=\displaystyle \frac{{\rm{d}}t}{0}\\ & = & \displaystyle \frac{{\rm{d}}u}{0}=\displaystyle \frac{{\rm{d}}v}{0}.\end{array}\end{eqnarray}$Solving equation (13) leads to the following group invariants(14)$\begin{eqnarray}\xi =x,\eta =t,u=f(\xi ,\eta ),v=g(\xi ,\eta ).\end{eqnarray}$Substituting (14) into (1) we obtain reduced (1+1)-dimensional PDEs as(15)$\begin{eqnarray}\left\{\begin{array}{l}{g}_{\eta }-{g}_{\xi }f+{f}_{\xi }=0,\\ {g}_{\xi \xi \xi }=0.\end{array}\right.\end{eqnarray}$Then, one group invariant solution for system (1) is given by(16)$\begin{eqnarray}\left\{\begin{array}{rcl}u(x,y,t) & = & -\displaystyle \frac{1}{4{F}_{4}{\left(t\right)}^{\tfrac{5}{2}}}({{\rm{e}}}^{\tfrac{{F}_{4}(t){x}^{2}}{2}}(2\sqrt{2\pi }{F}_{4}{\left(t\right)}^{2}{F}_{6}^{\prime} (t)H\\ & & -2\sqrt{2\pi }{F}_{4}(t){F}_{5}(t){F}_{5}^{\prime} (t)H\\ & & +\sqrt{2\pi }{F}_{5}{\left(t\right)}^{2}{F}_{5}^{\prime} (t)H\\ & & +\sqrt{2\pi }{F}_{4}(t){F}_{4}^{\prime} (t)H\\ & & -\,4{F}_{4}{\left(t\right)}^{\tfrac{5}{2}}{{\rm{e}}}^{{F}_{5}(t)x}{F}_{7}(t))-2{F}_{4}{\left(t\right)}^{\tfrac{3}{2}}{F}_{4}^{\prime} (t)x\\ & & -4{F}_{4}{\left(t\right)}^{\tfrac{3}{2}}{F}_{5}^{\prime} (t)\\ & & +2{F}_{5}(t){F}_{4}^{\prime} (t)\sqrt{{F}_{4}(t)}),\\ v(x,y,t) & = & \displaystyle \frac{1}{2}{F}_{4}(t){x}^{2}+{F}_{5}(t)x+{F}_{6}(t),\end{array}\right.\end{eqnarray}$where ${F}_{4}(t),{F}_{5}(t),{F}_{6}(t)$ and ${F}_{7}(t)$ are an arbitrary functions with respect to t and H denotes the equation$\begin{eqnarray*}H={{\rm{e}}}^{\tfrac{{F}_{5}(t)(2{F}_{4}(t)x+{F}_{5}(t))}{2{F}_{4}(t)}}\mathrm{erf}(\displaystyle \frac{{F}_{1}(t)x+{F}_{2}(t)}{\sqrt{2{F}_{1}(t)}}),\end{eqnarray*}$where $\mathrm{erf}(x)=\tfrac{2}{\sqrt{\pi }}{\int }_{0}^{x}{{\rm{e}}}^{-{t}^{2}}{\rm{d}}t$.

Example 3 ${{\rm{\Omega }}}_{14}$ The characteristic equation of ${{\rm{\Omega }}}_{14}={V}_{2}+{V}_{4}=\tfrac{\partial }{\partial y}$ is(17)$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}x}{0} & = & \displaystyle \frac{{\rm{d}}y}{1}=\displaystyle \frac{{\rm{d}}t}{0}\\ & = & \displaystyle \frac{{\rm{d}}u}{0}=\displaystyle \frac{{\rm{d}}v}{1}.\end{array}\end{eqnarray}$Solving equation (17) leads to the following group invariants(18)$\begin{eqnarray}\xi =x,\eta =t,u=f(\xi ,\eta ),v=y+g(\xi ,\eta ).\end{eqnarray}$Substituting (18) into (1) we obtain the same reduced (1+1)-dimensional PDEs as equation(15), so the corresponding invariant solution is the same as equation (16).

Example 4 ${{\rm{\Omega }}}_{16}$ The characteristic equation of ${{\rm{\Omega }}}_{16}\,={V}_{1}+{V}_{4}=\tfrac{\partial }{\partial x}+\tfrac{\partial }{\partial v}$ is(19)$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}x}{1} & = & \displaystyle \frac{{\rm{d}}y}{0}=\displaystyle \frac{{\rm{d}}t}{0}\\ & = & \displaystyle \frac{{\rm{d}}u}{0}=\displaystyle \frac{{\rm{d}}v}{1}.\end{array}\end{eqnarray}$

Solving equation (19) leads to the following group invariants(20)$\begin{eqnarray}\xi =y,\eta =t,u=f(\xi ,\eta ),v=x+g(\xi ,\eta ).\end{eqnarray}$Substituting (20) into (1) we obtain reduced the (1+1)-dimensional PDEs as(21)$\begin{eqnarray}\left\{\begin{array}{l}{g}_{\eta }-f=0,\\ {f}_{\xi \eta }=0.\end{array}\right.\end{eqnarray}$Then, one group invariant solution for system (1) is given by$\begin{eqnarray*}\left\{\begin{array}{l}u(x,y,t)={F}_{8}^{\prime} (t)+{F}_{9}(y),\\ v(x,y,t)={F}_{8}(t)+{{tF}}_{9}(y)+{F}_{10}(y)+x,\end{array}\right.\end{eqnarray*}$where ${F}_{8}(t)$ is an arbitrary function with respect to t and ${F}_{9}(y),{F}_{10}(y)$ are arbitrary functions with respect to y.

4. Nonlinear Self-adjointness and Conservation Laws

In this section, we prove the nonlinear self-adjointness of (2+1)-dimensional KdV system and find the conservation laws provided by the symmetry (3) using Ibragimov’s method [7, 8]. Definition 1 [8]. The system of $\overline{m}$ deferential equations(22)$\begin{eqnarray}{F}_{\overline{\alpha }}(x,u,{u}_{(1)},\cdots ,{u}_{(s)})=0,\overline{\alpha }=1,2,\cdots ,\overline{m},\end{eqnarray}$with m dependent variables $u=({u}^{1},\cdots ,{u}^{m})$ is said to be nonlinearly self-adjoint if the adjoint equations(23)$\begin{eqnarray}\begin{array}{rcl}{F}_{\alpha }^{* }(x,u,v,{u}_{(1)},{v}_{(1)}\cdots ,{u}_{(s)},{v}_{(s)}) & = & \displaystyle \frac{\delta ({v}^{\overline{\beta }}{F}_{\overline{\beta }})}{\delta {u}^{\alpha }}=0,\\ \alpha & = & 1,2,\cdots ,m,\end{array}\end{eqnarray}$upon a substitution(24)$\begin{eqnarray}{v}^{\overline{\alpha }}={\varphi }^{\overline{\alpha }}(x,u),\overline{\alpha }=1,2,\cdots ,\overline{m},\end{eqnarray}$satisfy(25)$\begin{eqnarray}\begin{array}{l}{F}_{\alpha }^{* }(x,u,\varphi (x,u),{u}_{(1)},{\varphi }_{(1)}\cdots ,{u}_{(s)},{\varphi }_{(s)})\\ \quad ={\lambda }_{\alpha }^{\overline{\beta }}{F}_{\overline{\beta }}(x,u,{u}_{(1)},\cdots ,{u}_{(s)}),\\ \quad \alpha =1,2,\cdots ,m,\end{array}\end{eqnarray}$where $\tfrac{\delta }{\delta {u}^{\alpha }}$ denotes the Euler–Lagrange operator$\begin{eqnarray*}\displaystyle \frac{\delta }{\delta {u}^{\alpha }}=\displaystyle \frac{\partial }{\partial {u}^{\alpha }}+{\sum _{j=1}}^{\infty }{\left(-1\right)}^{j}{D}_{{i}_{1}}{D}_{{i}_{2}}\cdots {D}_{{i}_{j}}\displaystyle \frac{\partial }{\partial {u}_{{i}_{1}{i}_{2}\cdots {i}_{j}}}.\end{eqnarray*}$v and φ are the m-dimensional vectors$\begin{eqnarray*}v=({v}^{1},\cdots ,{v}^{\overline{m}}),\varphi =({\varphi }^{1},\cdots ,{\varphi }^{\overline{m}}).\end{eqnarray*}$${\lambda }_{\alpha }^{\overline{\beta }}$ are undetermined coefficients and ${\varphi }_{(\sigma )}$ are derivatives of (24)$\begin{eqnarray*}{\varphi }_{(\sigma )}=\{{D}_{{i}_{1}}{D}_{{i}_{2}}\cdots {D}_{{i}_{\sigma }}({\varphi }^{\overline{\alpha }}(x,u))\},\end{eqnarray*}$and equation (24) means that not all components ${\varphi }^{\overline{\alpha }}$ of φ vanish simultaneously.

Theorem 2 equation (1)(26)$\begin{eqnarray}{\rm{\Delta }}=\left\{\begin{array}{l}{F}_{1}={v}_{t}+{{uv}}_{x}+{u}_{x}=0\\ {F}_{2}={u}_{{yt}}+{u}_{x}{u}_{y}+{{uu}}_{{xy}}+{v}_{{xxx}}=0\end{array}\right.\end{eqnarray}$is nonlinear self-adjoint.

Proof The formal Lagrangian for this equation is$\begin{eqnarray*}L=z({v}_{t}+{{uv}}_{x}+{u}_{x})+w({u}_{{yt}}+{u}_{x}{u}_{y}+{{uu}}_{{xy}}+{v}_{{xxx}}),\end{eqnarray*}$where z and w are new dependent variables, equations (25) are written as$\begin{eqnarray*}\begin{array}{rcl}{F}_{1}^{* } & = & \displaystyle \frac{\delta L}{\delta u}=\displaystyle \frac{\partial L}{\partial u}-{D}_{x}(\displaystyle \frac{\partial L}{\partial {u}_{x}})-{D}_{y}(\displaystyle \frac{\partial L}{\partial {u}_{y}})\\ & & +{D}_{y}{D}_{t}(\displaystyle \frac{\partial L}{\partial {u}_{{yt}}})+{D}_{x}{D}_{y}(\displaystyle \frac{\partial L}{\partial {u}_{{xy}}})\\ & = & {{zv}}_{x}-{z}_{x}+{w}_{{yt}}+{{uw}}_{{xy}}.\\ {F}_{2}^{* } & = & \displaystyle \frac{\delta L}{\delta v}=-{D}_{x}(\displaystyle \frac{\partial L}{\partial {v}_{x}})-{D}_{t}(\displaystyle \frac{\partial L}{\partial {v}_{t}})\\ & & -{D}_{x}{D}_{x}{D}_{x}(\displaystyle \frac{\partial L}{\partial {v}_{{xxx}}})\\ & = & -{{uz}}_{x}-{{zu}}_{x}-{z}_{t}-{w}_{{xxx}}.\end{array}\end{eqnarray*}$By using equation (24), we substitute $z(x,t,y)$ and $w(x,t,y)$ to ${\varphi }^{1}(x,t,y,u,v)$ and ${\varphi }^{2}(x,t,y,u,v)$ respectively. Correspondingly, their derivatives are substituted$\begin{eqnarray*}\begin{array}{rcl}{z}_{x} & = & {D}_{x}({\varphi }^{1})={\varphi }_{x}^{1}+{\varphi }_{u}^{1}{u}_{x}+{\varphi }_{v}^{1}{v}_{x},\\ {z}_{t} & = & {D}_{t}({\varphi }^{1})={\varphi }_{t}^{1}+{\varphi }_{u}^{1}{u}_{t}+{\varphi }_{v}^{1}{v}_{t},\\ {w}_{{yt}} & = & {D}_{y}{D}_{t}({\varphi }^{2}),\qquad \ \ {w}_{{xy}}={D}_{x}{D}_{y}({\varphi }^{2}),\\ {w}_{{xxx}} & = & {D}_{x}{D}_{x}{D}_{x}({\varphi }^{2}),\end{array}\end{eqnarray*}$where D_i is total differentiations operator. Then we obtain the equations(27)$\begin{eqnarray}{F}_{1}^{* }{| }_{z={\varphi }^{1},w={\varphi }^{2}}={\lambda }_{11}{F}_{1}+{\lambda }_{12}{F}_{2}.\end{eqnarray}$(28)$\begin{eqnarray}{F}_{2}^{* }{| }_{z={\varphi }^{1},w={\varphi }^{2}}={\lambda }_{21}{F}_{1}+{\lambda }_{22}{F}_{2}.\end{eqnarray}$Equating the coefficients for v_t and u_yt on both sides of equations (27) and equations (28), we obtain(29)$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{11} & = & {\varphi }_{{yv}}^{2},\qquad {\lambda }_{12}={\varphi }_{u}^{2},\\ {\lambda }_{21} & = & -{\varphi }_{v}^{1},\qquad {\lambda }_{22}=0.\end{array}\end{eqnarray}$Now we calculate the coefficients for the derivative of u and v with respect to x, t and y, taking into account equations (29) and arrive at the following equations:$\begin{eqnarray*}\begin{array}{l}{\varphi }_{u}^{1}={\varphi }_{u}^{2}={\varphi }_{v}^{2}=0,\\ {\varphi }^{1}-{\varphi }_{v}^{1}=0,\\ {\varphi }_{{ty}}^{2}-{\varphi }_{x}^{1}+u{\varphi }_{{xy}}^{2}=0,\\ {\varphi }_{t}^{1}+u{\varphi }_{x}^{1}+{\varphi }_{{xxx}}^{2}=0.\end{array}\end{eqnarray*}$Solving these equations, we obtain$\begin{eqnarray*}{\varphi }^{1}={G}_{1}(y){{\rm{e}}}^{v},{\varphi }^{2}={G}_{2}(y)+\displaystyle \frac{1}{2}{G}_{3}(t){x}^{2}+{G}_{4}(t)x+{G}_{5}(t),\end{eqnarray*}$where ${G}_{i}(y),i=1,2$ and ${G}_{i}(t),i=3,4,5$ are arbitrary functions respectively respect to y and t. Moreover, ${\varphi }^{\alpha }\ne 0,\alpha =1,2$ satisfy the definition of nonlinear self-adjointness.

Theorem 3 [8] Let the system of differential equations (22) be nonlinearly self-adjoint. Specifically, let the adjoint system (23) to (22) be satisfied for all solutions of equations (22) upon a substitution (24)(30)$\begin{eqnarray}{v}^{\overline{\alpha }}={\varphi }^{\overline{\alpha }}(x,u),\overline{\alpha }=1,2,\cdots ,\overline{m}.\end{eqnarray}$Then any Lie point, contact or Lie-Bäcklund symmetry$\begin{eqnarray*}V={\xi }^{i}(x,u,{u}_{(1)}\cdots )\displaystyle \frac{\partial }{\partial {x}^{i}}+{\eta }^{\alpha }(x,u,{u}_{(1)}\cdots )\displaystyle \frac{\partial }{\partial {u}^{\alpha }},\end{eqnarray*}$as well as a nonlocal symmetry of equations (22) leads to a conservation law$\begin{eqnarray*}{D}_{i}{C}^{i}=0,\end{eqnarray*}$constructed by the following formula:$\begin{eqnarray*}\begin{array}{rcl}{C}^{i} & = & {\xi }^{i}L+{W}^{\alpha }[\displaystyle \frac{\partial L}{\partial {u}_{i}^{\alpha }}-{D}_{j}(\displaystyle \frac{\partial L}{\partial {u}_{{ij}}^{\alpha }})\\ & & +{D}_{j}{D}_{k}(\displaystyle \frac{\partial L}{\partial {u}_{{ijk}}^{\alpha }})-\cdots ]\\ & & +{D}_{j}({W}^{\alpha })[\displaystyle \frac{\partial L}{\partial {u}_{{ij}}^{\alpha }}-{D}_{k}(\displaystyle \frac{\partial L}{\partial {u}_{{ijk}}^{\alpha }})\\ & & +\cdots ]+{D}_{j}{D}_{k}({W}^{\alpha })[(\displaystyle \frac{\partial L}{\partial {u}_{{ijk}}^{\alpha }})-\cdots ],\end{array}\end{eqnarray*}$where$\begin{eqnarray*}{W}^{\alpha }={\eta }^{\alpha }-{\xi }^{j}{u}_{j}^{\alpha },\end{eqnarray*}$and L is the formal Lagrangian for the system(22)$\begin{eqnarray*}L={v}^{\overline{\beta }}{F}_{\overline{\beta }}.\end{eqnarray*}$

Above we have proved the nonlinear self-adjointness of the (2+1)-dimensional KdV system. According to this theorem, we can find the conservation laws. Consider the Lie point symmetry $\begin{eqnarray*}\begin{array}{rcl}V & = & \xi \displaystyle \frac{\partial }{\partial x}+\eta \displaystyle \frac{\partial }{\partial y}+\tau \displaystyle \frac{\partial }{\partial t}+\phi \displaystyle \frac{\partial }{\partial u}+\psi \displaystyle \frac{\partial }{\partial v},\end{array}\end{eqnarray*}$and the formal Lagrangian$\begin{eqnarray*}\begin{array}{rcl}L & = & z({v}_{t}+{{uv}}_{x}+{u}_{x})+w(\displaystyle \frac{1}{2}({u}_{{yt}}+{u}_{{ty}})+{u}_{x}{u}_{y}\\ & & +\displaystyle \frac{1}{2}u({u}_{{xy}}+{u}_{{yx}})+{v}_{{xxx}}).\end{array}\end{eqnarray*}$The corresponding conserved vector can be written as$\begin{eqnarray*}\begin{array}{rcl}{C}^{1} & = & \xi L+{W}^{1}[\displaystyle \frac{\partial L}{\partial {u}_{x}}-{D}_{y}\displaystyle \frac{\partial L}{\partial {u}_{{xy}}}]\\ & & +{D}_{y}({W}^{1})\displaystyle \frac{\partial L}{\partial {u}_{{xy}}}\\ & & +{W}^{2}[\displaystyle \frac{\partial L}{\partial {v}_{x}}+{D}_{x}{D}_{x}\displaystyle \frac{\partial L}{\partial {v}_{{xxx}}}]\\ & & -{D}_{x}({W}^{2}){D}_{x}\displaystyle \frac{\partial L}{\partial {v}_{{xxx}}}\\ & & +{D}_{x}{D}_{x}({W}^{2})\displaystyle \frac{\partial L}{\partial {v}_{{xxx}}},\\ {C}^{2} & = & \eta L\,+\,{W}^{1}[\displaystyle \frac{\partial L}{\partial {u}_{y}}-{D}_{x}\displaystyle \frac{\partial L}{\partial {u}_{{yx}}}\\ & & -{D}_{t}\displaystyle \frac{\partial L}{\partial {u}_{{yt}}}]+{D}_{x}({W}^{1})\displaystyle \frac{\partial L}{\partial {u}_{{yx}}}\\ & & +{D}_{t}({W}^{1})\displaystyle \frac{\partial L}{\partial {u}_{{yt}}},\\ {C}^{3} & = & \tau L+{W}^{1}[\displaystyle \frac{\partial L}{\partial {u}_{t}}-{D}_{y}\displaystyle \frac{\partial L}{\partial {u}_{{ty}}}]\\ & & +{D}_{y}({W}^{1})\displaystyle \frac{\partial L}{\partial {u}_{{ty}}}+{W}^{2}\displaystyle \frac{\partial L}{\partial {v}_{t}},\end{array}\end{eqnarray*}$where$\begin{eqnarray*}\begin{array}{rcl}{W}^{1} & = & \phi -\xi {u}_{x}-\eta {u}_{y}-\tau {u}_{t},\\ {W}^{2} & = & \psi -\xi {v}_{x}-\eta {v}_{y}-\tau {v}_{t}.\end{array}\end{eqnarray*}$Denoting $x={x}^{1},y={x}^{2},t={x}^{3},u={u}^{1},v={u}^{2}$.

For ${V}_{1}=\tfrac{\partial }{\partial x}$. We have $\xi =1,\eta =\tau =\phi =\psi =0$ and ${W}^{1}=-{u}_{x},{W}^{2}=-{v}_{x}$. The conserved vector is obtained$\begin{eqnarray*}\begin{array}{rcl}{C}^{1} & = & \displaystyle \frac{1}{2}\left(2{{zv}}_{t}+{{uu}}_{x}{w}_{y}+w(2{u}_{{ty}}+{u}_{x}{u}_{y}+{{uu}}_{{xy}})\right.\\ & & \left.+2{w}_{x}{v}_{{xx}}-2{v}_{x}{w}_{{xx}}\right),\\ {C}^{2} & = & -{u}_{x}(-\displaystyle \frac{1}{2}{w}_{t}+{{wu}}_{x}+\displaystyle \frac{1}{2}(-{{wu}}_{x}-{{uw}}_{x}))\\ & & -\displaystyle \frac{1}{2}{{wu}}_{x,t}-\displaystyle \frac{1}{2}{{uwu}}_{{xx}},\\ {C}^{3} & = & \displaystyle \frac{1}{2}{u}_{x}{w}_{y}-{{zv}}_{x}-\displaystyle \frac{1}{2}{{wu}}_{{xy}}.\end{array}\end{eqnarray*}$

For ${V}_{2}=\tfrac{\partial }{\partial y}$. We have $\eta =1,\xi =\tau =\phi =\psi =0$ and ${W}^{1}=-{u}_{y},{W}^{2}=-{v}_{y}$. Then the conserved vector is obtained$\begin{eqnarray*}\begin{array}{rcl}{C}^{1} & = & -\displaystyle \frac{1}{2}{u}_{y}(2z+{{wu}}_{y}-{{uw}}_{y})-\displaystyle \frac{1}{2}{{uwu}}_{{yy}}+{w}_{x}{v}_{{xy}}\\ & & -{v}_{y}({uz}+{w}_{{xx}})-{{wv}}_{{xxy}},\\ {C}^{2} & = & -\displaystyle \frac{1}{2}(2z({v}_{t}+{u}_{x}+{{uv}}_{x})+{u}_{y}({w}_{t}+{{wu}}_{x}+{{uw}}_{x})\\ & & +w({u}_{{ty}}+{{uu}}_{{xy}}+2{v}_{{xxx}})),\\ {C}^{3} & = & -{{zv}}_{y}+\displaystyle \frac{1}{2}{u}_{y}{w}_{y}-\displaystyle \frac{1}{2}{{wu}}_{{yy}}.\end{array}\end{eqnarray*}$

For ${V}_{3}=x\tfrac{\partial }{\partial x}+4y\tfrac{\partial }{\partial y}+u\tfrac{\partial }{\partial u}$. We have $\xi =x,\eta =4y,\tau =0,\phi =u,\psi =0$ and ${W}^{1}=u-{{xu}}_{x}-4{{yu}}_{y},{W}^{2}\,=-{{xv}}_{x}-4{{yv}}_{y}$. Then the conserved vector is obtained$\begin{eqnarray*}\begin{array}{rcl}{C}^{1} & = & -\displaystyle \frac{1}{2}(2z+{{wu}}_{y}-{{uw}}_{y})(-u+4{{yu}}_{y}+{{xu}}_{x})\\ & & +\,{xz}({v}_{t}+{u}_{x}+{{uv}}_{x})\\ & & -\displaystyle \frac{1}{2}{uw}(3{u}_{y}+4{{yu}}_{{yy}}+{{xu}}_{{xy}})\\ & & -(4{{yv}}_{y}+{{xv}}_{x})({uz}+{w}_{{xx}})+{w}_{x}(4{{yv}}_{{xy}}+{{xv}}_{{xx}})\\ & & +{xw}({u}_{{ty}}+{u}_{y}{u}_{x}+{{uu}}_{{xy}}\\ & & +{v}_{{xxx}})-w(4{{yv}}_{{xxy}}+{{xv}}_{{xxx}}),\\ {C}^{2} & = & \displaystyle \frac{1}{2}(8{yz}({v}_{t}+{u}_{x}+{{uv}}_{x})-(u-4{{yu}}_{y}-{{xu}}_{x})({w}_{t}+{{uw}}_{x})\\ & & +w({u}_{t}+4{{yu}}_{{ty}}+{{uu}}_{x}+4{{yu}}_{y}{u}_{x}\\ & & -{{xu}}_{x}^{2}+4{{yuu}}_{{xy}}-{{xu}}_{{xt}}-{{xuu}}_{{xx}}+8{{yv}}_{{xxx}})),\\ {C}^{3} & = & -\displaystyle \frac{1}{2}{w}_{y}(u-4{{yu}}_{y}-{{xu}}_{x})-z(4{{yv}}_{y}+{{xv}}_{x})\\ & & -\displaystyle \frac{1}{2}w(3{u}_{y}+4{{yu}}_{{yy}}+{{xu}}_{{xy}}).\end{array}\end{eqnarray*}$

For ${V}_{4}=\tfrac{\partial }{\partial v}$. We have $\xi =\eta =\tau =\phi =0,\psi =1$ and ${W}^{1}=0,{W}^{2}=1$. Therefore the conserved vector is obtained$\begin{eqnarray*}\begin{array}{rcl}{C}^{1} & = & {zu}+{w}_{{xx}},\\ {C}^{2} & = & 0,\\ {C}^{3} & = & z.\end{array}\end{eqnarray*}$

For ${V}_{5}=x\tfrac{\partial }{\partial x}+2t\tfrac{\partial }{\partial t}-u\tfrac{\partial }{\partial u}$. We have $\xi =x,\eta =0,\tau =2t,\phi =-u,\psi =0$ and ${W}^{1}=-u-{{xu}}_{x}-2{{tu}}_{t},{W}^{2}\,=-{{xv}}_{x}-2{{tv}}_{t}$. Thus the conserved vector is obtained$\begin{eqnarray*}\begin{array}{rcl}{C}^{1} & = & -\displaystyle \frac{1}{2}(2z+{{wu}}_{y}-{{uw}}_{y})(u+2{{tu}}_{t}+{{xu}}_{x})\\ & & +\,{xz}({v}_{t}+{u}_{x}+{{uv}}_{x})\\ & & -\displaystyle \frac{1}{2}{uw}({u}_{y}+2{{tu}}_{{ty}}+{{xu}}_{{xy}})\\ & & -(2{{tv}}_{t}+{{xv}}_{x})({uz}+{w}_{{xx}})+{w}_{x}(2{{tv}}_{{xt}}+{{xv}}_{{xx}})\\ & & +{xw}({u}_{{ty}}+{u}_{y}{u}_{x}+{{uu}}_{{xy}}+{v}_{{xxx}})\\ & & -w(2{{tv}}_{{xxt}}+{{xv}}_{{xxx}}),\end{array}\end{eqnarray*}$

$\begin{eqnarray*}\begin{array}{rcl}{C}^{2} & = & \displaystyle \frac{1}{2}((u+2{{tu}}_{t}+{{xu}}_{x})({w}_{t}+{{uw}}_{x})-w(2{{tu}}_{{tt}}\\ & & +3{{uu}}_{x}+{{xu}}_{x}^{2}+{u}_{t}(3+2{{tu}}_{x})\\ & & +{{xu}}_{{xt}}+2{{tuu}}_{{xt}}+{{xuu}}_{{xx}})),\\ {C}^{3} & = & \displaystyle \frac{1}{2}({w}_{y}(u+2{{tu}}_{t}+{{xu}}_{x})-2z(2{{tv}}_{t}+{{xv}}_{x})\\ & & -w({u}_{y}+2{{tu}}_{{ty}}+{{xu}}_{{xy}})+4t(z({v}_{t}+{u}_{x}+{{uv}}_{x})\\ & & +w({u}_{{ty}}+{u}_{y}{u}_{x}+{{uu}}_{{xy}}+{v}_{{xxx}}))).\end{array}\end{eqnarray*}$

For ${V}_{6}=\tfrac{\partial }{\partial t}$. We have $\tau =1,\xi =\eta =\phi =\psi =0$ and ${W}^{1}=-{u}_{t},{W}^{2}=-{v}_{t}$. Then the conserved vector is obtained$\begin{eqnarray*}\begin{array}{rcl}{C}^{1} & = & -\displaystyle \frac{1}{2}(2z+{{wu}}_{y}-{{uw}}_{y}){u}_{t}-\displaystyle \frac{1}{2}{{uwu}}_{{ty}}\\ & & +{w}_{x}{v}_{{xt}}-{v}_{t}({uz}+{w}_{{xx}})-{{wv}}_{{xxt}},\\ {C}^{2} & = & \displaystyle \frac{1}{2}({u}_{t}({w}_{t}-{{wu}}_{x}+{{uw}}_{x})-w({u}_{{tt}}+{{uu}}_{{xt}})),\\ {C}^{3} & = & \displaystyle \frac{1}{2}{{wu}}_{{ty}}+{{zu}}_{x}+{{wu}}_{y}{u}_{x}\\ & & +{w}_{y}({{tu}}_{t}+\displaystyle \frac{1}{2}{{xu}}_{x})\\ & & +u(\displaystyle \frac{1}{2}{w}_{y}+{{zv}}_{x}+{{wu}}_{{xy}})+{{wv}}_{{xxx}},\end{array}\end{eqnarray*}$Here z and w are two arbitrary solutions of the adjoint equation (27) and (28), hence provide an infinite number of conservation laws.

5. Conclusion

In this paper, we obtain the Lie point symmetry of the (2+1)-dimensional KdV system applying invariant analysis and construct a one-dimensional optimal system combining the Ovsiannikov’s global matrix method and Ibragimov’s method. Then the symmetry reductions are performed and some group invariant solutions are obtained based on the one-dimensional optimal system most of which are listed in table 2. Furthermore, it is proved that the (2+1)-dimensional KdV system embraces nonlinear self-adjointness and infinite number of conservation laws using Ibragimov’s method. We believe that these results could be useful for the investigation of the phenomenon in which the (2+1)-dimensional KdV system is derived. Meanwhile, there also exist many interesting properties of the (2+1)-dimensional KdV system to be discovered in the future.

Acknowledgments

The authors would like to thank the editors and referees for their useful suggestions which have significantly improved this paper. This work is supported by the National Natural Science Foundation of China (Grant No. 11 771 352, 11 871 396), the Natural Science Foundation of Shaanxi Province (Grant No. 2020JM-431).