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Lie symmetry analysis, Lie-Bäcklund symmetries, explicit solutions, and conservation laws of Drinfeld-Sokolov-Wilson system
- Yufeng Zhang^{1}Email author and
- Zhonglong Zhao^{2}
Received: 10 August 2017
Accepted: 9 October 2017
Published: 23 October 2017
Abstract
The symmetry analysis method is used to study the Drinfeld-Sokolov-Wilson system. The Lie point symmetries of this system are obtained. An optimal system of one-dimensional subalgebras is derived by using Ibragimov’s method. Based on the optimal system, similarity reductions and explicit solutions of the system are presented. The Lie-Bäcklund symmetry generators are also investigated. Furthermore, the method of constructing conservation laws of nonlinear partial differential equations with the aid of a new conservation theorem associated with Lie-Bäcklund symmetries is presented. Conservation laws of the Drinfeld-Sokolov-Wilson system are constructed by using this method.
Keywords
MSC
1 Introduction
The Lie symmetry method was initiated by Lie [1] in the second half of the nineteenth century. It has become one of the most powerful methods to study nonlinear partial differential equations (NLPDEs). The core idea of the Lie symmetry method is that one-parameter groups of transformations acting on the space of the independent and dependent variables leave the NLPDEs unchanged [2–5]. Most problems in science and engineering can be represented by NLPDEs [6–9]. Application of the Lie symmetry method for constructing the explicit solutions of the NLPDEs can always be regarded as one of the most important fields of mathematical physics. Many important properties of NLPDEs such as symmetry reductions, conservation laws, and explicit solutions by using symmetries can be considered successively [10–12].
Studying conservation laws is helpful in analyzing NLPDEs in the physical point of view [13]. Some studies have indicated that conservation laws play an important role in the numerical integration of PDEs [14]. Conservation laws are also helpful in solving equations by means of the double reduction theory [15, 16]. In order to construct conservation laws various methods have been developed, such as Noether’s theorem [17], the partial Noether approach [18], the multiplier approach [19], and a new conservation theorem [20, 21]. Noether’s theorem and the partial Noether approach establish a relationship between symmetries and conservation laws for NLPDEs. Nevertheless, these methods do not work on a nonlinear equation without a Lagrangian. It is notable that a new conservation theorem was proposed by Ibragimov [20]. The conservation laws of a nonlinear equation and its adjoint equation can be constructed by using formal Lagrangian.
The DSWS has attracted the attention of many scholars. Some conservation laws of DSWS were obtained with the aid of the multiplier approach [22]. Then the double reduction analysis was employed to study the reductions of some conservation laws for DSWS [23]. This system has an infinite number of conservation laws, has a Lax representation, and is a member of KP hierarchy [24, 25], which indicates the integrality of this system. The scaling invariant Lax pairs of the DSWS were derived by Hickman and Hereman et al. [26]. The homotopy analysis method was applied to obtain the approximate solutions of DSWS [27]. Matjila et al. derived the exact solutions of system (1) by using the \(( {G'/G} ) \)-expansion function method. They also constructed conservation laws using Noether’s approach [28]. We gave some symmetry reductions and conservation laws of this system [29].
This paper is arranged as follows. In Section 2, we derive the Lie point symmetries of the DSWS using Lie group analysis and find the transformed solutions. In Section 3, a new optimal system of subalgebras of system (1) is constructed by using a concise method. The new optimal system contains five operators. Then in Section 4, based on the optimal system, the similarity reduced equations and the explicit solutions of system (1) are investigated systematically. In Section 5, the method of constructing conservation laws of NLPDEs with the aid of the new conservation theorem associated with Lie-Bäcklund symmetries is presented. In Section 6, the conservation laws of the DSWS are constructed. Finally, the conclusions are given in the last section.
2 Lie point symmetries
Theorem 1
3 Optimal system of subalgebras
Over the past few decades, the problem of classifying the subgroup was studied by many researchers. In order to demonstrate the classification of the group invariant solutions, Ovsiannikov put forward the concept of optimal systems of subalgebras of a Lie algebra [2]. Then this method extended to many examples of optimal systems of subgroups for the Lie group of mathematical physics models by Winternitz et al. [31, 32]. Notably, Olver [4] used a different technique for constructing a one-dimensional optimal system, which was based on a commutator table and adjoint representation. Furthermore, Ibragimov presented a concise method to get the optimal system in their paper [33]. Zhao and Han extended this method to the Heisenberg equation [11], the AKNS system [12], and the Broer-Kaup system [34]. This method only relies on a commutator table. In this section, we shall use the Ibragimov method to construct an optimal system of one-dimensional subalgebra of the Lie algebra \({L_{3}}\) for the DSWS.
Commutator table
\(\boldsymbol{[X_{i}, X_{j}]}\) | \(\boldsymbol{X_{1}}\) | \(\boldsymbol{X_{2}}\) | \(\boldsymbol{X_{3}}\) |
---|---|---|---|
\(X_{1}\) | 0 | 0 | \(3X_{1}\) |
\(X_{2}\) | 0 | 0 | \(X_{2}\) |
\(X_{3}\) | \(-3X_{1}\) | \(-X_{2}\) | 0 |
Theorem 2
Proof
- (I)\({l^{2}} \ne 0\). We assume \({l^{2}} = 1\) and make \({l^{1}} = \pm 1\) by the transformations (19). In addition, taking into account the possibility \({l^{1}} = 0\), we obtain the following representatives for the optimal system$$ {X_{2}}, \quad {X_{2}} - {X_{1}}, \quad {X_{2}} + {X_{1}}. $$(24)
- (II)
Let \({l_{2}} = 0\). If \({l^{1}} \ne 0\) we can set \({l^{1}} = 1\) and obtain the vector \(( {1,0,0} ) \). This gives rise to the operator \({X_{1}}\).
4 Symmetry reductions and explicit solutions of the DSWS
Reduction of the DSWS
Case | Similarity variables | Reduced equations |
---|---|---|
(1) \(X_{1}\) | ξ = x, u(x,t)=F(ξ), v(x,t)=H(ξ). | \(H' = 0, 3kFH' + 3bF'H - aH''' = 0\). |
(2) \(X_{2}\) | ξ = t, u(x,t)=F(ξ), v(x,t)=H(ξ). | \(F' = 0\), \(H' = 0\). |
(3) \(X_{3}\) | \(\xi = x{t^{-{{1 \over 3}}}}\), \(u ( {x,t} ) = {t^{ - { {2 \over 3}}}}F ( \xi )\), \(v ( {x,t} ) = {t^{ - { {2 \over 3}}}}H ( \xi ) \). | \(\xi F' - 6HH' + 2F = 0\), \(- 9kFH' - 9bF'H + \xi H' + 3aH''' + 2H = 0\). (A) |
(4) \(X_{2}-X_{1}\) | ξ = x + t, u(x,t)=F(ξ), v(x,t)=H(ξ). | \(2HH' + F' = 0,3bF'H + 3kFH' - aH''' + H' = 0\). |
(5) \(X_{2}+X_{1}\) | ξ = x − t, u(x,t)=F(ξ), v(x,t)=H(ξ). | \(2HH' - F' = 0\), \(3bF'H + 3kFH' - aH''' - H' = 0\). (B) |
4.1 Explicit solutions of system (1) using the simplest equation method
4.2 Explicit solutions of system (1) using generalized tanh method
4.3 Explicit power series solutions of system (1)
Remark 1
The power series method is a useful approach to solve higher order variable coefficient ordinary differential equations. A large number of solutions for ordinary differential equations are constructed by utilizing the method by Liu et al. [45, 46]. Moreover, we can show the convergence of the power series solution (42) just as the method from the paper [46]. If we regard the series solution (42) to a particular section, we get the polynomial solution. In addition, we may obtain the approximate solutions of system (1) by using Newton’s interpolating series [47].
5 Construction of conservation laws using Lie-Bäcklund symmetries
In this section, we briefly present the notations and theorem which are useful for constructing conservation laws with the aid of Lie-Bäcklund symmetries. For more detailed information, the reader is referred to the literature [20, 48–51].
Theorem 3
Remark 2
6 Conservation laws of the DSWS
- (I)For the Lie-Bäcklund symmetry generator \({U_{1}} = {u_{x}}\frac{ \partial }{{\partial u}} + {v_{x}}\frac{\partial }{{\partial v}}\), the Lie characteristic functions are \({W^{1}} = {u_{x}}\) and \({W^{2}} = {v_{x}}\). Using equation (60), we obtain the following components of the conserved vector:$$ \begin{aligned} &T_{1}^{t} = \phi {u_{x}} + \tau {v_{x}},\\ &T_{1}^{x} = 3b\tau {u_{x}}v + 2\phi v{v_{x}} + 3k\tau u{v_{x}} - a{\tau _{xx}}{v_{x}} + a{\tau _{x}}{v_{xx}} - a\tau {v_{xxx}}. \end{aligned} $$(61)
- (II)The Lie-Bäcklund symmetry generator \({U_{2}} = 2v{v_{x}}\frac{ \partial }{{\partial u}} + ( {3b{u_{x}}v + 3ku{v_{x}} - a{v_{xxx}}} ) \frac{\partial }{{\partial v}}\) has the Lie characteristic functions \({W^{1}} = 2v{v_{x}}\) and \({W^{2}} = {3b{u_{x}}v + 3ku{v _{x}} - a{v_{xxx}}}\). Using equation (60), we obtain the following components of the conserved vector:$$\begin{aligned} \begin{gathered} T_{2}^{t} = 2\phi v{v_{x}} + ( {3b{u_{x}}v + 3ku{v_{x}} - a{v _{xxx}}} ) \tau , \\ T_{2}^{x} = 6b\tau {v^{2}}{v_{x}} + (3b{u_{x}}v + 3ku{v_{x}} - a{v_{xxx}})(3k\tau u + 2\phi v - a{\tau _{xx}}) \\ \hphantom{T_{2}^{x} =}{} + a{\tau _{x}}(3b{u_{xx}}v + 3b{u_{x}}{v_{x}} + 3k{u_{x}}{v_{x}} + 3ku{v_{xx}} - a{v_{xxxx}}) \\ \hphantom{T_{2}^{x} =}{}- a\tau ( 3b{u_{xxx}}v + 6b{u_{xx}}{v_{x}} + 3b{u_{x}}{v_{xx}} + 3k{u_{xx}}{v_{x}} + 6k{u_{x}}{v_{xx}} \\ \hphantom{T_{2}^{x} =}{} + 3ku{v_{xxx}} - a{v_{xxxxx}} ). \end{gathered} \end{aligned}$$(62)
- (III)Finally, for the Lie-Bäcklund symmetry generatorthe Lie characteristic functions are \({W^{1}} = 6tv{v_{x}} - {u_{xx}} - 2u\) and \({W^{2}} = 9bt{u_{x}}v + 9ktu{v_{x}} - 3at{v_{xxx}} - {v _{xx}} - 2v\). Using equation (60), we obtain the following components of the conserved vector:$$ {U_{3}} = ( {6tv{v_{x}} - {u_{xx}} - 2u} ) \frac{\partial }{ {\partial u}} + ( {9bt{u_{x}}v + 9ktu{v_{x}} - 3at{v_{xxx}} - {v_{xx}} - 2v} ) \frac{\partial }{{\partial v}}, $$$$\begin{aligned} \begin{aligned} &T_{3}^{t} = \phi ( {6tv{v_{x}} - {u_{xx}} - 2u} ) + ( 9bt{u_{x}}v + 9ktu{v_{x}} - 3at{v_{xxx}} - {v_{xx}} - 2v ) \tau , \\ &T_{3}^{x} = 3b\tau v ( {6tv{v_{x}} - {u_{xx}} - 2u} ) \\ &\hphantom{T_{3}^{x} =}{}+ ( {9bt{u_{x}}v + 9ktu{v_{x}} - 3at{v_{xxx}} - {v_{xx}} - 2v} ) ( {3k\tau u + 2\phi v}- a{\tau _{xx}} ) \\ &\hphantom{T_{3}^{x} =}{}+ a{\tau _{x}}({9bt{u_{xx}}v + 9bt{u_{x}}{v_{x}} + 9kt{u_{x}}{v_{x}} + 9ktu{v_{xx}} - 3at{v_{xxxx}} - {v_{xxx}} - 2{v_{x}}}) \\ &\hphantom{T_{3}^{x} =}{}- a\tau ( 9bt{u_{xxx}}v + 18bt{u_{xx}} {v_{x}} + 9bt {u_{x}} {v_{xx}}+ 9kt{u_{xx}} {v_{x}} + 18kt{u_{x}} {v_{xx}} \\ &\hphantom{T_{3}^{x} =}{}+ 9ktu v_{xxx}- 3at{v_{xxxxx}} - {v_{xxxx}} - 2{v_{xx}}). \end{aligned} \end{aligned}$$(63)
7 Conclusions
Lie symmetry analysis has been employed to investigate Lie point symmetries of the Drinfeld-Sokolov-Wilson system. The symmetries \(X_{1}\), \(X_{2}\), and \(X_{3}\) form a three-dimensional Lie algebra \(L_{3}\). By using Ibragimov’s method, we have derived an optimal system of one-dimensional subalgebra. It is proved that the optimal system has five operators. Based on the optimal system, we have considered the symmetry reductions and group invariant solutions of the DSWS. To the best of our knowledge, very little work has been devoted to constructing conservation laws of NLPDEs by using Lie-Bäcklund symmetries. Lie-Bäcklund symmetries of the DSWS have been derived. The method of constructing conservation laws of NLPDEs with the aid of a new conservation theorem associated with Lie-Bäcklund symmetries has been presented. Conservation laws of the DSWS have been constructed by using this method.
Declarations
Acknowledgements
This work is supported by the Fundamental Research Funds for the Central University (No. 2017XKZD11).
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