Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation

Abstract In this paper, Lie symmetry analysis is performed for the equation derived from $(2+1)$-dimensional higher order Broer-Kaup equation. Meanwhile, the optimal system and similarity reductions based on the Lie group method are obtained. Furthermore, the conservation law is studied via the Ibragimov’s method.


Introduction
Nonlinear partial di erential equations (PDEs) arising in many physical elds like the condense matter physics, plasma physics, uid mechanics and optics and so on. In order to investigate the exact solution of PDEs, a fruitful techniques have been developed, such as traveling wave transformations, inverse scattering method [1], Darboux and Bäcklund transformations [2], Lie symmetry analysis [3][4][5]. Lie symmetry analysis is a very useful method to nd the new solutions of PDEs, which was distribution by Sophus Lie ( − ). In addition on the base of symmetries, the integrability of the nonlinear PDEs, such as group classi cation, optimal system and conservation laws, can be considered. Lie groups, as a type of trandformation groups, can tranfer one solution to another one of a given PDE. In other words, if we get one solution of a PDE, we can obtain the other ones via the symmetry of the PDE. Based on this, we will investigate the Lie symmetry analysis of the given PDE.
Noether's theorem [6] establishes a connection between symmetries of di erential equations and conservation laws. However, there are other methods to study the conservation laws, such as partial Noether's approach, multiplier approach and Ibragimov's method. As stated in [7], the former three methods are not applicable to the nonlinear PDEs that do not admit a Lagrangian. In order to overcome these di culties, Ibragimov's method was proposed [8]. Especially state, on the contribution of Lie symmetry method, signi cant researches have been done on the integrability of the nonlinear PDEs, group classi cation, optimal system, reduced solutions and conservation laws, such as [9][10][11][12][13][14] and [15][16][17][18][19][20][21] published this year and last year.
The ( + )-dimensional higher order Broer-Kaup equation was considered in [22] and [23], whose expression is as follows: Such that (1.1) becomes a single di erential equation: (1.2) For (1.2), we consider its special case. That is, U = U(x, t) is regarded as ( + )-dimensional and replaced by u, then (1.2) becomes For convenience to cite later, we call (1.3) to be Li-Mei system, which is equivalent to The exact traveling wave solutions have been investigated in [24]. However, to the best of our knowledge, the Lie symmetry, optional system and conservation law of Li-Mei equation have not been researched, which is the original intention of this work. This paper is organized as follows. In section 2, we perform Lie symmetry analysis of Li-Mei system. In section 3, the optimal system and similarity reductions are studied. Section 4 distributes to studying the conservation law in the method of Ibragimov's and construction the conserved vectors.

Lie symmetries of Li-Mei equation (1.3)
Lie symmetries analysis will be performed of Eq. (1.3) in this section. Consider a one-parameter Lie group of transformations: With a small parameter ε . The vector eld associated with the above transformation group can assumed as: Thus the third prolongation pr ( ) V is: where only the terms involved in (1.3) appear in (2.3). In (2.3), ϕ x , ϕ t , ϕ xx and ϕ xxx are all undetermined functions, which are given by the following formulae.
where Dx, D t are denoted the total derivatives with respect to x and t, respectively. The determining equation of Eq. (1.3) arises from the following invariance condition: By (2.8), we have the following symmetry condition: ) whenever it appears, and comparing the coe cients of the various monomials in the rst, second and third order partial derivatives, and solving the system, we obtain the expression of ξ (x, t, u), τ(x, t, u) and ϕ(x, t, u).
where c , c , c are arbitrary constants.
Hence the in nitesimal generators of Eq. (1.3) can be listed as follows By solving the following ordinary di erential equations with initial condition: (2.13) We therefore obtain the group transformation which is generated by the in nitesimal generators V , V , V , respectively: Here G , G , G are all one-dimensional Lie groups generated by their own generators g i,ε , whose operation is manifested by (2.14),(2.15),(2.16), respectively.
It is trivial that V , V , V form a -dimensional Lie algebra L with the following Lie bracket:

Theorem 1. The vector elds V , V and V supply a representation of the Lie algebra
The de nition of representations of Lie algebras see [25].
Proof. It is su ce if we take the representation space to be the set of all the analytic functions and the linear mapping ρ : Remark 3. The vector elds V and V have trivial prolongation. However, the prolongation of V can be computed: It is easy to check pr ( ) V (∆) = − · ∆, which is called the symmetry invariance of di erential equation (1.3).
We are now to take an example to illustrate the applications of Lie symmetry analysis. We take u t = uxx as an example rather than Eq. (1.3) since it is di cult to nd the analytical solution. The vector elds of this . Under the operation of Lie group generated by V -V , we can check that are all the solutions of u t = uxx.

Optimal system of one-dimensional subalgebras
The more technical matters arose in order to classify the subalgebra of Lie algebra generated by Lie point symmetries, for instance [26] and [3]. A concise method to get the optimal system was presented by Ibragimov in 2010 [27]. In this section we shall construct an optimal system of one-dimensional subalgebra. and

2)
where both ν and µ are arbitrary constants.
Proof. Suppose W and V are two vector eld and dW dε = adV| W , W( ) = w .
By solving this ODE we have W(ε) = Ad(exp(εV))W , by summing the Lie series [3] Ad(exp(εV))W = ∞ n= ε n n! (adV) n (W ) (3.4) In view of (3.4), we obtain Ad(exp(εV i ))V i = V i , i = , , ; Ad(exp(εV ))V = V , Ad(exp(εV ))V = V − εV ; Ad(exp(εV ))V = V , Ad(exp(εV ))V = V − εV ; Ad(exp(εV ))V = e ε V , Ad(exp(εV ))V = e ε V . For an arbitrary nonzero vector our task is to simplify as many of the coe cients a i as possible through the applications of adjoint maps to V. Case 1. a ≠ . Scaling V if necessary, we can assume that a = . By making use of (3.5) and acting on such a V by Ad(exp(ε a V )), we can make the coe cient of V vanish: Next we act on V by Ad(exp(εa V )), to cancel the coe cient of V . Hence V is equivalent to V under the adjoint representation.
Case 2. a = . Subcase 1. a ≠ , a ≠ . Without losing generality, we can assume that a = . One can easily gure out that the adjoint representation induced by any combinations of V , V , V shall make a V + V invariant. In other words, any one-dimensional subalgebra generated by V is equivalent to the subalgebra generated by a V + V . Subcase 2. a = , a ≠ . Similarly to the discussion of Subcase 1, we can conclude that V is equivalent to V under the adjoint representation. The other optimal system can be obtained similarly.

Similarity reductions and exact solutions for Eq. (1.3)
In the preceding section, we got the optimal system of Eq. where ζ = x − νt, which is a traveling wave transformation. By substituting (4.1) into Eq. (1.3), we reduce this equation to the following ODE where f = df dζ , ν ≠ . The traveling wave solutions were obtained in [24]. ( ) For the generator V , we have  (4.4) where f = df dζ . For optimal system II, we only discuss the similarity reductions of V and V + µV . ( ) For the generator V , we have u = f (ζ ), (4.5) where ζ = x. By substituting (4.5) into Eq. (1.3), we reduce this equation to the following ODE where f = df dζ . ( ) For the linear combination V + µV , we have where ζ = µx − t, which follows that this ODE where f = df dζ and µ ≠ .    In the above, we sketch the graphs of f (ζ ) in Eqs.

Nonlinear self-adjointness and conservation law
First of all we show that Li-Mei equation is nonlinearly self-adjoint.
admitted by the system of Eq. (5.1)gives rise to a conservation law, where the components C i of the conserved vector C = (C , · · · , C n ) are determined by For the generator V = ξ ∂ ∂x + τ ∂ ∂t + ϕ ∂ ∂u , we have W = ϕ − ξux − τu t , we therefore obtain the following components of conserved vector Taking the formal Lagrangian L given by (5.6) into (5.13) and (5.14), we can simplify the expressions of C x and C t as follows For the generator V = ∂ ∂x , it has the Lie characteristic function W = −ux. By using of the formulae (5.15) and (5.16), it can give rise to the following components of the conserved vector For the generator V = ∂ ∂t , we have W = −u t , the formulae (5.15) and (5.16) yield the following components of the conserved vector For the generator V = x ∂ ∂x + t ∂ ∂t − u ∂ ∂u , we have W = −u − xux − tu t , the formulae (5.15) and (5.16) imply the following components of the conserved vector C x = −(u + xux + tu t )( u v + vux − uvx + vxx) − ( ux + xuxx + tu xt )( uv − vx) − v( uxx + xuxxx + tu xxt ), These vectors involve an arbitrary solution v of the adjoint equation (5.7) and hence provide an in nite number of conservation laws.

Conclusions
In this paper, we have obtained the symmetries and the corresponding Lie algebras of Li-Mei system by using Lie symmetry analysis method. Meanwhile, the optimal system and its similarity reductions are investigated. Furthermore, we proved that it is nonlinearly self-adjoint. Finally, the conserved vectors were constructed via the Ibragimov's method.
The vector elds generate the equation under consideration supply a representation of a Lie algebra. However, for a given nitely dimensional Lie algebra, such as nine types of simply Lie algebras, how to get its representation via vector elds? If we have already obtained the vector elds, can we get the di erential equation which generates the vector eld? If the di erential equation is obtained, is it unique? All of them are the aims that we will study in the near future.