Lie Symmetry Analysis, Self-Adjointness and Conservation Law for a Type of Nonlinear Equation

: In the present paper, we mainly focus on the symmetry of the solutions of a given PDE via Lie group method. Meanwhile we transfer the given PDE to ODEs by making use of similarity reductions. Furthermore, it is shown that the given PDE is self-adjoining, and we also study the conservation law via multiplier approach.


Introduction
When investigating the classification of the solutions of differential equations, Sophus Lie introduced the notion of the continuous transformation group, which was named the Lie group by the following people in memory of him.Afterwards, the application of Lie groups has been developed by lots of mathematicians, such as Ovsiannikov [1], Ibragimov [2], Olver [3] and Bluman et al. [4,5].Symmetry is one of the intrinsic features of a partial differential equation (PDE).Lie group analysis is one of the most powerful methods to study the symmetry of their solutions.Based on the symmetries of a PDE, many other properties of the equation, such as exact solutions to conservation laws can be successively considered.In any case, another method to solve the nonlinear evolution equation (such as Burgers equation) is to use Hopf cole transformation in a high order spectral column setting [6,7].
The conservation laws (Cls) have drawn great attention from the mathematical physicists.In the past few decades, many methods for dealing with the Cls were derived, the most famous being Noether's approach [8], multiplier approach and Ibragimov's method [9].Moreover, Cls for nonlinearly self-adjoint systems can be constructed systematically by pairs of symmetries and adjoint symmetries.In particular, ref. [10] supplies the most general connection between Cls and pairs of symmetries and adjoint symmetries for non-self-adjoint systems.On the contribution of the Lie symmetry method, significant studies have been performed on the integrability of the nonlinear PDEs, group classification, optimal system, reduced solutions and conservation laws, such as [11][12][13][14][15][16][17][18][19][20][21][22][23], and references therein.
In [24,25], a new completely integrable equation was proposed.It was shown that this equation has no smooth solitons and it has bi-Hamiltonian structure and Lax pair.At the end of [25], Qiao introduced a more general PDE with a constant k ∈ R.
, it is a Harry-Dym case; if k = 2, it is reduced to (1).Unless explicitly stated, throughout this paper we always assume k = 0, −1.Sakovich in [26] supplied a transformation which was used to derive smooth soliton solutions of the new equation from the known rational and soliton solutions of the old one.To the authors' knowledge, the Lie symmetry analysis, self-adjointness and conservation laws of (2) have not been studied.In this paper, we mainly focus on the above aspects on (2) for k = 0, −1.
If we perform the transformation u(x, t) = 1 m(x,t) , Equation ( 2) is reduced to From now on, we will focus on the study of Equation ( 2), which is equivalent to that of Equation (2).

Lie Symmetries of Equation (2)
In this section, we shall investigate the Lie symmetry analysis of Equation ( 2).First of all, let us consider the Lie group of point transformations with a small parameter ε 1.The vector field associated with the above transformation group of transformations can expressed as where ξ(x, t, u), τ(x, t, u) and φ(x, t, u) are coefficient functions of the vector field to be determined.
Theorem 1.For the arbitrary real parameter k, if k = 0, −1, the complete group classification of Equation ( 2) is as follows: (1) (2 Proof.The third prolongation of V is where the explicit expressions of unknown functions φ t , φ x , φ xx , φ xxx are as follows In ( 9)-( 12), D x and D t are both total derivatives with respect to x, t, respectively.We denote the left hand side of Equation ( 2) by ∆.In view of pr (3) V(∆)| ∆=0 = 0, we obtain We substitute ( 9)-( 12) into (8) and replace u t by whenever it appears in (13) and divide it into several cases to discuss it.Firstly we assume k = 1, 2. By solving (13), one can get following forms for the infinitesimal elements ξ, τ, φ: If k = 1 2 , Equation ( 14) implies ξ = c, τ = −(k + 1)at + b, φ = au, which yields the vector fields in (6).
Secondly, we assume k = 1 or k = 2, similar process yields the vector fields in (6).
It is necessary to check that the vector fields form a Lie algebra, respectively.Taking (7) as an example, We denote this Lie algebra by L. If we drop the relation in ( 7) of V 1 , • • • , V 5 , it will be an abstract Lie algebra.The vector fields corresponding to V 1 , • • • , V 5 supply a Lie algebra representation of L (see [27]).In addition, L can be decomposed as L = s ⊕ n, where s = span{V 1 , V 4 , V 5 } denotes the simple ideal of L, n = span{V 2 , V 3 } stands for the nilpotent subalgebra of L.
As we know, infinitesimal generators V i generate the one-parameter transform groups, by solving the following ordinary differential equations with the initial conditions: and case k = 1 2 Remark 1.Any solution of Equation ( 2) when k = 1 2 is invariant under the operation of G 1 , G 2 , G 3 and their products in ( 15)- (17).
Any solution of Equation ( 2) when k = 1 2 is invariant under the operation of G 1 -G 5 and their products in ( 18)- (22).
is also invariant under the operation of the prolongations of V 1 -V 3 .We take V 3 as an example, since the prolongations of V 1 and V 2 are trivial.

Similarity Reductions for Equation (2)
In this section, we shall construct the similarity variables so as to deal with the symmetry reduction, which transfer the PDE into ODE.Considering the fact that the vector fields of k = 1 2 and k = 1 2 are partly the same, we shall discuss them together.(1) For the generator V 1 , we assume ζ = t, u = f (ζ) and obtain the trivial solution f = c, where c is an arbitrary nonzero constant.
(3) For the generator V 3 , we have where ζ = x.Substituting (25) into Equation (2), we arrive at where f = d f dζ .(4) For the generator V 4 , we have where ζ = t.By substituting (27) into Equation ( 2), we get the trivial solution f = c, where c is an arbitrary nonzero constant.
(5) For the generator V 5 , we have where ζ = t, we get the trivial solution f = c, where c is an arbitrary nonzero constant.
The ODEs ( 24) and ( 26) hold for arbitrary k, including 1 2 , but the similarity reductions of V 4 and V 5 only belong to the equation when k = 1 2 .

Nonlinear Self-Adjointness and Conservation Law
For given PDEs R β (x, u, u (1) , • • • , u (k) ) = 0, (29) define the Euler-Lagrange operator and the formal Lagrangian The adjoint equations is given by Theorem 2. The differential Equation ( 2) is self-adjoining if and only if k = 1.