Symmetry Reductions, Exact Solutions, and Conservation Laws of a Modified Hunter-Saxton Equation

We study a modified Hunter-Saxton equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the underlying equation are derived. We utilize the Lie algebra admitted by the equation to obtain the optimal system of one-dimensional subalgebras of the Lie algebra of the equation. These subalgebras are then used to reduce the underlying equation to nonlinear third-order ordinary differential equations. Exact traveling wave group-invariant solutions for the equation are constructed by integrating the reduced ordinary differential equations. Moreover, using the variational method, we construct infinite number of nonlocal conservation laws by the transformation of the dependent variable of the underlying equation.


Introduction
The nonlinear partial differential equation was proposed by Hunter and Saxton [1]. They showed that the weakly nonlinear waves are described by (1), where ( , ) describes the director field of a nematic liquid crystal, is a space variable in a reference frame moving with the linearized wave velocity, and is a slow time variable. Liquid crystals are fluids made up of long rigid molecules. Equation (1) is a high-frequency limit of the Camassa-Holm equation [2]. Hunter and Zheng [3] showed that it is bivariational, bi-Hamiltonian, and a member of the Harry Dym hierarchy of integrable flows. It is well known that (1) under reciprocal transformation reduces to the Liouville equation [2]. Also, an interesting geometric interpretation of the Hunter-Saxton equation [4,5] is that, for spatially periodic functions, it describes the geodesic flow on the homogeneous space Vir(S)/Rot( ) of the Virasoro group Vir( ) modulo the rotations Rot( ), with respect to the rightinvariant homogeneouṡ1 metric: ⟨ , ⟩ = ∫ .
In this paper the modified Hunter-Saxton equation [6] is studied from the Lie group analysis standpoint. We first obtain the Lie point symmetry generators and then utilize the Lie algebra admitted by the equation to obtain the optimal system of one-dimensional subalgebras of the Lie algebra of the equation. These subalgebras are later used to reduce the modified Hunter-Saxton equation to nonlinear third-order ordinary differential equations. We then construct exact traveling wave group-invariant solutions for the equation by integrating the reduced ordinary differential equations. Finally, using the variational method, we construct infinite number of nonlocal conservation laws by the transformation of the dependent variable of the underlying equation. The outline of the paper is as follows. In Section 2, we briefly discuss some main operator identities and their relationship. In Section 3, we present the Lie point symmetry generators of (2). In Section 4, the optimal system of onedimensional subalgebras of the Lie symmetry algebra of (2) is constructed. Moreover, using the optimal system of subalgebras, symmetry reductions and exact group-invariant 2 Abstract and Applied Analysis solutions of (2) are obtained. In Section 5, nonlocal conservation laws of (2) are constructed using Noether's theorem. Finally, in Section 6, concluding remarks are presented.

Preliminaries
In this section we present the notations that will be used in the sequel. For details, the reader is referred to [7,8].
Consider a th-order system of partial differential equations (PDEs) of independent variables = ( 1 , 2 , . . . , ) and dependent variables = where (1) , (2) , . . . , ( ) denote the collections of all first, second,. . ., th-order partial derivatives; that is, = ( ), = ( ), . . ., respectively, with the total differentiation operator with respect to being defined by and the summation convention is used whenever appropriate. The conserved vector = ( 1 , 2 , . . . , ) of (3), where each ∈ A, A is the space of all differential functions, satisfies the equation along the solutions of (3). The Lie point symmetry generator in the form of infinite formal sum is given by where and are functions of and and are independent of derivatives of , and the additional coefficients are determined uniquely by the prolongation formulae = ( ) + , in which is the Lie characteristic function defined by = − . Let = ( , , (1) , . . . , ( ) ) ∈ A, ≤ be a Lagrangian associated with a Noether symmetry operator . If 1 , . . . , are point-dependent gauge terms, then the Noether symmetry operator is determined by and the conserved vectors, , corresponding to each are obtained via Noether's theorem (see [9]).

Lie Point Symmetries of (2)
In this section, we discuss in brief the Lie symmetry group method to obtain Lie point symmetry generators admitted by (2). For detailed account of this method see [10][11][12][13][14] and the references therein. A vector field is a generator of point symmetry of (2), if where the operator [3] is the third prolongation of the operator defined by with the coefficients , , , and being given by Here, and are the total derivative operators defined by The coefficient functions , , and are independent of the derivatives of ; thus equating the coefficients of like derivatives of in the determining equation (11) yields the following over determined system of linear PDEs: Solving the determining equations (15) for , , and , we obtain the three-dimensional Lie algebra spanned by the following Lie point symmetry generators admitted by (2):    Table 3: Subalgebras, group invariants, and group-invariant solutions of (2).

Symmetry Reductions and Group-Invariant Solutions of (2)
Here we utilize the Lie point symmetry generators (16) of (2) found in Section 3 to obtain symmetry reduction and construct exact group-invariant solutions for (2). We first present the optimal system of one-dimensional subalgebras of the Lie algebra admitted by (2). The onedimensional subalgebras are then used to reduce (2) to ordinary differential equations (ODEs). Exact group-invariant solutions for the underlying equation (2) are constructed by integrating the reduced ODEs.
The results on the classification of the Lie point symmetries of (2) are summarized in the Tables 1, 2, and 3. The commutator table of the Lie point symmetries of (2) and the adjoint representations of the symmetry group of (2) on its Lie algebra are given in Tables 1 and 2, respectively. Tables 1  and 2 are then used to construct the optimal system of onedimensional subalgebras for (2) which are given in Table 3 (for more details of the approach see [10]). Case 2. The group-invariant solution arising from the subalgebra 1 + 2 reduces (2) to the following nonlinear thirdorder ODE: (ℎ − ) ℎ + 2ℎ ℎ + ℎ = 0, where "prime" denotes differentiation with respect to . Integrating (17) twice with respect to , we obtain where 1 and 2 are arbitrary constants of integration. Separating the variables in (18), integrating, and reverting back to our original variables, one obtains the solution of (2) in quadrature where 3 is an arbitrary constant of integration. One can obtain two particular solutions for (17) given by where = 1 − 2 , = − , and 1 , are arbitrary constants. Hence, we obtain the two special group-invariant solutions for (2) given by Likewise, one can obtain the following three group-invariant steady state solutions of (2) for the case when = 0: Here, 1 , 2 , and 3 are arbitrary constants of integration.
Case 3. Substitution of the group-invariant solution corresponding to the subalgebra 3 + 2 into (2) results in the reduced nonlinear third-order ODE where "prime" denotes differentiation with respect to .

Conservation Laws of (2)
In this section, we obtain nonlocal conservation laws of (2) using the variational approach. Equation (2) does not have a Lagrangian as it is an evolution equation. By making a substitution = V , (2) becomes

Equation (24) has a usual Lagrangian
If = + + V is the Noether symmetry operator, then from the Noether operator determining equation (8) we obtain Expansion of (26) and then equating the coefficients of the various monomials in the first-and second-order partial derivatives of V to zero yield the following determining equations: The solution of (27) is = 1 , = 2 , = ( ), 1 = ( , ), and 2 = −1/2 ( )V + ( , ), where 1 , 2 are arbitrary constants, ( ) is an arbitrary function, and + = 0. We set = = 0. Thus, we obtain the following Noether symmetry operators associated with the Lagrangian (25) for (24): Hence, by invoking (9), we obtain the following conserved vectors corresponding to the Noether symmetry operators (28): (i) 1 = , 1 = 0, 2 = 0, Hence, from (5), we have Taking these terms across and including them into the conserved flows, we get If we let̃1 one can readily verify that the new components of the conserved vector̃satisfy the equatioñ| (24) = 0.
(ii) Consider 2 = , 1 = 0, 2 = 0, Invoking (5), we obtain Taking the terms across and adding them into the conserved flows yield Abstract and Applied Analysis we obtain the modified new components of the conserved vector̃so that the equation,̃| (24) = 0, is satisfied.
In this case, we have infinite number of conservation laws.
Using (5), we have Taking the term across and adding it into the conserved flows, we obtain If we choosẽ1 we obtaiñ| (24) = 0.

Concluding Remarks
In this paper, we studied the modified Hunter-Saxton equation (2) using the Lie symmetry group of infinitesimal transformations of the equation. We found that the underlying equation admits a three-dimensional Lie algebra spanned by the vector fields of translations in time and space and the scaling of time and the dependent variable. We obtained the optimal system of one-dimensional subalgebras of the Lie algebra of the equation. These subalgebras were then used to reduce the underlying equation to nonlinear third-order ordinary differential equations. Exact group-invariant solutions called traveling wave solutions were constructed by integrating the reduced ODEs. Furthermore, we constructed infinite number of nonlocal conservation laws for the underlying equation by the transformation of the dependent variable of the equation and making use of Noether's theorem.