On the Conservation Laws and Exact Solutions of a Modified Hunter-Saxton Equation

It is well known that in order to obtain the physical meanings of the equation considered below, conservation laws are the key instruments. They can be observed in a variety of fields such as obtaining the numerical schemas, Lyapunov stability analysis, and numerical integration. In the literature there exist a lot ofmethods (see, [1–7]). A detailed reviewof existing methods in the literature can be found in [8]. In addition, we observe some valuable software computer packages in this area [9, 10]. In this work, we study the modified Hunter-Saxton (MHS) equation


Introduction
It is well known that in order to obtain the physical meanings of the equation considered below, conservation laws are the key instruments.They can be observed in a variety of fields such as obtaining the numerical schemas, Lyapunov stability analysis, and numerical integration.In the literature there exist a lot of methods (see, [1][2][3][4][5][6][7]).A detailed review of existing methods in the literature can be found in [8].In addition, we observe some valuable software computer packages in this area [9,10].
In this work, we study the modified Hunter-Saxton (MHS) equation which is a third order nonlinear partial differential equation (PDE).This equation has been first suggested by Hunter and Saxton [11] for the theoretical modeling of nematic liquid crystals.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 [11,12].Geometric interpretations and integrability properties of (1) are studied by some authors [13,14].Johnpillai and Khalique [12] showed that the underlying equation admits three parameter Lie-point symmetry generators.Using these generators they obtained an optimal system of one-dimensional subalgebras.Symmetry reductions and exact solutions are obtained.Moreover, using the variational method, they constructed an infinite number of nonlocal conservation laws by the transformation of the dependent variable of the underlying equation.In [15], Nadjafikhah and Ahangari investigated the Lie symmetries and conservation laws of second order nonlinear hyperbolic Hunter-Saxton equation (HSE).The conservation laws of the HSE are computed via three different methods including Boyer's generalization of Noether's theorem, first homotopy method, and second homotopy method.
In this work, we investigate local conservation laws of (1).For this aim, we consider Ibragimov's nonlocal conservation and Steudel's multiplier methods, respectively.In addition, we obtain some reductions and exact solutions using the relationship between conservation laws and Lie-point symmetries [16].
The outline of the paper is as follows.In Section 2, we discuss some main operator identities and their relationship.Then, in Section 3, we briefly give nonlocal conservation, multiplier, and double reduction methods.In Section 4, local symmetry generators are constructed with two distinct methods.In this section symmetry reductions and exact solutions are also obtained.Finally, in Section 5, conclusions are presented.

Preliminaries
We briefly present notation to be used and recall basic definitions and theorems which utilize below [2,7,16].Consider the th-order system of PDEs of  independent variables  = ( 1 ,  2 , . . .,   ) and  dependent variables  = ( 1 ,  2 , . . .,   ):   (, ,  (1) , . . .,  () ) = 0,  = 1, . . ., , where  () is the collection of th-order partial derivatives,    =   (  ),    =     (  ), . .., respectively, with the total differentiation operator with respect to   given by in which the summation convention is used.The Lie-point generator is where   and   are functions of only independent and dependent functions.The operator ( 4) is an abbreviated form of the infinite formal sum where the additional coefficients can be determined from the prolongation formulae The Noether operators associated with a Lie-point generator  are in which   is the Lie characteristic function The conserved vector of (2), where each   ∈ ,  is the space of all differential functions, satisfies the equation along the solution of (2).

Conservation Laws Methods
With this formal Lagrangian, adjoint equation is constructed.Here / is the Euler-Lagrange operator and defined by Theorem 1 (see [7]).Every Lie-point, Lie-Bäcklund, and nonlocal symmetry of (2) gives a conservation law for the equation under consideration.The conserved vector components are determined with where Lagrangian (formal Lagrangian) function is given by ,   are the coefficient functions of the associated generator (4).
The conserved vectors obtained from ( 14) involve the arbitrary solutions  of the adjoint equation ( 12), and hence one obtains an infinite number of conservation laws for (1) by choosing .Definition 2. We say that ( 2) is strictly self-adjoint if the adjoint equation ( 12) becomes equivalent to (2) after the substitution  = :    =  (, , ,   , . . .,   ) , with  being generic coefficient.

The Multiplier Method.
A multiplier Λ  (, ,   , . ..) has the property that holds identically.Here we will consider multipliers of third order; that is, Λ  = Λ  (, , ,   ,   ,   ).The right hand side of ( 17) is a divergence expression.The determining equation for the multiplier Λ  is Once the multipliers are obtained, the conserved vectors are calculated via a homotopy formula [5,17].All the multipliers can be calculated with the aid of (18) for which the equation can be expressed as a local conservation law [9].

Double Reduction Method.
Let  be any Lie-point symmetry, and   are the components of conserved vector.If  and  satisfy then  is associated with .We define a nonlocal variable V by   = V  ,   = −V  .Taking the similarity variables , ,  with the generator  = /, we have in similarity variables so that the conservation law is rewritten as Using the chain rule, we have so that and so Using the above linear algebraical system, we can get The components   ,   depend on (, , ,  (1) ,  (2) , . . .,  (−1) ) which means that   ,   depend on (, , ,   ,   , . . .,  (−1) ) for solutions invariant under .Therefore (21) becomes (  /) +     = 0.
Equation (2) of order  with two independent variables, which admits a symmetry  that is associated with a conserved vector , is reduced to an ODE of order  − 1, namely,   = , where   is given by (26) for solutions invariant under .

Main Results
Firstly we use the nonlocal conservation method given by Ibragimov.Equation (1) admits the following three Lie-point symmetry generators [12]: Equation ( 1) does not have the usual Lagrangian.The Lagrangian for ( 1) is The adjoint equation for ( 1) is  * (, , , , . . .,   ) and we can get the adjoint equation where  is the adjoint variable.Let us investigate the quasiself-adjointness of (1).We make the ansatz of  = ().
Taking into account (29) of  * and using (16) Equation ( 31) should be satisfied identically in all variables   ,   ,   , . ... Comparing the coefficients of   in both sides of (31) we can easily obtain  = −  .Then we equate all coefficients of linear and nonlinear mixed derivatives terms and get () =  1  +  2 .

Advances in Mathematical Physics
The conserved components of (1), associated with a symmetry, can be obtained from (14) where  is Lie characteristic function.According to (31), we can determine  at two cases  1 = 1,  2 = 0 and  1 = 0,  2 = 1 has an infinite number of solutions.The conservation laws associated with the generators (27) are below.Firstly we take  = .
Case 1. Now, let us make calculations for the operator  1 = / in detail.For this operator, the infinitesimals are   = 1,   = 0, and  = 0 and we get  = −  and the corresponding conserved vector of (1) as It is readily seen that in this case we obtain null conserved vectors by the definition of conservation laws.
Case 2. In this case for the generator  2 = / (  = 0,   = 1, and  = 0), we calculate  = −  and the conserved quantities of (1) as The divergence condition becomes We observe that extra terms emerge.By some adjustments, these terms can be absorbed as into the conservation law.Taking these terms across and including them into the conserved flows, we get The modified conserved quantities are now labeled T , where   ( T ) +   ( T ) = 0, modulo the equation.It is readily seen that in this case we obtain null conserved vectors by the definition of conservation laws.
Case 3. Let us find the conservation law provided by  3 = (/) − (/) (  = 0,   = , and  = −).In this case we have  = − −   and (32) yield the conservation laws (9) with The divergence of (38) is After some adjustments the nontrivial conserved quantities are as follows: For the second case (, ) = 1 the corresponding conservation laws are as follows.
Case 4. For the generator  1 = / and Lie characteristic function  = −  , we get the following conserved vectors: Again, like in Case 1 we obtain the null conserved vectors.
Case 5.In this case for the generator  2 = / (  = 0,   = 1, and  = 0), we calculate  = −  and the conserved quantities of (1) as After adjustment according to divergence we get modified conserved vectors Again, like in Case 2 we obtain the null conserved vectors.
Case 6. Lastly we consider the generator  3 = (/) − (/), where   = 0,   = , and  = −.In this case we have  = − −   and (32) yield the conservation laws ( 9) with We calculate the divergence Following the same line we find that the modified nontrivial conserved vectors are Now, we will derive the conservation laws of the MHS equation by the multiplier method.The third order multiplier for (1) is Λ(, , ,   ,   ,   ,   ) and the corresponding determining equation is Expanding and then separating (47) with respect to different combinations of derivatives of  yields the following overdetermined system for the multipliers: The solution of system (48) can be expressed as where  1 ,  2 are constants.Corresponding to the above multiplier, we have the following conserved vectors of (49): The multiplier approach gave two local conservation laws for the MHS equation.Now, we will derive the exact group-invariant solution of (1) using the relationship between local conservation laws and Lie-point symmetries.Equation ( 1 We set  =  1 +  2 .Then the canonical coordinates of  are  = ,  =  −  and .Since  = (  ,   ) is associated with , we have to find the value of   .Using (26) we obtain the following conserved vector: We can substitute the variables   =  and   = (/) in (52).After using these variables, (52) reduces to first order ordinary differential equation (ODE): We can solve (53) by separation of variables and the solution gives rise to which constitutes the solution of the MHS equation.

Conclusion
In this work, we studied conservation laws, symmetry reductions, and exact solutions of MHS equation.Utilizing nonlocal conservation and multiplier method, we constructed four distinct local conservation laws (see (40), (46), and (50)).It is clear that by using Ibragimov's nonlocal conservation method one can obtain infinite nonlocal conservation laws.Then, using the double reduction method we reduced the MHS equation to second order ODE in the canonical variables (see (52)).Exact group-invariant solutions were constructed by integrating the reduced ODE.
as follows: