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

Abstract

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.

1. Introduction

The nonlinear partial differential equation 𝑢_𝑡+𝑢𝑢_𝑥𝑥𝑥=12𝑢²_𝑥𝑥 (1) was proposed by Hunter and Saxton [1]. They showed that the weakly nonlinear waves are described by (1), where 𝑢(𝑥,𝑡) describes the director field of a nematic liquid crystal, 𝑥 is a space variable in a reference frame moving with the linearized wave velocity, and 𝑡 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(𝑆) of the Virasoro group Vir(𝑆) modulo the rotations Rot(𝑆), with respect to the right-invariant homogeneous 𝐻̇¹ metric: ⟨𝑓,𝑔⟩=∫_𝑆𝑓_𝑥𝑔_𝑥𝑑𝑥.

In this paper the modified Hunter-Saxton equation [6] 𝑢_𝑡−2𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥−𝑢_𝑥𝑥𝑡=0 (2) 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 one-dimensional subalgebras of the Lie symmetry algebra of (2) is constructed. Moreover, using the optimal system of subalgebras, symmetry reductions and exact group-invariant 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.

2. 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 𝑘th-order system of partial differential equations (PDEs) of 𝑛 independent variables 𝑥=(𝑥¹,𝑥²,…,𝑥^𝑛) and 𝑚 dependent variables 𝑢=(𝑢¹,𝑢²,…,𝑢^𝑚); namely, 𝐸^𝛼𝑥,𝑢,𝑢₍₁₎,…,𝑢_(𝑘)=0,𝛼=1,…,𝑚, (3) where 𝑢₍₁₎,𝑢₍₂₎,…,𝑢_(𝑘) denote the collections of all first, second,…, 𝑘th-order partial derivatives; that is, 𝑢^𝛼_𝑖=𝐷_𝑖(𝑢^𝛼), 𝑢^𝛼_𝑖𝑗=𝐷_𝑗𝐷_𝑖(𝑢^𝛼),…, respectively, with the total differentiation operator with respect to 𝑥^𝑖 being defined by 𝐷_𝑖=𝜕_𝑥𝑖+𝑢^𝛼_𝑖𝜕_𝑢𝛼+𝑢^𝛼_𝑖𝑗𝜕_𝑢^𝛼_𝑗+⋯,𝑖=1,…,𝑛, (4) and the summation convention is used whenever appropriate.

The conserved vector 𝑇=(𝑇¹,𝑇²,…,𝑇^𝑛) of (3), where each 𝑇^𝑖∈𝒜, 𝒜 is the space of all differential functions, satisfies the equation 𝐷_𝑖𝑇^𝑖=0 (5) along the solutions of (3).

The Lie point symmetry generator in the form of infinite formal sum is given by 𝑋=𝜉^𝑖𝜕_𝑥𝑖+𝜂^𝛼𝜕_𝑢𝛼+_𝑠≥1𝜁^𝛼_{𝑖1𝑖2⋯𝑖𝑠}𝜕_𝑢^𝛼_{𝑖1𝑖2𝑠⋯𝑖}, (6) where 𝜉 and 𝜂 are functions of 𝑥 and 𝑢 and are independent of derivatives of 𝑢, and the additional coefficients are determined uniquely by the prolongation formulae 𝜁^𝛼_𝑖=𝐷_𝑖𝑊^𝛼+𝜉^𝑗𝑢^𝛼_𝑖𝑗,𝜁^𝛼_{𝑖1⋯𝑖𝑠}=𝐷_𝑖1⋯𝐷_𝑖𝑠𝑊^𝛼+𝜉^𝑗𝑢^𝛼_{𝑗𝑖1⋯𝑖𝑠},𝑠>1, (7) in which 𝑊^𝛼 is the Lie characteristic function defined by 𝑊^𝛼=𝜂^𝛼−𝜉^𝑗𝑢^𝛼_𝑗.

Let 𝐿=𝐿(𝑥,𝑢,𝑢₍₁₎,…,𝑢_(𝑙))∈𝒜, 𝑙≤𝑘 be a Lagrangian associated with a Noether symmetry operator 𝑋. If 𝐵¹,…,𝐵^𝑛 are point-dependent gauge terms, then the Noether symmetry operator 𝑋 is determined by 𝑋𝐿+𝐿𝐷_𝑖𝜉^𝑖=𝐷_𝑖𝐵^𝑖,𝑖=1,…,𝑛, (8) and the conserved vectors, 𝑇^𝑖, corresponding to each 𝑋𝑇^𝑖=𝐵^𝑖−𝐿𝜉^𝑖−𝑊^𝛼𝜕𝐿𝜕𝑢^𝛼_𝑖−𝐷_𝑗𝑊^𝛼−𝑊^𝛼𝐷_𝑗𝜕𝐿𝜕𝑢^𝛼_𝑖𝑗,𝑖=1,…,𝑛 (9) are obtained via Noether’s theorem (see [9]).

3. 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 [10–14] and the references therein.

A vector field 𝑋=𝜏(𝑡,𝑥,𝑢)𝜕_𝑡+𝜉(𝑡,𝑥,𝑢)𝜕_𝑥+𝜂(𝑡,𝑥,𝑢)𝜕𝑢 (10) is a generator of point symmetry of (2), if 𝑋^[3]𝑢_𝑡−2𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥−𝑢_{𝑥𝑥𝑡(2)}=0, (11) where the operator 𝑋^[3] is the third prolongation of the operator 𝑋 defined by 𝑋^[3]=𝑋+𝜁_𝑡𝜕_𝑢𝑡+𝜁_𝑥𝜕_𝑢𝑥+𝜁_𝑥𝑥𝑥𝜕_{𝑢𝑥𝑥𝑥}+𝜁_𝑥𝑥𝑡𝜕_{𝑢𝑥𝑥𝑡}, (12) with the coefficients 𝜁_𝑡,  𝜁_𝑥,  𝜁_𝑥𝑥𝑥, and 𝜁_𝑥𝑥𝑡 being given by 𝜁_𝑡=𝐷_𝑡𝜂−𝑢_𝑡𝐷_𝑡(𝜏)−𝑢_𝑥𝐷_𝑡(𝜉),𝜁_𝑥=𝐷_𝑥𝜂−𝑢_𝑡𝐷_𝑥(𝜏)−𝑢_𝑥𝐷_𝑥(𝜉),𝜁_𝑥𝑥=𝐷_𝑥𝜁_𝑥−𝑢_𝑥𝑡𝐷_𝑥(𝜏)−𝑢_𝑥𝑥𝐷_𝑥(𝜉),𝜁_𝑥𝑥𝑥=𝐷_𝑥𝜁_𝑥𝑥−𝑢_𝑥𝑥𝑡𝐷_𝑥(𝜏)−𝑢_𝑥𝑥𝑥𝐷_𝑥(𝜉),𝜁_𝑥𝑥𝑡=𝐷_𝑡𝜁_𝑥𝑥−𝑢_𝑥𝑥𝑡𝐷_𝑡(𝜏)−𝑢_𝑥𝑥𝑥𝐷_𝑡(𝜉). (13) Here, 𝐷_𝑡 and 𝐷_𝑥 are the total derivative operators defined by 𝐷_𝑡=𝜕_𝑡+𝑢_𝑡𝜕_𝑢𝐷+⋯,_𝑥=𝜕_𝑥+𝑢_𝑥𝜕_𝑢+⋯. (14) The coefficient functions 𝜏, 𝜉, and 𝜂 are independent of the derivatives of 𝑢; thus equating the coefficients of like derivatives of 𝑢 in the determining equation (11) yields the following over determined system of linear PDEs: 𝜏=𝜏(𝑡),𝜉=𝜉(𝑡,𝑥),𝜂_𝑢𝑢𝜉=0,_𝑥𝑥−2𝜂_𝑥𝑢=0,2𝜉_𝑥−𝜂_𝑥𝑥𝑢𝜏=0,_𝑡−3𝜉_𝑥+𝜂_𝑢+𝜂_𝑥𝑥𝑢𝜂=0,_𝑡−𝜂_𝑡𝑥𝑥−𝑢𝜂_𝑥𝑥𝑥𝜉=0,_𝑡−𝑢𝜏_𝑡+3𝑢𝜉_𝑥−𝑢𝜂_𝑥𝑥𝑢−𝜂=0,2𝜉_𝑡𝑥+3𝑢𝜉_𝑥𝑥−2𝜂_𝑥−𝜂_𝑡𝑢−3𝑢𝜂_𝑥𝑢𝜉=0,_𝑡𝑥𝑥+𝑢𝜉_𝑥𝑥𝑥−𝜉_𝑡−2𝜂_𝑥𝑥−2𝜂_𝑡𝑥𝑢−3𝑢𝜂_𝑥𝑥𝑢=0. (15) Solving the determining equations (15) for 𝜏, 𝜉, and 𝜂, we obtain the three-dimensional Lie algebra spanned by the following Lie point symmetry generators admitted by (2): 𝑋₁=𝜕_𝑡,𝑋₂=𝜕_𝑥,𝑋₃=𝑡𝜕_𝑡−𝑢𝜕_𝑢. (16)

4. 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 one-dimensional 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 one-dimensional subalgebras for (2) which are given in Table 3 (for more details of the approach see [10]).

Table 1 

Commutator table of the Lie algebra of (2).

Table 2 

Adjoint table of the Lie algebra of (2).

Table 3 

Subalgebras, group invariants, and group-invariant solutions of (2).

Case 1. In this case, the group-invariant solution corresponding to the symmetry generator 𝑋₂ leads to the trivial solution 𝑢(𝑥,𝑡)=𝐶, where 𝐶 is a constant.

Case 2. The group-invariant solution arising from the subalgebra 𝑋₁+𝜖𝑋₂ reduces (2) to the following nonlinear third-order ODE: (ℎ−𝜖)ℎ^′′′+2ℎ^′ℎ^′′+𝜖ℎ^′=0,𝜖≠0, (17) where “prime” denotes differentiation with respect to 𝛾. Integrating (17) twice with respect to 𝛾, we obtain ℎ^′2=𝜖ℎ²+𝐾₁ℎ+𝐾₂,𝜖−ℎ (18) where 𝐾₁ and 𝐾₂ 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 𝜖−𝑢𝜖𝑢²+𝐾₁𝑢+𝐾₂𝑑𝑢=𝐾₃±(𝑥−𝜖𝑡), (19) where 𝐾₃ is an arbitrary constant of integration. One can obtain two particular solutions for (17) given by ℎ𝛾=1𝐵𝛿±𝐵𝛾2²−𝐴𝐵,ℎ𝛾𝜖=−4𝛿±𝛾², (20) where 𝐴=𝐶₁−𝜖², 𝐵=−𝜖, and 𝐶₁, 𝛿 are arbitrary constants. Hence, we obtain the two special group-invariant solutions for (2) given by 𝑢()1𝑥,𝑡=−𝜖𝛿±𝜖(𝜖𝑡−𝑥)2²−𝐶₁−𝜖²,𝑢()𝜖𝑥,𝑡=−4[𝛿±(𝑥−𝜖𝑡)]². (21) Likewise, one can obtain the following three group-invariant steady state solutions of (2) for the case when 𝜖=0: 𝑢(𝑥,𝑡)=√𝐶₁𝐶₃,𝑢()=±𝑥𝑥,𝑡9𝐶₂4𝐶₃±𝑥^21/3,𝐶₁𝑢𝐶₁𝑢+𝐶₂−𝐶₂log𝐶₁𝐶₁𝑢+𝐶₂+𝐶₁√𝑢=𝐶₁^3/2𝐶₃±𝑥. (22) Here, 𝐶₁,  𝐶₂, and 𝐶₃ are arbitrary constants of integration.

Case 3. Substitution of the group-invariant solution corresponding to the subalgebra 𝑋₃+𝑎𝑋₂ into (2) results in the reduced nonlinear third-order ODE (ℎ−𝑎)𝐻^′′′−ℎ^′′+2ℎ^′ℎ^′′+𝑎ℎ^′+ℎ=0, (23) where “prime” denotes differentiation with respect to 𝛾.

5. 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 𝑢=𝑣_𝑥, (2) becomes 𝑣_𝑥𝑡−2𝑣_𝑥𝑥𝑣_𝑥𝑥𝑥−𝑣_𝑥𝑣_{𝑥𝑥𝑥𝑥}−𝑣_{𝑥𝑥𝑥𝑡}=0. (24) Equation (24) has a usual Lagrangian 1𝐿=−2𝑣_𝑡𝑣_𝑥+𝑣_𝑥𝑣²_𝑥𝑥+𝑣_𝑥𝑡𝑣_𝑥𝑥. (25) If 𝑋=𝜏𝜕_𝑡+𝜉𝜕_𝑥+𝜂𝜕_𝑣 is the Noether symmetry operator, then from the Noether operator determining equation (8) we obtain −12𝑣_𝑥𝜁_𝑡−12𝑣_𝑡+𝑣²_𝑥𝑥𝜁_𝑥−𝑣_𝑥𝑣_𝑥𝑥+12𝑣_𝑥𝑡𝜁_𝑥𝑥−12𝑣_𝑥𝑥𝜁_𝑥𝑡+𝜏_𝑡+𝜏_𝑣𝑣_𝑡𝐿+𝜉_𝑥+𝜉_𝑣𝑣_𝑥𝐿=𝐵¹_𝑡+𝐵¹_𝑣𝑣_𝑡+𝐵²_𝑥+𝐵²_𝑣𝑣_𝑥. (26) 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: 𝜏_𝑥=𝜏_𝑣=0,𝜉_𝑡=𝜉_𝑣𝜂=0,_𝑣=0,4𝜉_𝑥−𝜏_𝑡−3𝜂_𝑣𝜂=0,_𝑣−𝜉_𝑥,𝜉_𝑡−𝜂_𝑥𝐵=0,_1𝑣1=−2𝜂_𝑥,𝐵²_𝑣1=−2𝜂_𝑡,𝐵¹_𝑡+𝐵²_𝑥=0. (27) The solution of (27) is 𝜏=𝑐₁, 𝜉=𝑐₂, 𝜂=𝑎(𝑡), 𝐵¹=𝑐(𝑡,𝑥), and 𝐵²=−1/2𝑎^′(𝑡)𝑣+𝑑(𝑡,𝑥), where 𝑐₁, 𝑐₂ are arbitrary constants, 𝑎(𝑡) is an arbitrary function, and 𝑐_𝑡+𝑑_𝑥=0. We set 𝑐=𝑑=0. Thus, we obtain the following Noether symmetry operators associated with the Lagrangian (25) for (24): 𝑋₁=𝜕_𝑡,𝐵¹=0,𝐵²𝑋=0,₂=𝜕_𝑥,𝐵¹=0,𝐵²𝑋=0,₃=𝑎(𝑡)𝜕_𝑣,𝐵¹=0,𝐵²1=−2𝑎^′(𝑡)𝑣. (28) Hence, by invoking (9), we obtain the following conserved vectors corresponding to the Noether symmetry operators (28):

(i) 𝑋₁=𝜕_𝑡, 𝐵¹=0, 𝐵²=0, 𝑇¹=12𝑣_𝑥𝑣²_𝑥𝑥+12𝑣_𝑡𝑣_𝑥𝑥𝑥,𝑇²1=−2𝑣²_𝑡+12𝑣_𝑡𝑣²_𝑥𝑥+𝑣_𝑡𝑣_𝑥𝑥𝑡+𝑣_𝑡𝑣_𝑥𝑣_𝑥𝑥𝑥−12𝑣_𝑡𝑡𝑣_𝑥𝑥−𝑣_𝑥𝑣_𝑡𝑥𝑣_𝑥𝑥−12𝑣²_𝑡𝑥. (29) Hence, from (5), we have 𝐷_𝑡𝑇¹+𝐷_𝑥𝑇²=−𝑣_𝑡𝑣_𝑥𝑡−2𝑣_𝑥𝑥𝑣_𝑥𝑥𝑥−𝑣_𝑥𝑣_{𝑥𝑥𝑥𝑥}−𝑣_{𝑥𝑥𝑥𝑡}+12𝑣_𝑡𝑣_{𝑥𝑥𝑥𝑡}−12𝑣_𝑥𝑥𝑣_𝑥𝑡𝑡=12𝑣_𝑡𝑣_{𝑥𝑥𝑥𝑡}−12𝑣_𝑥𝑥𝑣_𝑥𝑡𝑡=𝐷_𝑡12𝑣_𝑡𝑣_𝑥𝑥𝑥−𝐷_𝑥12𝑣_𝑥𝑥𝑣_𝑡𝑡. (30) Taking these terms across and including them into the conserved flows, we get 𝐷_𝑡𝑇¹−12𝑣_𝑡𝑣_𝑥𝑥𝑥+𝐷_𝑥𝑇²+12𝑣_𝑥𝑥𝑣_𝑡𝑡=0. (31) If we let 𝑇¹=𝑇¹−12𝑣_𝑡𝑣_𝑥𝑥𝑥=12𝑣_𝑥𝑣²_𝑥𝑥,𝑇²=𝑇²+12𝑣_𝑥𝑥𝑣_𝑡𝑡1=−2𝑣²_𝑡+12𝑣_𝑡𝑣²_𝑥𝑥+𝑣_𝑡𝑣_𝑥𝑥𝑡+𝑣_𝑡𝑣_𝑥𝑣_𝑥𝑥𝑥−𝑣_𝑥𝑣_𝑡𝑥𝑣_𝑥𝑥−12𝑣²_𝑡𝑥, (32) one can readily verify that the new components of the conserved vector 𝑇 satisfy the equation 𝐷_𝑖𝑇^𝑖|₍₂₄₎=0.

(ii) Consider 𝑋₂=𝜕_𝑥, 𝐵¹=0, 𝐵²=0, 𝑇¹1=−2𝑣²_𝑥+12𝑣_𝑥𝑣_𝑥𝑥𝑥−12𝑣²_𝑥𝑥,𝑇²=𝑣_𝑥𝑣_𝑥𝑥𝑡+𝑣²_𝑥𝑣_𝑥𝑥𝑥−12𝑣_𝑥𝑡𝑣_𝑥𝑥. (33) Invoking (5), we obtain 𝐷_𝑡𝑇¹+𝐷_𝑥𝑇²=−𝑣_𝑥𝑣_𝑥𝑡−2𝑣_𝑥𝑥𝑣_𝑥𝑥𝑥−𝑣_𝑥𝑣_{𝑥𝑥𝑥𝑥}−𝑣_{𝑥𝑥𝑥𝑡}+12𝑣_𝑥𝑣_{𝑥𝑥𝑥𝑡}−12𝑣_𝑥𝑥𝑣_𝑥𝑥𝑡=12𝑣_𝑥𝑣_{𝑥𝑥𝑥𝑡}−12𝑣_𝑥𝑥𝑣_𝑥𝑥𝑡=𝐷_𝑥12𝑣_𝑥𝑣_𝑥𝑥𝑡−𝐷_𝑡12𝑣²_𝑥𝑥. (34) Taking the terms across and adding them into the conserved flows yield 𝐷_𝑡𝑇¹+12𝑣²_𝑥𝑥+𝐷_𝑥𝑇²−12𝑣_𝑥𝑣_𝑥𝑥𝑡=0. (35) Considering 𝑇¹=𝑇¹+12𝑣²_𝑥𝑥=12𝑣_𝑥𝑣_𝑥𝑥𝑥−12𝑣²_𝑥,𝑇²=𝑇²−12𝑣_𝑥𝑣_𝑥𝑥𝑡=12𝑣_𝑥𝑣_𝑥𝑥𝑡+𝑣²_𝑥𝑣_𝑥𝑥𝑥−12𝑣_𝑥𝑡𝑣_𝑥𝑥, (36) we obtain the modified new components of the conserved vector 𝑇 so that the equation, 𝐷_𝑖𝑇^𝑖|₍₂₄₎=0, is satisfied.

(iii) Consider 𝑋₃=𝑎(𝑡)𝜕_𝑣, 𝐵¹=0, 𝐵²=−(1/2)𝑎(𝑡)𝑣, 𝑇¹=12𝑎(𝑡)𝑣_𝑥−12𝑎(𝑡)𝑣_𝑥𝑥𝑥,𝑇²1=−2𝑎^′(𝑡)1𝑣+2𝑎(𝑡)𝑣_𝑡−12𝑎(𝑡)𝑣²_𝑥𝑥−𝑎(𝑡)𝑣_𝑥𝑥𝑡−𝑎(𝑡)𝑣_𝑥𝑣_𝑥𝑥𝑥. (37) In this case, we have infinite number of conservation laws. Using (5), we have 𝐷_𝑡𝑇¹+𝐷_𝑥𝑇²=𝑎(𝑡)𝑣_𝑥𝑡−2𝑣_𝑥𝑥𝑣_𝑥𝑥𝑥−𝑣_𝑥𝑣_{𝑥𝑥𝑥𝑥}−𝑣_{𝑥𝑥𝑥𝑡}−12𝑎^′(𝑡)𝑣_𝑥𝑥𝑥+𝑎(𝑡)𝑣_{𝑥𝑥𝑥𝑡}1=−2𝑎^′(𝑡)𝑣_𝑥𝑥𝑥+𝑎(𝑡)𝑣_{𝑥𝑥𝑥𝑡}=𝐷_𝑡−12𝑎(𝑡)𝑣_𝑥𝑥𝑥. (38) Taking the term across and adding it into the conserved flows, we obtain 𝐷_𝑡𝑇¹+12𝑎(𝑡)𝑣_𝑥𝑥𝑥+𝐷_𝑥𝑇²=0. (39) If we choose 𝑇¹=𝑇¹+12𝑎(𝑡)𝑣_𝑥𝑥𝑥=12𝑎(𝑡)𝑣_𝑥,𝑇²=𝑇², (40) we obtain 𝐷_𝑖𝑇^𝑖|₍₂₄₎=0.

6. 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.