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

Authors: Sait San; Emrullah Yaşar
Journal: Advances in Mathematical Physics
 (2014)

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## Abstract

We study the modified Hunter-Saxton equation which arises in modelling of nematic liquid crystals. We obtain local conservation laws using the nonlocal conservation method and multiplier approach. In addition, using the relationship between conservation laws and Lie-point symmetries, some reductions and exact solutions are obtained.


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## Body

## 1. 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, [1–7]). 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(1)ut-2uxuxx-uxxxu-uxxt=0,
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 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 [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 Section2, 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.


## 2. Preliminaries

We briefly present notation to be used and recall basic definitions and theorems which utilize below [2, 7, 16]. Consider the kth-order system of PDEs of n independent variables x=(x1,x2,…,xn) and m dependent variables u=(u1,u2,…,um):
(2)Eα(x,u,u(1),…,u(k))=0,α=1,…,m,
where u(i) is the collection of ith-order partial derivatives, uiα=Di(uα),  uijα=DjDi(uα),…, respectively, with the total differentiation operator with respect to xi given by
(3)Di=∂∂xi+uiα∂∂uα+uijα∂∂ujα+⋯,i=1,…,n,
in which the summation convention is used. The Lie-point generator is
(4)X=ξi∂∂xi+ηα∂∂uα,
where ξi and ηα are functions of only independent and dependent functions. The operator (4) is an abbreviated form of the infinite formal sum
(5)X=ξi∂∂xi+ηα∂∂uα+∑s≥1‍ζi1i2⋯isα∂∂ui1i2⋯isα,
where the additional coefficients can be determined from the prolongation formulae
(6)ζiα=Di(ηα)-ξjujiα,ζi1⋯isα=Di1⋯Dis(ζi1⋯is-1α)-ξjuji1⋯isα,s>1.
The Noether operators associated with a Lie-point generator X are
(7)Ni=ξi+Wαδδuiα+∑s≥1∞Di1⋯Dis∂∂ui1⋯isα,i=1,2,…,n,
in which Wα is the Lie characteristic function
(8)Wα=ηα-ξjujα.
The conserved vector of (2), where each Ti∈A, A is the space of all differential functions, satisfies the equation
(9)DiT|(2)i=0,
along the solution of (2).


## 3. Conservation Laws Methods

### 3.1. Nonlocal Conservation Method

We will denote independent variablesx=(x1,x2) with x1=x, x2=t, one dependent variable u together with its derivatives up to p arbitrary order. The pth-order PDE
(10)E(x,u,u1,…,up)=0
has always formal Lagrangian. Formal Lagrangian is multiplication of a new adjoint variable, w(x,t), with a given equation. Namely,
(11)L=wαEα.
With this formal Lagrangian,
(12)E*=δLδuα
adjoint equation is constructed. Here δ/δu is the Euler-Lagrange operator and defined by
(13)δδuα=∂∂uα+∑s≥1∞(-1)sDi1⋯Dis∂∂ui1⋯isα,α=1,…,m.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
(14)Ti=ξiL+Wα[∂L∂ui-Dj(∂L∂uij)+DjDk(∂L∂uijk)000000000000-DjDkDm(∂L∂uijkm)]+Dj(Wα)[∂L∂uij-Dk(∂L∂uijk)+DkDm(∂L∂uijkm)]+DjDk(Wα)[∂L∂uijk-Dm(∂L∂uijkm)]+DjDkDm(Wα)[(∂L∂uijkm)],
where Lagrangian (formal Lagrangian) function is given by
(15)L=wαEα(x,u,u(1),…,u(k)).ξi, ηα are the coefficient functions of the associated generator (4).The conserved vectors obtained from (14) involve the arbitrary solutions w of the adjoint equation (12), and hence one obtains an infinite number of conservation laws for (1) by choosing w.Definition 2.
We say that (2) is strictly self-adjoint if the adjoint equation (12) becomes equivalent to (2) after the substitution w=u:
(16)δLδuα=λE(x,t,u,ux,…,uxxt),
with λ being generic coefficient.Definition 3.
We say that (2) is quasi-self-adjoint if the adjoint equation (12) becomes equivalent to (2) after the substitution w=ϕ(u), ϕ(u)≠0.


### 3.2. The Multiplier Method

A multiplierΛα(x,u,ux,…) has the property that
(17)ΛαEα=DiTi
holds identically. Here we will consider multipliers of third order; that is, Λα=Λα(x,t,u,ux,uxx,uxxx). The right hand side of (17) is a divergence expression. The determining equation for the multiplier Λα is
(18)δ(ΛαEα)δuα=0.
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].


### 3.3. Double Reduction Method

LetX be any Lie-point symmetry, and Ti are the components of conserved vector. If X and T satisfy
(19)X(Ti)+TiDj(ξj)-TjDj(ξi)=0,i=1,2,
then X is associated with T. We define a nonlocal variable v by Tt=vx, Tx=-vt. Taking the similarity variables r,  s,  ϖ with the generator X=∂/∂s, we have in similarity variables
(20)Tr=vs,Ts=-vr,
so that the conservation law is rewritten as
(21)DrTr+DsTs=0.
Using the chain rule, we have
(22)Dx=Dx(s)Ds+Dx(r)Dr,Dt=Dt(s)Ds+Dt(r)Dr,
so that
(23)vx=vsDx(s)+vrDx(r)vt=vsDt(s)+vrDt(r)
and so
(24)Tt=TrDx(s)-TsDx(r),Tx=TrDx(s)-TsDt(r).
Using the above linear algebraical system, we can get
(25)Ts=TtDt(s)+TxDx(s)Dt(r)Dx(s)-Dx(r)Dt(s),(26)Tr=TtDt(r)+TxDx(r)Dt(r)Dx(s)-Dx(r)Dt(s).The componentsTx, Tt depend on (t,x,u,u(1),u(2),…,u(q-1)) which means that Ts,  Tr depend on (s,r,ϖ,ϖr,ϖrr,…,ϖr(q-1))for solutions invariant under X. Therefore (21) becomes (∂Ts/∂s)+DrTr=0.ForT associated with X we have XTr=0 and XTs=0. Thus Tr and Tr are invariant under X. This means ∂Ts/∂s=0 and ∂Tr/∂s=0 so that Tr(r,ϖ,ϖr,ϖrr,…,ϖr(q-1))=k, where k is constant.Equation (2) of order q with two independent variables, which admits a symmetry X that is associated with a conserved vector T, is reduced to an ODE of order q-1, namely, Tr=k, where Tr is given by (26) for solutions invariant under X.


## 3.1. Nonlocal Conservation Method

We will denote independent variablesx=(x1,x2) with x1=x, x2=t, one dependent variable u together with its derivatives up to p arbitrary order. The pth-order PDE
(10)E(x,u,u1,…,up)=0
has always formal Lagrangian. Formal Lagrangian is multiplication of a new adjoint variable, w(x,t), with a given equation. Namely,
(11)L=wαEα.
With this formal Lagrangian,
(12)E*=δLδuα
adjoint equation is constructed. Here δ/δu is the Euler-Lagrange operator and defined by
(13)δδuα=∂∂uα+∑s≥1∞(-1)sDi1⋯Dis∂∂ui1⋯isα,α=1,…,m.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
(14)Ti=ξiL+Wα[∂L∂ui-Dj(∂L∂uij)+DjDk(∂L∂uijk)000000000000-DjDkDm(∂L∂uijkm)]+Dj(Wα)[∂L∂uij-Dk(∂L∂uijk)+DkDm(∂L∂uijkm)]+DjDk(Wα)[∂L∂uijk-Dm(∂L∂uijkm)]+DjDkDm(Wα)[(∂L∂uijkm)],
where Lagrangian (formal Lagrangian) function is given by
(15)L=wαEα(x,u,u(1),…,u(k)).ξi, ηα are the coefficient functions of the associated generator (4).The conserved vectors obtained from (14) involve the arbitrary solutions w of the adjoint equation (12), and hence one obtains an infinite number of conservation laws for (1) by choosing w.Definition 2.
We say that (2) is strictly self-adjoint if the adjoint equation (12) becomes equivalent to (2) after the substitution w=u:
(16)δLδuα=λE(x,t,u,ux,…,uxxt),
with λ being generic coefficient.Definition 3.
We say that (2) is quasi-self-adjoint if the adjoint equation (12) becomes equivalent to (2) after the substitution w=ϕ(u), ϕ(u)≠0.


## 3.2. The Multiplier Method

A multiplierΛα(x,u,ux,…) has the property that
(17)ΛαEα=DiTi
holds identically. Here we will consider multipliers of third order; that is, Λα=Λα(x,t,u,ux,uxx,uxxx). The right hand side of (17) is a divergence expression. The determining equation for the multiplier Λα is
(18)δ(ΛαEα)δuα=0.
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].


## 3.3. Double Reduction Method

LetX be any Lie-point symmetry, and Ti are the components of conserved vector. If X and T satisfy
(19)X(Ti)+TiDj(ξj)-TjDj(ξi)=0,i=1,2,
then X is associated with T. We define a nonlocal variable v by Tt=vx, Tx=-vt. Taking the similarity variables r,  s,  ϖ with the generator X=∂/∂s, we have in similarity variables
(20)Tr=vs,Ts=-vr,
so that the conservation law is rewritten as
(21)DrTr+DsTs=0.
Using the chain rule, we have
(22)Dx=Dx(s)Ds+Dx(r)Dr,Dt=Dt(s)Ds+Dt(r)Dr,
so that
(23)vx=vsDx(s)+vrDx(r)vt=vsDt(s)+vrDt(r)
and so
(24)Tt=TrDx(s)-TsDx(r),Tx=TrDx(s)-TsDt(r).
Using the above linear algebraical system, we can get
(25)Ts=TtDt(s)+TxDx(s)Dt(r)Dx(s)-Dx(r)Dt(s),(26)Tr=TtDt(r)+TxDx(r)Dt(r)Dx(s)-Dx(r)Dt(s).The componentsTx, Tt depend on (t,x,u,u(1),u(2),…,u(q-1)) which means that Ts,  Tr depend on (s,r,ϖ,ϖr,ϖrr,…,ϖr(q-1))for solutions invariant under X. Therefore (21) becomes (∂Ts/∂s)+DrTr=0.ForT associated with X we have XTr=0 and XTs=0. Thus Tr and Tr are invariant under X. This means ∂Ts/∂s=0 and ∂Tr/∂s=0 so that Tr(r,ϖ,ϖr,ϖrr,…,ϖr(q-1))=k, where k is constant.Equation (2) of order q with two independent variables, which admits a symmetry X that is associated with a conserved vector T, is reduced to an ODE of order q-1, namely, Tr=k, where Tr is given by (26) for solutions invariant under X.


## 4. Main Results

Firstly we use the nonlocal conservation method given by Ibragimov. Equation (1) admits the following three Lie-point symmetry generators [12]:
(27)X1=∂∂x,X2=∂∂t,X3=t∂∂t-u∂∂u.
Equation (1) does not have the usual Lagrangian. The Lagrangian for (1) is
(28)L=w(ut-2uxuxx-uxxxu-uxxt)=0.
The adjoint equation for (1) is
(29)E*(t,x,u,w,…,wxxxx)=δδu[w(ut-2uxuxx-uxxxu-uxxt)]
and we can get the adjoint equation
(30)E*=wxuxx-wt+wxxux+wxxxu+wxxt=0,
where w is the adjoint variable. Let us investigate the quasi-self-adjointness of (1). We make the ansatz of w=ϕ(u). Taking into account (29) of E* and using (16) together with its consequences, w=ϕ(u), wt=ϕ′ut, wx=ϕ′ux, wxx=ϕ′′ux2+ϕ′uxx, wxxx=ϕ′′′ux3+ϕ′uxxx+3ϕ′′uxuxx,  and  wxxt=ϕ′′′utux2+2ϕ′′uxuxt+ϕ′′utuxx+ϕ′uxxt, we rewrite (30) in the following form:
(31)2ϕ′′uxuxx-ϕ′ut+ϕ′′ux3+uϕ′′′ux3+3uϕ′′uxuxx+uϕ′uxxx+ϕ′′′utux2+2ϕ′′uxuxt+ϕ′′utuxx+ϕ′uxxt=λ(ut-2uxuxx-uxxxu-uxxt).Equation (31) should be satisfied identically in all variables ut,ux,uxx,…. Comparing the coefficients of ut in both sides of (31) we can easily obtain λ=-ϕ′. Then we equate all coefficients of linear and nonlinear mixed derivatives terms and get ϕ(u)=c1u+c2.The conserved components of (1), associated with a symmetry, can be obtained from (14) as follows:
(32)Tt=ξtL+W(∂L∂ut+Dx2∂L∂uxxt)+Dx(W)(-Dx∂L∂uxxt)+Dx2(W)(∂L∂uxxt),Tx=ξxL+W(∂L∂ux-Dx∂L∂uxx+Dx2∂L∂uxxx+DxDt∂L∂utxx)+Dx(W)(∂L∂uxx-Dx∂L∂uxxx-Dt∂L∂utxx)+Dt(W)(-Dx∂L∂utxx)+DxDt(W)(∂L∂uxxt)+Dx2(W)(∂L∂uxxx),
where W is Lie characteristic function. According to (31), we can determine w at two cases c1=1, c2=0 and c1=0, c2=1 has an infinite number of solutions. The conservation laws associated with the generators (27) are below. Firstly we take w=u.Case 1.
Now, let us make calculations for the operatorX1=∂/∂x in detail. For this operator, the infinitesimals are ξx=1,  ξt=0, and η=0 and we get W=-ux and the corresponding conserved vector of (1) as
(33)T1x=-utu,T1t=uxu.
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 generatorX2=∂/∂t (ξx=0,  ξt=1, and  η=0), we calculate W=-ut and the conserved quantities of (1) as
(34)T2x=2utuxxu-uttux+utxxu2+uttxu,T2t=-2uxuxxu-u2uxxx+utuxx-utxux.
The divergence condition becomes
(35)DtTt+DxTx=-utx2+uttxxu-uttxux+utxxut.
We observe that extra terms emerge. By some adjustments, these terms can be absorbed as
(36)DtTt+DxTx=Dt(-utxux+utxxu)
into the conservation law. Taking these terms across and including them into the conserved flows, we get
(37)T2x~=2utuxxu-uttux+utxxu2+uttxu,T2t~=-2uxuxxu-u2uxxx+utuxx-utxxu.
The modified conserved quantities are now labeled Ti~, where Dt(Tt~)+Dx(Tx~)=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 byX3=t(∂/∂t)-u(∂/∂u) (ξx=0, ξt=t, and η=-u). In this case we have W=-u-tut and (32) yield the conservation laws (9) with
(38)T3x=3u2uxx+3uutx-3utux+2tuutuxx-tuxutt+tu2utxx+tuuttx,T3t=-2tuuxuxx-tu2uxxx-u2+2uuxx+tutuxx+ux2+tuxutx.
The divergence of (38) is
(39)DtTt+DxTx=Dx(tututx+3uutx+tuuxxt).
After some adjustments the nontrivial conserved quantities are as follows:
(40)T3x~=3u2uxx-3utux+2tuutuxx-tuxutt+tu2utxx-tututx,T3t~=-2tuuxuxx-tu2uxxx-u2+2uuxx+tutuxx+ux2+tuxutx.
For the second casew(x,t)=1 the corresponding conservation laws are as follows.Case 4.
For the generatorX1=∂/∂x and Lie characteristic function W=-ux,we get the following conserved vectors:
(41)T4x=ut,T4t=-ux.
Again, like in Case1 we obtain the null conserved vectors.Case 5.
In this case for the generatorX2=∂/∂t (ξx=0, ξt=1, and η=0), we calculate W=-ut and the conserved quantities of (1) as
(42)T5x=utuxx+utxux+utxxu+uttx,T5t=-2uxuxx-uuxxx.
After adjustment according to divergence we get modified conserved vectors
(43)T5x~=utuxx+utxux+utxxu+uttx,T5t~=-2uxuxx-uuxxx-uxxt.
Again, like in Case 2 we obtain the null conserved vectors.Case 6.
Lastly we consider the generatorX3=t(∂/∂t)-u(∂/∂u), where ξx=0, ξt=t, and  η=-u. In this case we have W=-u-tut and (32) yield the conservation laws (9) with
(44)T6x=2uuxx+tuxxut+ux2+tuxutx+ututxx+2utx+tuttx,T6t=-2tuxuxx-utuxxx-u+uxx.
We calculate the divergence(45)DtTt+DxTx=Dt(tuxxt+uxx).
Following the same line we find that the modified nontrivial conserved vectors are
(46)T6x~=2uuxx+tuxxut+ux2+tuxutx+tuutxx+2utx+tuttx,T6t~=-2tuxuxx-tuuxxx-u-tuxxt.
Now, we will derive the conservation laws of the MHS equation by the multiplier method. The third order multiplier for (1) is Λ(x,t,u,ux,uxx,uxxx,uxxt) and the corresponding determining equation is
(47)δδu[Λ(ut-2uxuxx-uxxxu-uxxt)]=0.
Expanding and then separating (47) with respect to different combinations of derivatives of u yields the following overdetermined system for the multipliers:(48)Λuu=0,Λt=0,Λx=0,Λux=0,Λuxx=0,Λuxxx=0,Λutxx=0.
The solution of system (48) can be expressed as
(49)Λ=c1u+c2,
where c1, c2 are constants. Corresponding to the above multiplier, we have the following conserved vectors of (49):
(50)T1x=-u2uxx-utxu2+uxut2,T1t=12u2-12uxxu,T2x=-uxxu-ux22,T2t=u-uxx.
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) admits the symmetry generators X1=∂/∂x, X2=∂/∂t associated with the conservation law
(51)Dt(u-uxx)+Dx(-uxxu-ux22)=0.
We set X=X1+αX2. Then the canonical coordinates of X are s=x, r=αx-t and u. Since T=(Tr,Ts) is associated with X, we have to find the value of Tr. Using (26) we obtain the following conserved vector:
(52)Tr=k=α2uurr+α22ur2.
We can substitute the variables ur=p and urr=p(dp/du) in (52). After using these variables, (52) reduces to first order ordinary differential equation (ODE):
(53)p(u)=±u(2ku+c1α2)uα.
We can solve (53) by separation of variables and the solution gives rise to
(54)r+c2=±α2k2ku2+α2uc1∓18×((α3c1ln⁡(12((1/2)α2c1+2ku)2k000000000000000+2ku2+α2uc112((1/2)α2c1+2ku)2k)2)×(k3/2)-1),00000000000000000000000000000000000000r=αx-t,
which constitutes the solution of the MHS equation.


## 5. 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.



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