Lie symmetry reductions and exact solutions to a generalized two-component Hunter-Saxton system

1.   Introduction

The Hunter-Saxton (HS) equation reads

where $u(x, t)$ depends on a time variable $t$ and a space variable $x$, $\kappa$ is a positive constant. This equation was derived as a model for propagation of orientation waves in a massive nematic liquid crystal director field ^[1]. In fact, it can be regarded as a short wave limit of the well known Camassa-Holm equation ^[2,3].

The two-component Hunter-Saxton (2-HS) equation ^[1] is

where $u(x, t)$ and $\rho(x, t)$ depend on variables $t$ and $x$, $\sigma, \kappa$ are positive constants. The 2-HS equation has attracted much attention and it has been studied extensively and some results were obtained, we can see ^[4,5].

Meanwhile, there is a generalized 2-HS system ^[6] as follow:

where $\alpha(\alpha\neq 1), \kappa$ are constants. The model with $(\alpha, \kappa) = (-1, -1)$ in system (1.3) appeared initially in the work of Lenells ^[7]. The author showed that system (1.3) is the geodesic equation on a manifold K which admits a $K\ddot{a}hler$ structure. The blow-up phenomena of system (1.3) was investigated in ^[4,8].

Our goal is to study exact solutions of system (1.3) by applying classical Lie group method ^{[9,10,11,12,13,14]}. Firstly, the vector field for the system (1.3) will be given by Lie symmetry analysis. Secondly, similarity variables and its symmetry reductions equations are obtained. Thirdly, by solving the reduced equations, some exact solutions of the system (1.3) will be presented. Finally, we give a conservation law of system (1.3).

2.   Lie symmetry analysis of the system (1.3)

First of all, let us consider a one-parameter Lie group of infinitesimal transformation:

with a small parameter $\epsilon\ll1$. The vector field associated with the above group of transformations can be written as

where the coefficient functions $\xi(x, t, u, \rho), \tau(x, t, u, \rho), \phi(x, t, u, \rho)$ and $\psi(x, t, u, \rho)$ of the vector field are to be determined later.

If the vector field (2.1) generates a symmetry of the system (1.3), then $V$ must satisfy the Lie symmetry condition

where $\mathrm{pr}^{(3)}V, \mathrm{pr}^{(1)}V$ denote the third and the first prolongation of $V$ respectively, and $\Delta_1 = u_{xxt}+uu_{xxx}+(1-\alpha)u_xu_{xx}-\kappa \rho\rho_x$, $\Delta_2 = \rho_t+u\rho_x-\alpha u_x\rho$ for system (1.3). Expanding (2.2), we find that the coefficient functions $\xi, \tau, \phi$ and $\psi$ must satisfy the symmetry condition

where $\phi, \psi, \phi^{x}, \psi^{x}, \psi^{t}, \phi^{xx}, \phi^{xxx}, \phi^{xxt}$ are the coefficient functions given by

where $D_{x}, D_{t}$ are the total derivatives with respect to $x$ and $t$ respectively.

Substituting (2.4) into (2.3), combined with system (1.3) and setting the coefficients of the various monomials in $u$ and $v$ and their partial derivatives equal to zero one obtains the determining equations for the symmetry group of (1.3) as follows

Solving these determining equations yields

where $F_1(t), F_2(t)$ are arbitrary functions of $t$, $C_1, C_2, C_3$ and $C_4$ are arbitrary constants.

Thus, the Lie algebra of infinitesimal symmetries of system (1.3) is spanned by the following vector fields

where $V_1$ and $V_2$ are the vector fields corresponding to the arbitrary functions $F_1(t)$ and $F_2(t)$ respectively.

The commutation relations of Lie algebra determined by $V_i(i = 1, 2, \cdots, 6)$, which are shown as

It is obvious that the vector fields $V_i(i = 1, 2, \cdots, 6)$ are closed under the Lie bracket.

3.   Symmetry reductions

In this section, we will get similarity variables and its symmetry reductions. By solving the reduced equations, some exact solutions of the system (1.3) will be presented.

Based on the infinitesimals (2.6), the similarity variables are found by solving the corresponding characteristic equations

Case 1 Let $C_1 = C_2 = F_1(t) = 0$, $C_3(\neq0)$ and $C_4$ be arbitrary constants, $F_2(t)$ is an arbitrary functions of $t$, then by solving the characteristic equation one can get the similarity variables

and the group-invariant solution is

Substituting the group-invariant solution (3.1) into system (1.3), we reduce equation (1.3) to the following ODE:

where $f' = df/d\omega, g' = dg/d\omega$.

Case 2 Let $C_1, C_3$ be arbitrary non-zero constants, $C_2 = C_4 = F_1(t) = F_2(t) = 0$, then by solving the characteristic equation one can get the similarity variables

and the group-invariant solution is

Substituting the group-invariant solution (3.3) into system (1.3), we reduce (1.3) to the following ODE:

where $f' = df/d\omega, g' = dg/d\omega$.

Case 3 Let $F_1(t) = kt, F_2(t) = 0$, $C_1, C_2, C_3, C_4$ and $k$ be constants which satisfy $C_2-\alpha k\neq0$ and $k+C_1+C_2\neq0$, then by solving the characteristic equation one can get the similarity variables

and the group-invariant solution is

Substituting the group-invariant solution (3.5) into system (1.3), we reduce (1.3) to the following ODE:

where $f' = df/d\omega, g' = dg/d\omega$.

4.   Exact solutions

In this section, we will derive the solutions of system (1.3) by using the symbolic computation ^[15,16,17]. Suppose that the solution of equation (3.2) is in the form

where $F(\omega)$ expresses the solution of the following generalized Riccati equation

and $r, p, q$ are real constants. Substituting (4.1) along with (4.2) into (3.2) and collecting all terms with the same power in $F^i(i = 0, 1, \cdots, 7)$ and setting the coefficients to zero yields a system of algebraic equations. Solving the algebraic equations and we can have the following results

with $b_2, p, q, r, C_3, C_4$ are constants and $\kappa$ is a positive constant.

The solutions of equation (4.2) are listed as follows:

(a) When $p^2-4qr > 0$ and $pq\neq0 \ (qr\neq0)$,

where $A, B$ are arbitrary constants.

(b) When $p^2-4qr < 0$ and $pq\neq0\ (qr\neq0)$,

where $A, B$ are arbitrary constants.

(c) When $r = 0$ and $pq\neq0$,

where $C$ is an arbitrary constant.

(d) When $p = r = 0$ and $q\neq0$,

where $C$ is an arbitrary constant.

Substituting (4.3) into (4.1) and (3.1), then we can obtain the following different exact solutions of system (1.3):

(a1) If $\Delta = p^2-4qr > 0$ and $pq\neq0\ (qr\neq0)$, then the solutions of system (1.3) with $\alpha = 2$ can be derived as

where $\omega = {x-\int\frac{F_2(t)}{C_3}dt}$.

If we take $F(t) = {\frac{F_2(t)+C_4}{C_3}}, \sqrt{\Delta} = 2c_1(c_1 > 0), b = {\frac{b_2}{4q^2}}$, then the above solution can be expressed as a simple form as

where $\omega = {x-\int F(t)dt}$, and $c_1(>0), b, \kappa$ are constants.

Similarly, we can derive the other solutions of system (1.3) as

where $A, B$ are arbitrary constants.

where $B^2-A^2 > 0$.

(a2) When $\Delta = p^2-4qr < 0$ and $pq\neq0\ (qr\neq0)$, if we denote $F(t) = {\frac{F_2(t)+C_4}{C_3}}, \sqrt{-\Delta} = 2c_1(c_1 > 0), b = {\frac{b_2}{4q^2}}$, then the solutions of system (1.3) with $\alpha = 2$ can be derived as

where $A, B$ are arbitrary constants and $A^2-B^2 > 0$.

where $A^2-B^2 > 0$.

(a3) When $r = 0$ and $pq\neq0$, if we denote $F(t) = {\frac{F_2(t)+C_4}{C_3}}, b = {\frac{b_2}{4q^2}}$, then the solutions of system (1.3) with $\alpha = 2$ can be derived as

where $\omega = {x-\int F(t)dt}$, $F(t)$ is an arbitrary function and $b, C$ are constants.

(a4) When $p = r = 0$ and $q\neq0$, if we denote $F(t) = {\frac{F_2(t)+C_4}{C_3}}, b = {\frac{b_2}{4q^2}}$, then the solution of system (1.3) with $\alpha = 2$ can be derived as

where $\omega = {x-\int F(t)dt}$, $F(t)$ is an arbitrary function and $b, C$ are constants.

In order to show the properties of the above solutions visually, we plot the 2D-graphs of some typical solutions. Some wave figures are given as follows (Figures 1–5):

For the solution (4.5), if we take the integration constant as 0 in $\omega = x-\int F(t)dt$, then we plot the solution for the plus sign in $u_1$ as

For the solution (4.6), if we take the integration constant as 0 in $\omega = x-\int F(t)dt$, then we plot the solution for the plus sign in $u_2$ as

For the solution (4.8), if we take the integration constant as 0 in $\omega = x-\int F(t)dt$, then we plot the solution for the plus sign in $u_4$ as

For the solution (4.10), if we take the integration constant as 0 in $\omega = x-\int F(t)dt$, then we plot the solution for the plus sign in $u_6$ as

For the solution (4.17), if we take the integration constant as $0$ in $\omega = x-\int F(t)dt$, then we plot the solution for the plus sign in $u_{13}$ as

Remark 1 If we take $F(t)$ as a constant, then all of the above solutions of system (1.3) are traveling wave solutions.

Remark 2 For the reduced equations (3.4) and (3.6), there exist a power series solutions ^[18,19]. We omit the details here for brevity.

5.   Conservation law

In this section, we use the direct multiplier method ^[20] to derive a conservation law for system (1.3). The zero-order multipliers $\Lambda_1(t, x, u, \rho)$, $\Lambda_2(t, x, u, \rho)$ for the system (1.3) are determined by

where $\frac{\delta}{\delta u}, \frac{\delta}{\delta \rho}$ are Euler-Lagrange operators defined by

Expanding (5.1) and splitting with respect to derivative of $u, \rho$, we obtain the following determining equations

Then we obtain the solution

where $A$ is an arbitrary constant, $H(t)$ is an arbitrary functions with respect to $t$. From the solution (5.4), we can see that system (1.3) has one zero-order multiplier in the form of $\Lambda_1 = H(t), \ \Lambda_2 = \rho^{-\frac{1+\alpha}{\alpha}}$. So a conservation law of system (1.3) is

6.   Conclusions

In this paper, a generalized 2-HS system is investigated by using the classical Lie group method. First, Lie symmetry analysis was performed for the generalized 2-HS system, and its infinitesimal generator, geometric vector fields and commutation table of Lie algebra were obtained. Then, all of the similarity variables and its symmetry reductions of this equation are obtained. And by solving the reduced equations, some new exact solutions including traveling wave solutions of this generalized 2-HS system are constructed successfully. These are new solutions for the generalized 2-HS system. Finally, a conservation law of the generalized 2-HS system are shown by using the multiplier method.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No.11461022) and Applied Basic Research Foundation of Yunnan Province (Nos. 2018FH001-013 and 2018FH001-014), the Science Research Foundation of Yunnan Education Bureau (No. 2018JS479) and the Second Batch of Middle and Young Aged Academic Backbone of Honghe University (No. 2015GG0207).

Conflict of interest

The authors declare that there are no conflict interests regarding the publication of this paper.