Generalized Nonlocal Symmetries of Two-Component Camassa–Holm and Hunter–Saxton Systems

Abstract

The two-component Camassa–Holm system and two-component Hunter–Saxton system are completely integrable models. In this paper, it is shown that these systems admit nonlocal symmetries by their geometric integrability. As an application, we obtain the recursion operator and conservation laws by using this kind of nonlocal symmetries.

1. Introduction

This paper mainly discusses nonlocal symmetries, conservation laws, and recursion operators of the two-component Camassa–Holm system [1,2] and the two-component Hunter–Saxton system [1,2]. These systems have Lax-pairs and bi-Hamiltonian structures, which are completely integrable systems. It has long been known that integrable systems have nonlocal symmetries, which is an interesting trait. The nonlocal symmetries of an integrable equation are related to the conservation law of the equations, the exact solutions, the Darboux transformation, and the integrability of the equations. Therefore, it is very important to study the nonlocal symmetries of integrable equations. As an application, we obtain the recursion operator and conservation laws by using this kind of nonlocal symmetries.

The famous Camassa–Holm equation [3,4]

$$u_t-u_{xxt}=-3u_{x}u+u{u}_{xxx}+2u_x{u}_{xx}$$

(1)

was first derived by Fokas and Fuchssteiner and Camassa and Holm. This model describes the unidirectional propagation of shallow water waves over a flat bottom. In fact, Fokas and Fuchssteiner [4] obtained the Camassa–Holm equation through the integrability of the KdV equation. Interestingly, Olver and Rosenau also obtained the Camassa–Holm equation by using the tri-Hamiltonian duality method [1].

If we introduce $m=u-u_{xx},$ then Equation (1) can be rewritten in the following form:

$$\begin{array}{c}{m}_t=-m_{x}u-2m{u}_x,\hfill \\ m=u-u_{xx}.\hfill \end{array}$$

(2)

Olver, Rosenau and Chen Liu Zhang obtained a two-component Camassa–Holm system, which is an extension of the Camassa–Holm equation,

$$\begin{array}{c}{m}_t+2m{u}_x+u{m}_x-\rho {\rho }_x=0,\hfill \\ {\rho }_t+{\left(\rho u\right)}_x=0,\hfill \end{array}$$

(3)

where $m=u-u_{xx}$. They point out that this system admits bi-Hamiltonian structure and Lax-pairs. The two-component Camassa–Holm system is also derived from the Green–Naghdi equation by Constantin and Ivanov [5]. The two-component Camassa–Holm system is a geodesic flow concerning the $H1$ metric on the semidirect product space $Diffs\left(S1\right)⋉C1\left(S1\right)$ [6,7]. Wave breaking phenomena of system (2) that have a certain initial value have been studied extensively. Various properties of the Camassa–Holm equation have been studied extensively [8,9,10]. Reyes shows that the Camassa–Holm equation has geometric integrability [11,12,13,14]. The Camassa–Holm equation can be obtained from a non-stretching invariant plane curve flow in Centro-Affine differential geometry by Chou and Qu in [15]. In Reference [16], Misiolek shows that the Camassa–Holm equation is a geodesic flow of a right-invariant on the Virasoro group. The two-component Camassa–Holm and Hunter–Saxton systems also have drawn much attention and have multi-peakon solitons [5]. As an extension of the Camassa–Holm equation, $\mu$ Camassa–Holm type equations also have geometric integrability and a bi-Hamiltonian structure, drawing much attention [17,18,19,20,21,22].

The outline of this paper is as follows. In Section 2, the generalized symmetries and their commutators of the two-component Camassa–Holm system are constructed. A recursion operator for the two-component Camassa–Holm system is obtained in Section 3. In Section 4, we construct generalized nonlocal symmetries and an infinite number of the two-component Hunter–Saxton system. Section 5 presents a concluding remark on this work.

2. Nonlocal Symmetries of the Two-Component Camassa–Holm System

2.1. Pseudo-Spherical Surface

Definition 1

([23]). A scalar differential equation $F(x,t,u,u_x,\cdots ,u_{x^n,t^m})=0$ in two independent variables $x,t$ is of pseudo-spherical type (or if it describes pseudo-spherical surfaces) if there exist one-forms ${\omega }^{\alpha }=0$, defined as

$${\omega }^{\alpha }=f_{\alpha 1}(x,t,u,\cdots ,u_{x^r,t^p})dx+f_{\alpha 2}(x,t,u,\cdots ,u_{x^s,t^q})dt,\alpha =1,2,3,$$

(4)

for which the coefficients $f_{\alpha ,\beta }$ are smooth functions that depend on $x,t$ and a finite number of derivatives of u, such that the one-forms ${\overline{\omega }}^{\alpha }={\omega }^{\alpha }\left(u(x,t)\right)$ satisfy the structure equations given by

$$\begin{array}{c}\hfill d{\overline{\omega }}^1={\omega }^3\wedge {\omega }^2,\\ \hfill d{\overline{\omega }}^2={\omega }^1\wedge {\omega }^3,\\ \hfill d{\overline{\omega }}^3={\omega }^1\wedge {\omega }^2,\end{array}$$

whenever $u=u(x,t)$ is a solution of $F(x,t,u,u_x,\cdots ,u_{x^n,t^m})=0$.

Definition 2

([23]). An equation $F(x,t,u,u_x,\cdots ,u_{x^n,t^m})=0$ is geometrically integrable if it describes a nontrial one-parameter family of pseudo-spherical surfaces.

Proposition 1

([23]). Let $F(x,t,u,u_x,\cdots ,u_{x^n,t^m})=0$ be a differential equation describing pseudo-spherical surfaces with associated one-forms ${\omega }^{\alpha }$. The following two Pfaffian systems are completely integrable whenever $u(x,t)$ is a solution of $F(x,t,u,u_x,\cdots ,u_{x^n,t^m})=0,$

$$\begin{array}{c}\hfill -2d\Gamma ={\omega }^3+{\omega }^2-2\Gamma {\omega }^1+{\Gamma }^2({\omega }^3-{\omega }^2),\end{array}$$

(5)

$$\begin{array}{c}\hfill -2d\Omega ={\omega }^3-{\omega }^2-2\Omega {\omega }^1+{\Omega }^2({\omega }^3+{\omega }^2).\end{array}$$

(6)

Moreover, the one-forms

$$\begin{array}{c}\hfill \theta ={\omega }^1-\Gamma ({\omega }^3-{\omega }^2),\end{array}$$

(7)

$$\begin{array}{c}\hfill \widehat{\theta }=-{\omega }^1+\Gamma ({\omega }^3+{\omega }^2)\end{array}$$

(8)

are closed whenever $u(x,t)$ is a solution of $F(x,t,u,u_x,\cdots ,u_{x^n,t^m})=0$ and Γ (respectively, Ω) is a solution of (5) (respectively, (6)).

Proposition 2

([23]). The two-component Camassa–Holm system describes pseudo-spherical surfaces; therefore, it is geometrically integrable.

For Proposition 2, we have the one-form [8]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\omega }_1=& \left[\left({\rho }^2-\frac{1}{4}\right){\lambda }^2+\lambda m+\frac{5}{4}\right]\mathrm{d}x+\left[\left(\frac{1}{4}-{\rho }^2\right)u{\lambda }^2\right.\hfill \\ \hfill & +\left(\frac{1}{2}{\rho }^2-um+\frac{1}{2}{u}_x-\frac{1}{8}\right)\lambda \left.-\frac{3}{4}u+\frac{5}{8}{\lambda }^{-1}\right]\mathrm{d}t,\hfill \\ \hfill {\omega }_2=& \lambda \mathrm{d}x-\left(\lambda u+u_x-\frac{1}{2}\right)\mathrm{d}t,\hfill \\ \hfill {\omega }_3=& \left[\left(\frac{1}{4}-{\rho }^2\right){\lambda }^2-\lambda m+\frac{3}{4}\right]\mathrm{d}x\hfill \\ \hfill & +\left[\left({\rho }^2-\frac{1}{4}\right)u{\lambda }^2-\left(\frac{1}{2}{\rho }^2-um+\frac{1}{2}{u}_x-\frac{1}{8}\right)\lambda \right.\hfill \\ \hfill & \left.-\frac{5}{4}u+\frac{3}{8}{\lambda }^{-1}\right]\mathrm{d}t,\hfill \end{array}\end{array}$$

(9)

which is associated with two-component Camassa–Holm system.

Theorem 1

([8]). The two-component Camassa–Holm system admits a quadratic pseudo-potential α, which is defined by the system

$$\begin{array}{c}\hfill {\alpha }_x=-{\alpha }^2+{\lambda }^2{\rho }^2+\lambda m+\frac{1}{4}\\ {\alpha }_t={\left[\left(\frac{1}{2\lambda }-u\right)\alpha +\frac{1}{2}{u}_x\right]}_x\hfill \end{array}$$

(10)

where $\lambda \ne 0,$ $m=u-u_{xx}$.

As an application, an infinite number of conservation laws with Equation (10) can be obtained. We expand $\alpha$ as

$$\alpha ={\displaystyle \sum _{n=-2}^{\infty }}{\alpha }_n(x,t){\lambda }^{-\frac{n}{2}}$$

(11)

Furthermore, we substitute the above equation into the conservation law described by Equation (10) and possess the parameter $\lambda$. The following system on ${\alpha }_j$ has been obtained:

$$\begin{array}{c}{\alpha }_{-2}^2(x,t)=\rho ,\hfill \\ -2{\alpha }_{-1}(x,t){\alpha }_{-2}(x,t)=0,\hfill \\ {\alpha }_{-2,x}^2=-{\alpha }_{-1}^2(x,t)-2{\alpha }_0(x,t){\alpha }_{-2}(x,t)+m,\hfill \\ \dots \hfill \end{array}$$

(12)

Thus, we calculate an infinite number of conservation laws of the two-component Camassa–Holm equation. The first three conservation densities are given by

$$\begin{array}{c}{H}_1=\int \rho \mathrm{d}x,\hfill \\ H_2=\int \frac{m}{\rho }\mathrm{d}x\hfill \\ H_3=\int \frac{{\rho }_x^3+{\rho }^2-\rho m^2}{{\rho }^3}\mathrm{d}x\hfill \end{array}$$

(13)

Now, if we set

$$\alpha ={\displaystyle \sum _{n=0}^{\infty }}{\alpha }_n(x,t){\lambda }^{\frac{n}{2}},$$

(14)

then Equation (10) can be rewritten as

$${\left({\displaystyle \sum _{n=0}^{\infty }}{\alpha }_n(x,t){\lambda }^{\frac{n}{2}}\right)}_x=-{\left({\displaystyle \sum _{n=0}^{\infty }}{\alpha }_n(x,t){\lambda }^{\frac{n}{2}}\right)}^2+\lambda m+{\lambda }^2{\rho }^2+\frac{1}{4}$$

(15)

The following system on ${\alpha }_j(x,t)$ has been obtained:

$$\begin{array}{cc}\hfill {\alpha }_{0,x}& =-{\alpha }_0{(x,t)}^2+\frac{1}{4},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\alpha }_{1,x}& =-2{\alpha }_0(x,t){\alpha }_1(x,t),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\alpha }_{2,x}& =-2{\alpha }_0(x,t){\alpha }_2(x,t)-{\alpha }_1{(x,t)}^2+m,\hfill \end{array}$$

(18)

$$\begin{array}{c}\hfill \cdots \end{array}$$

(19)

As a result, the first three conservation functionals are constructed,

$$\begin{array}{cc}\hfill H_1& =\int u\mathrm{d}x,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill H_2& =\int \left({\rho }^2-u^2-u_x^2\right)\mathrm{d}x,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill H_3& =\int (p^4-2p^2{u}^2+4p^{2}u{u}_x-2p^2{u}_x^2+u^4+6u^2{u}_x^2\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & -4u{u}_x^3+u_x^4+p^{2}u-p^2{u}_x-u^3-3u{u}_x^2+u_x^3){\mathrm{d}}_x.\hfill \end{array}$$

(23)

We obtain the same conservation densities as in [8], where the pseudo-potential function expands on “${\lambda }^n$” and “${\lambda }^{-n}$”.

2.2. Nonlocal Symmetries for the Two-Component Camassa–Holm System

Definition 3

([11]). Let N be a nonzero integer or $N=\infty$. An $N$—dimensional covering π of a (system of) partial differential equation(s) $F_a=0,a=1,\cdots ,k,$ is a triplet

$$\begin{array}{c}\hfill ({\alpha }^b,b=1,\cdots ,N;X_{ib},b=1,\cdots ,N,i=1,\cdots ,N;\widetilde{D_i},i=1,\cdots ,n)\end{array}$$

(24)

of variables ${\alpha }^b,$ smooth functions $X_{ib}$ depending on $x^i,u^{\alpha },{\alpha }_b$, and a finite number of partial derivatives of $u^{\alpha },$ and linear operators

$$\begin{array}{c}\hfill \widetilde{D_i}=D_i+{\displaystyle \sum _{b=1}^N}{X}_{ib}\frac{\partial }{\partial {\alpha }^b},\end{array}$$

(25)

such that equations

$$\begin{array}{c}\hfill \widetilde{D_i}\left(X_{jb}\right)=\widetilde{D_j}\left(X_{ib}\right),i,j=1,\cdots ,n,b=1,\cdots ,N\end{array}$$

(26)

hold whenever $u^{\alpha }\left(X^i\right)$ is a solution of $F_a=0$.

Definition 4

([11]). Let $F_a=0,a=1,\cdots ,k$ be a system of partial differential equations, with $\pi =({\alpha }^b,X_{i,b},\widetilde{D_i})$ a covering of $F_a=0$. A nonlocal π-symmetry of $F_a=0$ is a generalized symmetry

$$\begin{array}{c}\hfill X={\displaystyle \sum _a}{G}^a\frac{\partial }{\partial u^a}+{\displaystyle \sum _b}{H}^b\frac{\partial }{\partial u^b}\end{array}$$

(27)

of the augmented system

$$\begin{array}{cc}\hfill & F_a=0,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & \frac{\partial {\alpha }^b}{\partial x^i}=X_{ib},\hfill \end{array}$$

(29)

Now, a pseudo-potential $\alpha$ in Equation (10) has been defined. Then, the potential function $\delta$ is defined as

$$\begin{array}{cc}\hfill & {\delta }_x=\alpha ,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & {\delta }_t=\left(\frac{1}{2\lambda }-u\right)\alpha +\frac{1}{2}{u}_x\hfill \end{array}$$

(31)

Based on the infinitesimal criteria [24] for symmetries, the two-component Camassa–Holm system admits the following evolutionary vector field,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill W=& \alpha {\mathrm{e}}^{2\delta }\frac{\partial }{\partial u}-\left[2{\lambda }^2\rho {\rho }_x+\lambda m_x\right.\left.+\left(4{\lambda }^2{\rho }^2+4\lambda m\right)\alpha \right]{\mathrm{e}}^{2\delta }\frac{\partial }{\partial m},\hfill \\ \hfill & -\lambda \left({\rho }_x+2\alpha \rho \right){\mathrm{e}}^{2\delta }\frac{\partial }{\partial \rho }.\hfill \end{array}\end{array}$$

(32)

If we introduce $\beta (x,t)$, which satisfies the following system,

$$\begin{array}{cc}\hfill {\beta }_x& =-{\mathrm{e}}^{2\delta }\left(2{\lambda }^3{\rho }^2+{\lambda }^{2}m\right),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\beta }_t& ={\mathrm{e}}^{2\delta }\left(-\frac{{\alpha }^2}{2}+2{\lambda }^{3}u{\rho }^2+{\lambda }^{2}um-\frac{1}{2}{\lambda }^2{\rho }^2+\frac{1}{8}\right).\hfill \end{array}$$

(34)

then the two-component Camassa–Holm system admits the following nonlocal symmetry:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V=& \alpha {\mathrm{e}}^{2\delta }\frac{\partial }{\partial u}-\left[2{\lambda }^2\rho {\rho }_x+\lambda m_x\right.\left.+\left(4{\lambda }^2{\rho }^2+4\lambda m\right)\alpha \right]{\mathrm{e}}^{2\delta }\frac{\partial }{\partial m}\hfill \\ \hfill & -\lambda \left({\rho }_x+2\rho \alpha \right){\mathrm{e}}^{2\delta }\frac{\partial }{\partial \rho }-\left(2{\lambda }^3{\rho }^2+{\lambda }^{2}m\right){\mathrm{e}}^{2\delta }\frac{\partial }{\partial \alpha }\hfill \\ \hfill & +\beta \frac{\partial }{\partial \delta }+\left[{\beta }^2+\left(3{\lambda }^4{\rho }^2+{\lambda }^{3}m\right){\mathrm{e}}^{4\delta }\right]\frac{\partial }{\partial \beta }.\hfill \end{array}\end{array}$$

(35)

This result has been proved in [8].

Theorem 2.

The generalized symmetries for the augmented two-component Camassa–Holm system (10), (30), and (36) admit the following vector fields,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_1=& \left(-2u_{x}m-u{m}_x-\rho {\rho }_x\right)\frac{\partial }{\partial m}-u_t\frac{\partial }{\partial u}-{\left(\rho u\right)}_x\frac{\partial }{\partial \rho }\hfill \\ \hfill & +{\left(\left(\frac{1}{2\lambda }-u\right)\alpha +\frac{1}{2}{u}_x\right)}_x\frac{\partial }{\partial \alpha }+\left(\left(\frac{1}{2\lambda }-u\right)\alpha +\frac{1}{2}{u}_x\right)\frac{\partial }{\partial \delta }\hfill \\ \hfill & +{\mathrm{e}}^{2\delta }\left(-\frac{{\alpha }^2}{2}+2{\lambda }^{3}u{\rho }^2+{\lambda }^{2}um-\frac{1}{2}{\lambda }^2{\rho }^2+\frac{1}{8}\right)\frac{\partial }{\partial \beta },\hfill \\ \hfill V_2=& m_x\frac{\partial }{\partial m}+u_x\frac{\partial }{\partial u}+{\rho }_x\frac{\partial }{\partial \rho }+\left(-{\alpha }^2+{\lambda }^2{\rho }^2+\lambda m+\frac{1}{4}\right)\frac{\partial }{\partial \alpha }\hfill \\ \hfill & +\alpha \frac{\partial }{\partial \delta }+\left(-{\mathrm{e}}^{2\delta }\left(2{\lambda }^3{\rho }^2+{\lambda }^{2}m\right)\right)\frac{\partial }{\partial \beta },\hfill \\ \hfill V_3=& \frac{\partial }{\partial \delta }+2\beta \frac{\partial }{\partial \beta },\hfill \\ \hfill V_4=& \frac{\partial }{\partial \beta },\hfill \\ \hfill V_5=& \alpha {\mathrm{e}}^{2\delta }\frac{\partial }{\partial u}-\left[2{\lambda }^2\rho {\rho }_x+\lambda m_x\right.\left.+\left(4{\lambda }^2{\rho }^2+4\lambda m\right)\alpha \right]{\mathrm{e}}^{2\delta }\frac{\partial }{\partial m}\hfill \\ \hfill & -\lambda \left({\rho }_x+2\rho \alpha \right){\mathrm{e}}^{2\delta }\frac{\partial }{\partial \rho }-\left(2{\lambda }^3{\rho }^2+{\lambda }^{2}m\right){\mathrm{e}}^{2\delta }\frac{\partial }{\partial \alpha }\hfill \\ \hfill & +\beta \frac{\partial }{\partial \delta }+\left[{\beta }^2+\left(3{\lambda }^4{\rho }^2+{\lambda }^{3}m\right){\mathrm{e}}^{4\delta }\right]\frac{\partial }{\partial \beta }.\hfill \end{array}\end{array}$$

(36)

The proof of the above theorem is a straightforward computation.

Corollary 1.

The five nonlocal symmetries (36) generate a Lie algebra $\mathfrak{E}$, and their commutators are presented in Table 1.

Table 1. The commutation table of the two-component Camassa–Holm system’s nonlocal symmetry algebra.

	${\mathit{V}}_1$	${\mathit{V}}_2$	${\mathit{V}}_3$	${\mathit{V}}_4$	${\mathit{V}}_5$
$V_1$	0	0	0	0	0
$V_2$	0	0	0	0	0
$V_3$	0	0	0	−2$V_4$	2$V_5$
$V_4$	0	0	2$V_4$	0	$V_3$
$V_5$	0	0	$-2V_5$	$-V_3$	0

3. Recursion Operators for the Two-Component Camassa–Holm System

Set $\alpha$ as

$$\begin{array}{c}\hfill \alpha =\frac{{\psi }_x}{\psi }={(ln\psi )}_x,\end{array}$$

(37)

where $\alpha$ is determined by the pseudo-potential Equation (10). The function $\psi$ satisfies the second-order linear problem

$$\begin{array}{c}\hfill {\psi }_{xx}=\psi {\lambda }^2{\rho }^2+\psi \lambda m+\frac{\psi }{4}.\end{array}$$

(38)

The potential $\delta$ can be written as

$$\begin{array}{c}\hfill {\delta }_x=\alpha ,\end{array}$$

(39)

then, we have

$$\begin{array}{c}\hfill \delta =ln\psi .\end{array}$$

(40)

Now, the function $G=\alpha {\mathrm{e}}^{2\delta }$ can be rewritten as

$$\begin{array}{c}\hfill G(x,t)={\psi }_x\psi .\end{array}$$

(41)

Furthermore, we know that the shadow $G(x,t)$ satisfies

$$\begin{array}{cc}\hfill D_x^{-1}G& =\frac{\psi }{2},\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill D_{x}G& ={\psi }_{xx}\psi +{\psi }_x^2,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill D_{xx}G& ={\psi }_{xxx}\psi +3{\psi }_{xxx}{\psi }_x.\hfill \end{array}$$

(44)

Using Equation (38), we obtain

$$\begin{array}{c}\hfill 2\lambda {\psi }_{xx}{\psi }_x={\psi }_x\psi m+\frac{\lambda {\psi }_x\psi }{2}.\end{array}$$

(45)

In contrast, from Equation (43), ${\psi }_{xx}\psi$ can be written as

$$\begin{array}{c}\hfill \lambda {\psi }_{xx}{\psi }_x=\frac{G_x}{6}-\frac{\lambda {\psi }_{xxx}\psi }{3}.\end{array}$$

(46)

In the above equation, the term $\psi {\psi }_{xxx}$ can be obtained by taking the x-derivative of Equation (38) and multiplying the resulting equation by $2\lambda {\psi }_x$,

$$\begin{array}{c}\hfill 2\lambda {\psi }_{xxx}\psi ={\psi }_x\psi m+{\psi }^2{m}_x+\frac{\lambda {\psi }_x\psi }{2}\end{array}$$

(47)

Then, we have

$$\begin{array}{c}\hfill 2\lambda {\psi }_{xx}{\psi }_x=\frac{G_{xx}}{3}-\frac{\left(D_x^{-1}G(x,t)\right)m_x}{3\lambda }-\frac{G(x,t)m}{6\lambda }-\frac{G(x,t)}{12}.\end{array}$$

(48)

The function $G(x,t)$ satisfies the following equation by replacing (45) in Equation (48):

$$\begin{array}{c}\hfill 0=\frac{G_{xx}}{3}-\frac{\left(D_x^{-1}G(x,t)\right)m_x}{3\lambda }-\frac{G(x,t)m}{6\lambda }-\frac{G(x,t)}{12}-\frac{G(x,t)(\lambda +2m)}{4\lambda }\end{array}$$

(49)

As a result, Equation (49) can lead to

$$\begin{array}{c}\hfill \lambda G={\left(D_x^2-1\right)}^{-1}\left(m_x{D}_x^{-1}+2m\right)G.\end{array}$$

(50)

The pseudo-differential operator

$$\begin{array}{c}\hfill Q={\left(D_x^2-1\right)}^{-1}\left(m_x{D}_x^{-1}+2m\right)\end{array}$$

(51)

is precisely the recursion operator for the two-component Camassa–Holm system.

4. Nonlocal Symmetries of the Two-Component Hunter–Saxton System

The two-component Hunter–Saxton system

$$\begin{array}{cc}\hfill & m_t+2m{u}_x+u{m}_x-\rho {\rho }_x=0,\hfill \\ \hfill & {\rho }_t+{\left(\rho u\right)}_x=0,\hfill \\ \hfill & m=u-u_{xx}\hfill \end{array}$$

(52)

describes a pseudo-spherical surface with the following associated one-form:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\omega }_1=& \left[\left({\rho }^2-\frac{1}{4}\right){\lambda }^2+\lambda m+1\right]\mathrm{d}x+\left[\left(\frac{1}{4}-{\rho }^2\right)u\right.\hfill \\ \hfill & +\left(\frac{1}{2}{\rho }^2-um+\frac{1}{2}{u}_x-\frac{1}{8}\right)\lambda \left.-u+\frac{1}{2}{\lambda }^{-1}\right]\mathrm{d}t\hfill \\ \hfill {\omega }_2=& \lambda \mathrm{d}x-\left(\lambda u+u_x-\frac{1}{2}\right)\mathrm{d}t\hfill \\ \hfill {\omega }_3=& \left[\left(\frac{1}{4}-{\rho }^2\right){\lambda }^2-\lambda m+1\right]\mathrm{d}x\hfill \\ \hfill & +\left[\left({\rho }^2-\frac{1}{4}\right)u{\lambda }^2-\left(\frac{1}{2}{\rho }^2-um+\frac{1}{2}{u}_x-\frac{1}{8}\right)\lambda \right.\hfill \\ \hfill & \left.-u+\frac{1}{2}{\lambda }^{-1}\right]\mathrm{d}t.\hfill \end{array}\end{array}$$

(53)

Theorem 3.

The two-component Hunter–Saxton system admits the quadratic pseudo-potential α defined by the system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_x=& -{\alpha }^2+{\lambda }^2{\rho }^2+\lambda m,\hfill \\ \hfill {\alpha }_t=& \left(u-\frac{1}{2\lambda }\right){\alpha }^2-u_x\alpha -{\lambda }^{2}u{\rho }^2+\lambda \left(\frac{{\rho }^2}{2}-um\right).\hfill \end{array}\end{array}$$

(54)

where $m=-u_{xx}$, and parameter $\lambda \ne 0.$

Now, we construct an infinite number of conservation laws of the two-component Hunter–Saxton system with the pseudo-potential $\alpha$. Set

$$\begin{array}{c}\hfill \alpha ={\displaystyle \sum _{n=-2}^{\infty }}{\alpha }_n(x,t){\lambda }^{-\frac{n}{2}}.\end{array}$$

(55)

Substituting this into Equation (52), one obtains the equations for the coefficient function ${\alpha }_{i,x},i=-2,-1,0,\cdots ,$

$$\begin{array}{c}\hfill {\displaystyle \sum _{n=-2}^{\infty }}{\left({\alpha }_n\right)}_x{\lambda }^{-\frac{n}{2}}=-{\left({\displaystyle \sum _{n=-2}^{\infty }}{\alpha }_n(x,t){\lambda }^{-\frac{n}{2}}\right)}^2+\lambda m+{\lambda }^2{\rho }^2\end{array}$$

(56)

Comparing the coefficients of ${\lambda }^{-i}$, we obtain the following equations for ${\alpha }_i,i=-2,-1,0,\cdots$:

$$\begin{array}{c}\hfill \begin{array}{c}{\alpha }_{-2}^2(x,t)=\rho ,\hfill \\ -2{\alpha }_{-1}(x,t){\alpha }_{-2}(x,t)=0,\hfill \\ {\alpha }_{-2,x}^2=-{\alpha }_{-1}^2(x,t)-2{\alpha }_0(x,t){\alpha }_{-2}(x,t)+m,\hfill \\ \dots \hfill \end{array}\end{array}$$

(57)

Then, the ${\alpha }_i,i=-2,-1,0,\cdots$ can be determined recursively. The first three conservation densities are presented as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{H}_1=\int \rho \mathrm{d}x,\hfill \\ H_2=\int \frac{m}{\rho }\mathrm{d}x\hfill \\ H_3=\int \frac{{\rho }_x^2-\rho m^2}{{\rho }^3}\mathrm{d}x.\hfill \end{array}\end{array}$$

(58)

If we set

$$\begin{array}{c}\hfill \alpha ={\displaystyle \sum _{n=0}^{\infty }}{\alpha }_n(x,t){\lambda }^{\frac{n}{2}},\end{array}$$

(59)

then one pbtains the equations for the coefficient functions ${\alpha }_{i,x},i=0,1,2,\cdots ,$

$$\begin{array}{c}\hfill {\left({\displaystyle \sum _{n=0}^{\infty }}{\alpha }_n(x,t){\lambda }^{\frac{n}{2}}\right)}_x=-{\left({\displaystyle \sum _{n=0}^{\infty }}{\alpha }_n(x,t){\lambda }^{\frac{n}{2}}\right)}^2+\lambda m+{\lambda }^2{\rho }^2,\end{array}$$

(60)

Comparing the coefficients of ${\lambda }^i,$ the following equations for ${\alpha }_i$ are obtained:

$$\begin{array}{cc}\hfill {\alpha }_{0,x}& =-{\alpha }_0{(x,t)}^2,\hfill \\ \hfill {\alpha }_{1,x}& =-2{\alpha }_0(x,t){\alpha }_1(x,t),\hfill \\ \hfill {\alpha }_{2,x}& =-2{\alpha }_0(x,t){\alpha }_2(x,t)-{\alpha }_1{(x,t)}^2+m,\hfill \\ \hfill & \cdots .\hfill \end{array}$$

Then, the ${\alpha }_i,i=0,1,2,\cdots$ can be determined, and conservation densities are presented as follows:

$$\begin{array}{cc}\hfill H_1& =\int \left({\rho }^2-u_x^2\right)\mathrm{d}x,\hfill \\ \hfill H_2& =\int (p^4-2p^2{u}^2+4p^{2}u{u}_x-2p^2{u}_x^2+u^4+6u^2{u}_x^2\hfill \\ \hfill & -4u{u}_x^3+p^{2}u-p^2{u}_x-3u{u}_x^2+u_x^3){\mathrm{d}}_x\hfill \\ \hfill & \cdots .\hfill \end{array}$$

Now, we consider the two-component Hunter–Saxton system’s nonlocal symmetries. System (52) can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{c}{\alpha }_x=-{\alpha }^2+{\lambda }^2{\rho }^2+\lambda m+\frac{1}{4}\hfill \\ {\alpha }_t={\left[\left(\frac{1}{2\lambda }-u\right)\alpha +\frac{1}{2}{u}_x\right]}_x\hfill \end{array}\end{array}$$

(61)

where $m=-u_{xx}$. Furthermore, the potential function $\delta (x,t)$ is defined as

$$\begin{array}{cc}\hfill & {\delta }_x=\alpha ,\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill & {\delta }_t=\left(\frac{1}{2\lambda }-u\right)\alpha +\frac{1}{2}{u}_x\hfill \end{array}$$

(63)

Theorem 4.

The following vector fields are the generalized symmetries for the augmented two-component Hunter–Saxton system (52) and (61)–(63).

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_1=& \left(-2u_{x}m-u{m}_x-\rho {\rho }_x\right)\frac{\partial }{\partial m}-u_t\frac{\partial }{\partial u}-{\left(\rho u\right)}_x\frac{\partial }{\partial \rho }\hfill \\ \hfill & +{\left(\left(\frac{1}{2\lambda }-u\right)\alpha +\frac{1}{2}{u}_x\right)}_x\frac{\partial }{\partial \alpha }+\left(\left(\frac{1}{2\lambda }-u\right)\alpha +\frac{1}{2}{u}_x\right)\frac{\partial }{\partial \delta }\hfill \\ \hfill & +{\mathrm{e}}^{2\delta }\left(-\frac{{\alpha }^2}{2}+2{\lambda }^{3}u{\rho }^2+{\lambda }^{2}um-\frac{1}{2}{\lambda }^2{\rho }^2\right)\frac{\partial }{\partial \beta },\hfill \\ \hfill V_2=& m_x\frac{\partial }{\partial m}+u_x\frac{\partial }{\partial u}+{\rho }_x\frac{\partial }{\partial \rho }+\left(-{\alpha }^2+{\lambda }^2{\rho }^2+\lambda m\right)\frac{\partial }{\partial \alpha }\hfill \\ \hfill & +\alpha \frac{\partial }{\partial \delta }+\left(-{\mathrm{e}}^{2\delta }\left(2{\lambda }^3{\rho }^2+{\lambda }^{2}m\right)\right)\frac{\partial }{\partial \beta },\hfill \\ \hfill V_3=& \frac{\partial }{\partial \delta }+2\beta \frac{\partial }{\partial \beta },\hfill \\ \hfill V_4=& \frac{\partial }{\partial \beta },\hfill \\ \hfill V_5=& \alpha {\mathrm{e}}^{2\delta }\frac{\partial }{\partial u}-\left[2{\lambda }^2\rho {\rho }_x+\lambda m_x\right.\left.+\left(4{\lambda }^2{\rho }^2+4\lambda m\right)\alpha \right]{\mathrm{e}}^{2\delta }\frac{\partial }{\partial m}\hfill \\ \hfill & -\lambda \left({\rho }_x+2\rho \alpha \right){\mathrm{e}}^{2\delta }\frac{\partial }{\partial \rho }-\left(2{\lambda }^3{\rho }^2+{\lambda }^{2}m\right){\mathrm{e}}^{2\delta }\frac{\partial }{\partial \alpha }\hfill \\ \hfill & +\beta \frac{\partial }{\partial \delta }+\left[{\beta }^2+\left(3{\lambda }^4{\rho }^2+{\lambda }^{3}m\right){\mathrm{e}}^{4\delta }\right]\frac{\partial }{\partial \beta }.\hfill \end{array}\end{array}$$

(64)

where

$$\begin{array}{cc}\hfill {\beta }_x& =-{\mathrm{e}}^{2\delta }\left(2{\lambda }^3{\rho }^2+{\lambda }^{2}m\right),\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill {\beta }_t& ={\mathrm{e}}^{2\delta }\left(-\frac{{\alpha }^2}{2}+2{\lambda }^{3}u{\rho }^2+{\lambda }^{2}um-\frac{1}{2}{\lambda }^2{\rho }^2\right).\hfill \end{array}$$

(66)

In addition, the following Corollary can be obtained.

Corollary 2.

The nonlocal symmetries (64) generate a Lie algebra $\mathfrak{E}$, and their commutators are presented in Table 2.

Table 2. The commutation table of the two-component Hunter–Saxton system’s nonlocal symmetry algebra.

	${\mathit{V}}_1$	${\mathit{V}}_2$	${\mathit{V}}_3$	${\mathit{V}}_4$	${\mathit{V}}_5$
$V_1$	0	0	0	0	0
$V_2$	0	0	0	0	0
$V_3$	0	0	0	−2$V_4$	2$V_5$
$V_4$	0	0	2$V_4$	0	$V_3$
$V_5$	0	0	$-2V_5$	$-V_3$	0

5. Concluding Remarks

We have shown that the two-component Camassa–Holm system and the two-component Hunter–Saxton system admit a class of nonlocal symmetries, and the recursion operator of the two-component Camassa–Holm system is constructed by using its potential variables. Thus, these kinds of nonlocal symmetries and recursion operators are related to the systems’ integrability. It is well-known that the Novikov equation and Degasperis–Procesi equation are geometric integrable models, and their spectral matrix is $3\times 3$. It is difficult to define a pseudo-potential function using the $3\times 3$ spectral matrix. However, it is interesting to investigate the existence of nonlocal symmetries and recursion operators, which will be our future study.