1 Introduction

The nonlinear partial differential equations (PDEs) are useful in analyzing nonlinear phenomena in engineering and scientific problems. In the past decades, many effective methods for investigating properties of PDEs have been developed, such as the bilinear method [1,2,3,4], Riemann–Bäcklund method [5,6,7], inverse scattering method [8, 9] algebraic geometry method [10,11,12] and Fokas method [13, 14]. Symmetry analysis method is one of the most effective method for analyzing PDEs [15,16,17,18,19,20]. Any symmetry transforms the solutions of a PDE to the solutions of the same equation. On the basis of the symmetry theory, one can construct conservation laws of PDEs. Many method for deriving conservation laws of PDEs have been developed, such as Noether’s approach [21,22,23] direct method [24,25,26,27], Ibragimovs method [28, 29] and the mixed method [30]. The direct method is able to find all conservation laws for any given system of PDEs. In contrast, Noether’s method is limited to variational systems, while Ibragimov’s method and the mixed method are merely special cases of the multiplier method [31,32,33]. The problem of finding all conservation laws for a given PDEs is equivalent to the problem of finding all infinitesimal symmetries. Therefore, there is no need to derive conservation laws with the aid of special methods [33]. Once a PDE’s conservation laws are constructed, the nonlocally related systems of this PDE can be established [25]. The nonlocally related systems are equivalent to the given PDE system [34]. Nonlocally related systems play an important role in finding the nonlocal symmetries and nonlocal conservation laws [35,36,37,38]. However, the conservation law-based method for constructing nonlocally related systems is not valid to the equation that has no nontrivial local conservation laws. It is notable that Bluman et al. proposed a symmetry-based method to find nonlocally related PDE systems [39]. Each point symmetry can yield a nonlocally related PDE system (inverse potential system). The symmetry-based method can also be used to construct trees of nonlocally related PDE system.

In the paper [40], based on the polynomial recursion formalism, Hou et al. derive the HS hierarchy. The first equation of this hierarchy is written as

$$\begin{aligned} U\left( {x,t,u} \right) = 0:\quad {u_{xxt}} + 4{u_x}{u_{xx}} + 2u{u_{xxx}} =0. \end{aligned}$$
(1)

This equation is an important physical model which can be used to describe the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field. The liquid crystal state is a distinct phase of matter observed between the solid and liquid states. The director field of the liquid crystal is usually floating [41]. Equation (1) is useful in studying the dynamics of director field since it can be used to model crucial point for nematic liquid crystals. Eq. (1) is Hunter–Saxton (HS) equation. HS equation is a short-wave limit of the Camassa–Holm equation [42]. This paper aims at constructing conservation laws and nonlocally related PDE systems of this equation.

This paper is organized as follows. In Sect. 2, the conservation laws of HS equation are constructed by using direct method. The conservation law-based method is employed to find the nonlocally related PDE systems of Eq. (1). Many new local and nonlocal symmetries for these systems are found. In Sect. 3, the equivalence transformations of two potential systems are investigated. In Sect. 4, the inverse potential systems arising from each Lie point symmetries are presented. A tree of inverse potential systems of Eq. (1) is also constructed. Finally, some conclusions are given in the last section.

2 Nonlocally related systems

Consider a k-order system of PDEs \({{\mathcal {R}}_\alpha }\left[ u \right] \) with n independent variables \(x = \left( {{x^1},{x^2},\dots ,{x^n}} \right) \) and m dependent variables \(u = \left( {{u^1},{u^2},\dots ,{u^m}} \right) \)

$$\begin{aligned} \begin{array}{*{20}{c}} {{{{{\mathcal {R}}}}_\alpha }\left[ u \right] = {{{{\mathcal {R}}}}_\alpha }\left( {x,u,{u_{\left( 1 \right) }}, \dots ,{u_{\left( k \right) }}} \right) = 0,}&\quad {\alpha = 1,2, \ldots ,m,} \end{array} \end{aligned}$$
(2)

where \({u_{\left( k \right) }}\) is kth-order derivative. A local divergence-type conservation law of the PDE system (2) is a divergence expression of the form

$$\begin{aligned} {D_i}{\Phi ^i}\left[ u \right] = {D_1}{\Phi ^1}\left[ u \right] + \cdots + {D_n}{\Phi ^n}\left[ u \right] \end{aligned}$$
(3)

in terms of total derivative operators holding on solutions of (2). There exists a set of conservation law multipliers

$$\begin{aligned} {\Lambda _\alpha }\left[ u \right] = {\Lambda _\alpha }\left( {x,u,\partial u, \dots ,{\partial ^l}u} \right) ,\quad \alpha = 1,2, \dots m, \end{aligned}$$
(4)

such that

$$\begin{aligned} {D_i}{\Phi ^i}\left[ u \right] \equiv {\Lambda _\alpha }\left[ u \right] {{\mathcal {R}}_\alpha }\left[ u \right] \end{aligned}$$
(5)

holds for arbitrary u.

For any divergence expression \({D_i}{\Phi ^i}\left[ u \right] \), one has

$$\begin{aligned} {E_{{u^j}}}\left( {{D_i}{\Phi ^i}\left[ u \right] } \right) \equiv 0,\quad j = 1,2, \dots m, \end{aligned}$$
(6)

where \({E_{{u^j}}} = \frac{\partial }{{\partial {u^j}}} - {D_i}\frac{\partial }{{\partial u_i^j}} + \cdots + {\left( { - 1} \right) ^s}{D_{{i_1}}} \cdots {D_{{i_s}}}\frac{\partial }{{\partial u_{{i_1} \cdots {i_s}}^j}} + \cdots \) is Euler operator with respect to \({u^j}\).

A set of local multipliers \({\Lambda _\alpha }\left( {x,u,\partial u, \dots ,{\partial ^l}u} \right) \) yields a divergence expression for PDE system (2) if and only if

$$\begin{aligned} {E_{{u^j}}}\left( {{\Lambda _\alpha }\left( {x,u,\partial u, \dots ,{\partial ^l}u} \right) {R_\alpha }\left( {x,u,\partial u, \dots ,{\partial ^k}u} \right) } \right) \equiv 0,\quad j = 1,2, \dots m \end{aligned}$$
(7)

holds for arbitrary u [16].

Consider the conservation law multipliers \(\Lambda [u] = \Lambda \left( {t,x,u} \right) \) to HS equation. Then

$$\begin{aligned} {E_u}\left( {\Lambda \left( {x,t,u} \right) \left( {{u_{xxt}} + 4{u_x}{u_{xx}} + 2u{u_{xxx}}} \right) } \right) \equiv \mathrm{{0}}\mathrm{{.}} \end{aligned}$$
(8)

Splitting with Eq. (8) respect to third derivatives of u yields the following determining system

$$\begin{aligned} \begin{array}{l} \begin{array}{l} - {\Lambda _{uu}} = 0,\; - 2{\Lambda _{xu}} = 0,\;\\ - {\Lambda _{txx}} - 2u{\Lambda _{xxx}} = 0,\;\\ - 6u{\Lambda _{xu}} - {\Lambda _{tu}} - 2{\Lambda _x} = 0, \end{array}\\ \begin{array}{l} - 6u{\Lambda _{xuu}} - {\Lambda _{tuu}} - 4{\Lambda _{xu}} = 0,\;\\ - 6u{\Lambda _{xxu}} - 2{\Lambda _{txu}} - 2{\Lambda _{xx}} = 0. \end{array} \end{array} \end{aligned}$$
(9)

The solution of the determining system (9) is given by

$$\begin{aligned} \Lambda \left( {x,t,u,{u_x},{u_t}} \right) = x{\mathcal {F}}^{\prime }(t) + u\left( { - 2{\mathcal {F}}\left( t \right) + {c_1}} \right) + {\mathcal {G}}\left( t \right) , \end{aligned}$$
(10)

where \({c_1}\) is an arbitrary constant and \({\mathcal {G}}(t)\) and \({\mathcal {F}}(t)\) are arbitrary differential functions about t. The solution yields three local conservation laws multipliers

$$\begin{aligned} \left( 1 \right) \;\Lambda = u,\quad \left( 2 \right) \;\Lambda = {{\mathcal {G}}}\left( t \right) ,\quad \left( 3 \right) \;\Lambda = x{{\mathcal {F}}}^{\prime }\left( t \right) - 2u{\mathcal F}\left( t \right) . \end{aligned}$$
(11)

Each multiplier determines a corresponding flux as Table 1 by using direct method with the aid of GeM [43, 44].

Table 1 Conservation laws of HS equation
Full size table

The three conservation laws in Table 1 result in the following potential systems

$$\begin{aligned}&U{V_1}\left\{ {x,t,u,{v_1}} \right\} = 0:\left\{ {\begin{array}{l} {{v_{1x}} = u{u_{xx}} + \frac{1}{2}u_x^2,}\\ {{v_{1t}} = {u_t}{u_x} - 2{u^2}{u_{xx}},} \end{array}} \right. \end{aligned}$$
(12)
$$\begin{aligned}&U{V_2}\left\{ {x,t,u,{v_2}} \right\} = 0: \left\{ {\begin{array}{l} {{v_{2x}} = {{{\mathcal {G}}}}\left( t \right) {u_{xx}},}\\ {{v_{2t}} = - \left( {2{{{\mathcal {G}}}}\left( t \right) u{u_{xx}} + {\mathcal {G}}\left( t \right) u_x^2 - {{{{\mathcal {G}}}}^\prime }\left( t \right) {u_x}} \right) ,} \end{array}} \right. \end{aligned}$$
(13)
$$\begin{aligned}&U{V_3}\left\{ {x,t,u,{v_3}} \right\} =\nonumber \\&\qquad 0:\left\{ \begin{array}{l} {{v_{3x}} = - {{{\mathcal {F}}}}\left( t \right) u_x^2 + {{{{\mathcal {F}}}}^\prime }\left( t \right) {u_x} - 2{{{\mathcal {F}}}}\left( t \right) u{u_{xx}} + x{{{{\mathcal {F}}}}^\prime }\left( t \right) {u_{xx}},}\\ \begin{array}{l} {{v_{3t}} = - \left( { - 4{{{\mathcal {F}}}}\left( t \right) {u^2}{u_{xx}} + x{{{{\mathcal {F}}}}^\prime }\left( t \right) u_x^2 + 2{{{\mathcal {F}}}}\left( t \right) {u_t}{u_x}} \right. }\\ \qquad \quad \left. { - x{{{{\mathcal {F}}}}^{\prime \prime }}\left( t \right) {u_x} - {{{{\mathcal {F}}}}^\prime }\left( t \right) {u_t} + 2{{{{\mathcal {F}}}}^\prime }\left( t \right) xu{u_{xx}}} \right) . \end{array} \end{array} \right. \end{aligned}$$
(14)

The three conservation laws in Table 1 yield up to \({2^3} - 1=7\) nonlocally related PDE systems. Therefore, the following theorem can be established.

Theorem 1

For the Hunter–Saxton equation, the set of locally inequivalent potential systems arising from multipliers depending on x, t and u is established by the following systems:

  • Three potential systems (12), (13) and (14) involving single potentials.

  • Three couplets \(U{V_1}{V_2}\left\{ {x,t,u,{v_1},{v_2}} \right\} \) [(12), (13)], \(U{V_1}{V_3}\left\{ {x,t,u,{v_1},{v_3}} \right\} \) [(12), (14)] and \(U{V_2}{V_3}\left\{ {x,t,u,{v_2},{v_3}} \right\} \) [(13), (14)].

  • One triplet \(U{V_1}{V_2}{V_3}\left\{ {x,t,u,{v_1},{v_2},{v_3}} \right\} \) [(12), (13), (14)].

A tree of nonlocally related PDE system for the Hunter–Saxton equation is presented in Fig. 1. In what follows, we shall investigate the Lie point and nonlocal symmetries of the nonlocally related PDE systems. On the basis of the Lie symmetry analysis, the Lie point symmetries of \(UV_1\) are given

$$\begin{aligned} \begin{array}{ll} {X_1} &{}= \dfrac{\partial }{{\partial t}},\quad {X_2} = \dfrac{\partial }{{\partial x}},\quad {X_3} = \dfrac{\partial }{{\partial v_1}},\\ {X_4} &{}= - t\dfrac{\partial }{{\partial t}} - x\dfrac{\partial }{{\partial x}} + v_1\dfrac{\partial }{{\partial v_1}},\\ {X_5} &{}= t\dfrac{\partial }{{\partial t}} + 2x\dfrac{\partial }{{\partial x}} + u\dfrac{\partial }{{\partial u}}. \end{array} \end{aligned}$$
(15)

The symmetry classification of system \(UV_2\) and \(UV_3\) are given by Tables 2 and 3 respectively. Table 4 presents the symmetry classification of other potential systems of the tree of nonlocally related system (Fig. 1).

Fig. 1
figure 1

A tree of nonlocally related PDE system for the Hunter–Saxton equation

Full size image
Table 2 Point symmetry classification of the potential system \(UV_2\)
Full size table

Theorem 2

If \({{{\mathcal {F}}}} ={e^t}\) the symmetry \(Z_{10}\) of the system \(UV_1V_3\) is the nonlocal symmetry of the system \(UV_1\).

Proof

For the Lie point symmetry \(Z_{10}\),

$$\begin{aligned} \begin{array}{l} {\xi ^1}\left( {x,t,u,{v_1},{v_3}} \right) = 2{e^t},\;{\xi ^2}\left( {x,t,u,{v_1},{v_3}} \right) = 2x{e^t},\\ {\eta ^1}\left( {x,t,u,{v_1},{v_3}} \right) = x{e^t},\; {\eta ^2}\left( {x,t,u,{v_1},{v_3}} \right) = {v_3},\\ {\eta ^3}\left( {x,t,u,{v_1},{v_3}} \right) = x{e^{2t}}. \end{array} \end{aligned}$$
(16)

Then

$$\begin{aligned} {\left( {\frac{{\partial {\xi ^1}}}{{\partial {v_3}}}} \right) ^2} + {\left( {\frac{{\partial {\xi ^2}}}{{\partial {v_3}}}} \right) ^2} + {\left( {\frac{{\partial {\eta ^1}}}{{\partial {v_3}}}} \right) ^2} + {\left( {\frac{{\partial {\eta ^2}}}{{\partial {v_3}}}} \right) ^2} = 1 > 0. \end{aligned}$$

So the symmetry \(Z_{10}\) is the nonlocal symmetry of the system \(UV_1\).\(\square \)

Remark 1

We can conclude that \(Z_{11}\) and \(Z_{12}\) are the nonlocal symmetries of system \(UV_3\) when \({{{\mathcal {F}}}} =\ln \left( t \right) \) as the same analysis as Theorem 2. In addition, \(Z_{15}\) is the nonlocal symmetry of \(UV_1\) and \(Z_{17}\) is the nonlocal symmetry of \(UV_3\). \(Z_{21}\) is the nonlocal symmetry of the system \(UV_1\) and \(UV_1V_2\). \(Z_{22}\) is the nonlocal symmetry of the system \(UV_2\), \(UV_1V_2\) and \(UV_2V_3\). Finally, \(Z_{23}\) is the nonlocal symmetry of the system \(UV_3\) and \(UV_2V_3\) when \({{{\mathcal {G}}}}={e^t}\) and \(\mathcal{F} =c\).

Remark 2

In this section, three local conservation laws of the Hunter–Saxton equation are constructed by limiting the multipliers to lowest-order. This class will miss some conservation laws. For the HS equation, the three-order multiplier \(\Lambda \left( {x,t,u,{u_x},{u_{xx}},{u_{xxx}}} \right) \) is \(x{{{{\mathcal {F}}}}^\prime }\left( t \right) - 2u{{{\mathcal {F}}}}\left( t \right) + {c_1}u + \mathcal{G}\left( t \right) + {c_2}\sqrt{{u_{xx}}}\). The term \(\sqrt{{u_{xx}}}\) will yield new conservation laws by using the direct method. However, the \(\sqrt{{u_{xx}}}\) is not a continuous function. It cannot split the flux continuously. For the multiplier \(\Lambda \left( {x,t,u,{u_x},{u_{xx}},{u_{xxx}},{u_{xxxx}},{u_{xxxxx}}} \right) \), it will appear new term \(\frac{{{u_{xxxx}}}}{{{{\left( {{u_{xx}}} \right) }^{{\textstyle {5 \over 2}}}}}} - \frac{5}{4}\frac{{u_{xxx}^2}}{{{{\left( {{u_{xx}}} \right) }^{{\textstyle {7 \over 2}}}}}}\). It is also hard to determine the conserved densities. Thus we don’t consider the high-order multiplier in this paper.

Remark 3

The obtained nonlocally related systems of theorem 1 are not exhaustive. It is a fact that linear combinations of the starting conservation laws may yield additional systems [45]. The most general form of potential system can be written as

$$\begin{aligned}\begin{array}{l} UV\left\{ {x,t,u,v} \right\} = 0:\\ \left\{ {\begin{array}{l} {v_x} = {c_1}\left( {u{u_{xx}} + \frac{1}{2}u_x^2} \right) + {c_2}\left( {{{{\mathcal {G}}}}\left( t \right) {u_{xx}}} \right) \\ \qquad + {c_3}\left( { - {{{\mathcal {F}}}}\left( t \right) u_x^2 + {{{{\mathcal {F}}}}^\prime }\left( t \right) {u_x} - 2{{{\mathcal {F}}}}\left( t \right) u{u_{xx}} + x{\mathcal{F}^\prime }\left( t \right) {u_{xx}}} \right) \\ \begin{array}{l} {v_t} = {c_1}\left( {{u_t}{u_x} - 2{u^2}{u_{xx}}} \right) + {c_2}\left( { - 2{{{\mathcal {G}}}}\left( t \right) u{u_{xx}} - {{{\mathcal {G}}}}\left( t \right) u_x^2 + {{{{\mathcal {G}}}}^\prime }\left( t \right) {u_x}} \right) \\ \quad \;\;\left. - {c_3}\left( { - 4{{{\mathcal {F}}}}\left( t \right) {u^2}{u_{xx}} + x{{{{\mathcal {F}}}}^\prime }\left( t \right) u_x^2 + 2{{{\mathcal {F}}}}\left( t \right) {u_t}{u_x}} \right. \right. \\ \left. \quad - x{{{{\mathcal {F}}}}^{\prime \prime }}\left( t \right) {u_x} - {{{{\mathcal {F}}}}^\prime }\left( t \right) {u_t} + 2{{{{\mathcal {F}}}}^\prime }\left( t \right) xu{u_{xx}} \right) . \end{array} \end{array}} \right. \end{array} \end{aligned}$$

Together with the potential systems in Theorem 1, they exhaust all possible inequivalent potential systems.

Table 3 Point symmetry classification of the potential system \(UV_3\)
Full size table
Table 4 Point symmetry classification of the potential system \(UV_1V_2\), \(UV_1V_3\), \(UV_2V_3\) and \(UV_1V_2V_3\)
Full size table

3 Equivalence transformations of potential systems \(UV_2\) and \(UV_3\)

An equivalence transformation transforms an equation that has arbitrary functions to an equation preserving the same differential structure but with different arbitrary functions [46, 47]. We shall use Lie’s infinitesimal criterion to derive the equivalence transformations of potential systems \(UV_2\) and \(UV_3\). For the system (13), the equivalence transformation is obtained by seeking an infinitesimal operator of the Lie algebra

$$\begin{aligned} E = \tau \frac{\partial }{{\partial t}} + \xi \frac{\partial }{{\partial x}} + {\zeta ^1}\frac{\partial }{{\partial u}} + {\zeta ^2}\frac{\partial }{{\partial {v_2}}} + {\zeta ^3}\frac{\partial }{{\partial {{\mathcal {G}}}}}. \end{aligned}$$
(17)

The one-parameter group of equivalence transformation is given by

$$\begin{aligned} \begin{array}{ll} {\tilde{t}} &{}= t + \varepsilon \tau \left( {t,x,u,{v_2}} \right) + O\left( {{\varepsilon ^2}} \right) ,\\ {\tilde{x}} &{}= x + \varepsilon \xi \left( {t,x,u,{v_2}} \right) + O\left( {{\varepsilon ^2}} \right) ,\\ {\tilde{u}} &{}= u + \varepsilon {\zeta ^1}\left( {t,x,u,{v_2}} \right) + O\left( {{\varepsilon ^2}} \right) ,\\ {{{\tilde{v}}}_2} &{}= {v_2} + \varepsilon {\zeta ^2}\left( {t,x,u,{v_2}} \right) + O\left( {{\varepsilon ^2}} \right) ,\\ \tilde{{{\mathcal {G}}}} &{}= {{{\mathcal {G}}}} + \varepsilon {\zeta ^3}\left( {t,x,u,{v_2},{{{\mathcal {G}}}}} \right) + O\left( {{\varepsilon ^2}} \right) , \end{array} \end{aligned}$$
(18)

where \(\varepsilon \) is the group parameter. The equivalence transformation operator (17) leaves not only the invariance of (13) but also the invariance of \({{{{\mathcal {G}}}}_x} = {{{{\mathcal {G}}}}_u} = {{{{\mathcal {G}}}}_{{v_2}}} = 0\). Then the invariance criterion yields an overdetermined system for \(\tau ,\,\xi ,\,{\zeta ^1},\,{\zeta ^2}\) and \({\zeta ^3}\). Solving this system one has following operators

$$\begin{aligned} \begin{array}{l} {E_1} = \dfrac{\partial }{{\partial t}},\quad {E_2} = \dfrac{\partial }{{\partial x}},\quad {E_3} = \dfrac{\partial }{{\partial {v_2}}},\\ {E_4} = {v_2}\dfrac{\partial }{{\partial {v_2}}} + {{{\mathcal {G}}}}\dfrac{\partial }{{\partial {{{\mathcal {G}}}}}},\\ {E_5} = - t\dfrac{\partial }{{\partial t}} + u\dfrac{\partial }{{\partial u}} - {{{\mathcal {G}}}}\dfrac{\partial }{{\partial {{{\mathcal {G}}}}}}, \end{array} \end{aligned}$$
(19)

where \({{{\mathcal {G}}}} = {{{\mathcal {G}}}}\left( t \right) \) is arbitrary function. Thus the five-parameter equivalence group associated with above five generators is given by

$$\begin{aligned} \begin{array}{l} {E_1}:\;{\tilde{t}} = t + {a_1},\;{\tilde{x}} = x,\;{\tilde{u}} = u,\;{{{\tilde{v}}}_2} = {v_2},\;\tilde{{{\mathcal {G}}}} = {{{\mathcal {G}}}},\\ {E_2}:\;{\tilde{t}} = t,\;{\tilde{x}} = x + {a_2},\;{\tilde{u}} = u,\;{{{\tilde{v}}}_2} = {v_2},\;\tilde{{{\mathcal {G}}}} = {{{\mathcal {G}}}},\\ {E_3}:\;{\tilde{t}} = t,\;{\tilde{x}} = x,\;{\tilde{u}} = u,\;{{{\tilde{v}}}_2} = {v_2} + {a_3},\;\tilde{{{\mathcal {G}}}} = {{{\mathcal {G}}}},\\ {E_4}:\;{\tilde{t}} = t,\;{\tilde{x}} = x,\;{\tilde{u}} = u,\;{{{\tilde{v}}}_2} = {e^{{a_4}}}{v_2},\;\tilde{{{\mathcal {G}}}} = {e^{{a_4}}}{{{\mathcal {G}}}},\\ {E_5}:\;{\tilde{t}} = {e^{ - {a_5}}}t,\;{\tilde{x}} = x,\;{\tilde{u}} = {e^{{a_5}}}u,\;{{{\tilde{v}}}_2} = {v_2},\;\tilde{{{\mathcal {G}}}} = {e^{ - {a_5}}}{{{\mathcal {G}}}}. \end{array} \end{aligned}$$
(20)

Therefore, the following theorem is established.

Theorem 3

Any transformation of the form

$$\begin{aligned} \begin{array}{l} {\tilde{t}} = {a_1} + {e^{ - {a_5}}}t,\quad {\tilde{x}} = {a_2} + x,\quad {\tilde{u}} = {e^{{a_5}}}u,\\ {{{\tilde{v}}}_2} = {a_3} + {e^{{a_4}}}{v_2},\quad \tilde{{{\mathcal {G}}}} = {e^{{a_4} - {a_5}}}{{{\mathcal {G}}}}, \end{array} \end{aligned}$$

where \({a_1}, \dots ,{a_5}\) are arbitrary constants, maps the potential systems \(UV_2\) (13) to the PDE system with same form

$$\begin{aligned} \left\{ {\begin{array}{l} {{{{\tilde{v}}}_{2{\tilde{x}}}} = \tilde{{{\mathcal {G}}}}\left( {{\tilde{t}}} \right) {{{\tilde{u}}}_{{\tilde{x}}{\tilde{x}}}},}\\ {{{{\tilde{v}}}_{2{\tilde{t}}}} = - \left( {2\tilde{{{\mathcal {G}}}}\left( {{\tilde{t}}} \right) {\tilde{u}}{{{\tilde{u}}}_{{\tilde{x}}{\tilde{x}}}} + \tilde{{{\mathcal {G}}}}\left( {{\tilde{t}}} \right) {\tilde{u}}_{{\tilde{x}}}^2 - {{\tilde{{{\mathcal {G}}}}}^\prime }\left( {{\tilde{t}}} \right) {{{\tilde{u}}}_{{\tilde{x}}}}} \right) .} \end{array}} \right. \end{aligned}$$

Then we can obtain the equivalence transformation theorem for the potential system \(UV_3\) (14) similar to the process of the derivation of Theorem 3.

Theorem 4

Any transformation of the form

$$\begin{aligned} \begin{array}{l} {\tilde{t}} = {a_1} + {e^{ - {a_3} + {a_4} + {a_5}}}t,\;{\tilde{x}} = {e^{ - {a_3} + 2{a_4} + {a_5}}}x,\;{\tilde{u}} = {e^{{a_4}}}u,\\ {{{\tilde{v}}}_1} = {a_2} + {e^{{a_3}}}{v_1},\;\tilde{{{\mathcal {F}}}} = {e^{{a_5}}}{{{\mathcal {F}}}}, \end{array} \end{aligned}$$

where \({a_1}, \dots ,{a_5}\) are arbitrary constants, maps the potential systems \(UV_3\) (14) to the PDE system with same form

$$\begin{aligned} \left\{ {\begin{array}{l} {{{{\tilde{v}}}_{3{\tilde{x}}}} = - \tilde{{{\mathcal {F}}}}\left( {{\tilde{t}}} \right) {\tilde{u}}_{{\tilde{x}}}^2 + {{\tilde{{{\mathcal {F}}}}}^\prime }\left( {{\tilde{t}}} \right) {{{\tilde{u}}}_{{\tilde{x}}}} - 2\tilde{{{\mathcal {F}}}}\left( {{\tilde{t}}} \right) {\tilde{u}}{{{\tilde{u}}}_{{\tilde{x}}{\tilde{x}}}} + {\tilde{x}}{{\tilde{{{\mathcal {F}}}}}^\prime }\left( {{\tilde{t}}} \right) {{{\tilde{u}}}_{{\tilde{x}}{\tilde{x}}}},}\\ {\begin{array}{l} {{{{\tilde{v}}}_{3{\tilde{t}}}} = - \left( { - 4\tilde{{{\mathcal {F}}}}\left( {{\tilde{t}}} \right) {{{\tilde{u}}}^2}{{{\tilde{u}}}_{{\tilde{x}}{\tilde{x}}}} + {\tilde{x}}{{\tilde{{{\mathcal {F}}}}}^\prime }\left( {{\tilde{t}}} \right) {\tilde{u}}_{{\tilde{x}}}^2 + 2\tilde{{{\mathcal {F}}}}\left( {{\tilde{t}}} \right) {{{\tilde{u}}}_{{\tilde{t}}}}{{{\tilde{u}}}_{{\tilde{x}}}}} \right. }\\ {\;\;\;\left. { - {\tilde{x}}{{\tilde{{{\mathcal {F}}}}}^{\prime \prime }}\left( {{\tilde{t}}} \right) {{{\tilde{u}}}_{{\tilde{x}}}} - {{\tilde{\mathcal{F}}}^\prime }\left( {{\tilde{t}}} \right) {{{\tilde{u}}}_{{\tilde{t}}}} + 2{{\tilde{{{\mathcal {F}}}}}^\prime }\left( {{\tilde{t}}} \right) {\tilde{x}}\tilde{u}{{{\tilde{u}}}_{{\tilde{x}}{\tilde{x}}}}} \right) .} \end{array}} \end{array}} \right. \end{aligned}$$

4 Inverse potential systems arising from Lie point symmetries

In this section, a symmetry-based method is employed to construct inverse potential systems of HS equation. The symmetry group will be generated by the vector field of the form

$$\begin{aligned} X = {\xi ^1}\left( {t,x,u} \right) \frac{\partial }{{\partial t}} + {\xi ^2}\left( {t,x,u} \right) \frac{\partial }{{\partial x}} + \eta \left( {t,x,u} \right) \frac{\partial }{{\partial u}}, \end{aligned}$$
(21)

then X (21) must satisfy Lie’s symmetry condition

$$\begin{aligned} pr^{\left( 3 \right) } X(\Delta )|_{\Delta = 0} = 0, \end{aligned}$$
(22)

where \(\Delta = {u_{xxt}} + 4{u_x}{u_{xx}} + 2u{u_{xxx}} = 0\). The Lie symmetry condition yields an overdetermined system of partial differential equations about \({\xi ^1},\;{\xi ^2}\) and \(\eta \)

$$\begin{aligned} \begin{array}{l} {\eta _{xx}} = 0,\quad {\eta _{xu}} = 0,\quad \xi _{tt}^1 = 2{\eta _x},\\ {\eta _{tu}} = 0,\;{\eta _{uu}} = 0,\quad \xi _x^1 = 0,\\ \xi _x^2 = \xi _t^1 + {\eta _u},\quad \xi _t^2 = - 2u{\eta _u} + 2\eta ,\\ \xi _u^1 = 0,\quad \xi _u^2 = 0. \end{array} \end{aligned}$$

Solving this system, one can get

$$\begin{aligned} \begin{array}{l} {\xi ^1} = {c_2}t + {c_3} + \int {2g^{\prime }\left( t \right) } dt,\\ {\xi ^2} = \left( {{c_2} + {c_1}} \right) x + {c_4} + \int {\left( {2g^{\prime }\left( t \right) x + 2f^{\prime }\left( t \right) } \right) } dt,\\ \eta = g^{\prime }\left( t \right) x + {c_1}u + f^{\prime }\left( t \right) . \end{array} \end{aligned}$$
(23)

where \(c_1\), \(c_2\), \(c_3\) and \(c_4\) are arbitrary constants and \(f\left( t \right) \) and \(g\left( t \right) \) are arbitrary differential functions. Hence the infinitesimal symmetries of (1) form the infinite dimensional Lie algebra L spanned by the following vector fields

$$\begin{aligned} \begin{array}{l} {X_1} = \dfrac{\partial }{{\partial t}},\quad {X_2} = \dfrac{\partial }{{\partial x}},\quad {X_3} = x\dfrac{\partial }{{\partial x}} + u\dfrac{\partial }{{\partial u}},\\ {X_4} = t\dfrac{\partial }{{\partial t}} + x\dfrac{\partial }{{\partial x}},\quad {X_5} = 2f\left( t \right) \dfrac{\partial }{{\partial x}} + f^{\prime }\left( t \right) \dfrac{\partial }{{\partial u}},\\ {X_6} = 2g\left( t \right) \dfrac{\partial }{{\partial t}} + 2g\left( t \right) x\dfrac{\partial }{{\partial x}} + g^{\prime }\left( t \right) x\dfrac{\partial }{{\partial u}}. \end{array} \end{aligned}$$
(24)

4.1 Inverse potential system from \(X_1\)

For the symmetry \(X_1\), it maps into the canonical form \({P} = \frac{\partial }{{\partial v}}\) by introducing canonical coordinates

$$\begin{aligned} \begin{array}{l} r = x,\\ s = u,\\ v(r,s) = t. \end{array} \end{aligned}$$
(25)

At the same time, the Eq. (1) is mapped to an invertibly equivalent equation

$$\begin{aligned} \begin{array}{l} \begin{array}{l} 12s{v_r}v_s^2v_{sr}^2 - 18s{v_s}v_r^2{v_{sr}}{v_{ss}} - 6sv_s^3{v_{sr}}{v_{rr}} \\ \quad - 6s{v_r}v_s^3{v_{srr}} { + 6sv_r^3v_{ss}^2 + 6sv_s^2{v_r}{v_{ss}}{v_{rr}} + 6sv_s^2v_r^2{v_{ssr}}} \end{array}\\ \quad -\, 2s{v_s}v_r^3{v_{sss}}+ 2sv_s^4{v_{rrr}} + 8v_s^2v_r^2{v_{sr}}\\ \quad - 4{v_s}v_r^3{v_{ss}} - 4{v_r}v_s^3{v_{rr}} - 2v_s^2v_{sr}^2 + 6{v_r}{v_s}{v_{sr}}{v_{ss}} + v_s^3{v_{srr}} \\ \quad -\, 3v_r^2v_{ss}^2 - v_s^2{v_{ss}}{v_{rr}} - 2{v_r}v_s^2{v_{ssr}}+ {v_s}v_r^2{v_{sss}} = 0. \end{array} \end{aligned}$$
(26)

Introducing the new variable \(\phi = {v_r}\) and \(\psi = {v_s}\), one can obtain the locally related intermediate system

$$\begin{aligned} \begin{array}{l} \begin{array}{l} {\begin{array}{l} {\phi = {v_r},}\\ {\psi = {v_s},}\\ 12s\phi {\psi ^2}\psi _r^2 - 18s\psi {\phi ^2}{\psi _r}{\psi _s} - 6s{\psi ^3}{\psi _r}{\phi _r} - 6s\phi {\psi ^3}{\psi _{rr}} + 6s{\phi ^3}\psi _s^2 \\ \quad + 6s{\psi ^2}\phi {\psi _s}{\phi _r} + 6s{\psi ^2}{\phi ^2}{\psi _{sr}} - 2s\psi {\phi ^3}{\psi _{ss}} \end{array}}\\ \quad +\, 2s{\psi ^4}{\phi _{rr}} + 8{\psi ^2}{\phi ^2}{\psi _r} - 4\psi {\phi ^3}{\psi _s} - 4\phi {\psi ^3}{\phi _r} - 2{\psi ^2}\psi _r^2\\ \quad + 6\phi \psi {\psi _r}{\psi _s} + {\psi ^3}{\psi _{rr}} - 3{\phi ^2}\psi _s^2 - {\psi ^2}{\psi _s}{\phi _r} \end{array}\\ \quad -\, 2\phi {\psi ^2}{\psi _{sr}} + \psi {\phi ^2}{\psi _{ss}} = 0. \end{array} \end{aligned}$$
(27)

Eliminating v from the system (27), one obtains an inverse potential system (\(\textit{IP}_1\)) of Eq. (1)

$$\begin{aligned} \begin{array}{l} \begin{array}{l} {\begin{array}{l} {{\phi _s} = {\psi _r},}\\ 12s\phi {\psi ^2}\psi _r^2 - 18s\psi {\phi ^2}{\psi _r}{\psi _s} - 6s{\psi ^3}{\psi _r}{\phi _r} - 6s\phi {\psi ^3}{\psi _{rr}}\\ \quad + 6s{\phi ^3}\psi _s^2 + 6s{\psi ^2}\phi {\psi _s}{\phi _r} + 6s{\psi ^2}{\phi ^2}{\psi _{sr}} - 2s\psi {\phi ^3}{\psi _{ss}} \end{array}}\\ \quad + 2s{\psi ^4}{\phi _{rr}} + 8{\psi ^2}{\phi ^2}{\psi _r} - 4\psi {\phi ^3}{\psi _s} - 4\phi {\psi ^3}{\phi _r} - 2{\psi ^2}\psi _r^2\\ \quad + 6\phi \psi {\psi _r}{\psi _s} + {\psi ^3}{\psi _{rr}} - 3{\phi ^2}\psi _s^2 - {\psi ^2}{\psi _s}{\phi _r} \end{array}\\ \quad -\, 2\phi {\psi ^2}{\psi _{sr}} + \psi {\phi ^2}{\psi _{ss}} = 0. \end{array} \end{aligned}$$
(28)

Due to the inverse potential system (28) is nonlocally related to the intermediate system (27), the inverse potential system (28) is nonlocally related to Eq. (1). The transformation (25) establishes a one-to-one mapping between the solutions of (28) and (1). As the process of construction of the nonlocally related system by using symmetry \(X_1\), one can construct the nonlocally related systems, which are based on \(X_2\) to \(X_6\).

4.2 Inverse potential system from \(X_2\)

For the symmetry \(X_2\), it maps into the canonical form \({P} = \frac{\partial }{{\partial v}}\) by introducing canonical coordinates

$$\begin{aligned} \begin{array}{l} r = t,\\ s = u,\\ v(r,s) = x. \end{array} \end{aligned}$$
(29)

At the same time, the Eq. (1) is mapped to an invertibly equivalent equation

$$\begin{aligned} v_s^2{v_{ssr}} - 3{v_s}{v_{sr}}{v_{ss}} + 3{v_r}v_{ss}^2 - 6sv_{ss}^2 - {v_r}{v_s}{v_{sss}} + 2s{v_s}{v_{sss}} + 4{v_s}{v_{ss}} = 0 \end{aligned}$$
(30)

Introducing the new variable \(\phi = {v_r}\) and \(\psi = {v_s}\), one can obtain the locally related intermediate system

$$\begin{aligned} \begin{array}{l} {\phi = {v_r},}\\ {\psi = {v_s},}\\ \begin{array}{l} v_s^2{v_{ssr}} - 3{v_s}{v_{sr}}{v_{ss}} + 3{v_r}v_{ss}^2 - 6sv_{ss}^2 - {v_r}{v_s}{v_{sss}} + 2s{v_s}{v_{sss}} + 4{v_s}{v_{ss}} = 0. \end{array} \end{array} \end{aligned}$$
(31)

Eliminating v from the system (31), one obtains an inverse potential system (\(\textit{IP}_2\)) of Eq. (1)

$$\begin{aligned} \begin{array}{l} {{\phi _s} = {\psi _r},}\\ \begin{array}{l} {\psi ^2}{\psi _{sr}} - 3\psi {\psi _r}{\psi _s} + 3\phi \psi _s^2 - 6s\psi _s^2 - \phi \psi {\psi _{ss}} + 2s\psi {\psi _{ss}} + 4\psi {\psi _s} = 0. \end{array} \end{array} \end{aligned}$$
(32)

4.3 Inverse potential system from \(X_3\)

For the symmetry \(X_3\), it maps into the canonical form \({P} = \frac{\partial }{{\partial v}}\) by introducing canonical coordinates

$$\begin{aligned} \begin{array}{l} r = t,\\ s = \dfrac{u}{x},\\ v(r,s) = x. \end{array} \end{aligned}$$
(33)

At the same time, the Eq. (1) is mapped to an invertibly equivalent equation

$$\begin{aligned}&{2sv_s^4 - {v_{sr}}v_s^3 + {v_{ss}}{v_r}v_s^2 - 4s{v_{ss}}v_s^2 + 3{v_s}{v_{sr}}{v_{ss}}} - v_s^2{v_{ssr}} - 3{v_r}v_{ss}^2 + 6sv_{ss}^2 \nonumber \\&\quad + \,{v_r}{v_s}{v_{sss}}-2s{v_s}{v_{sss}}+ 4v_s^3 - 4{v_s}{v_{ss}} = 0. \end{aligned}$$
(34)

Introducing the new variable \(\phi = {v_r}\) and \(\psi = {v_s}\), one can obtain the locally related intermediate system

$$\begin{aligned} \begin{array}{l} {\phi = {v_r},}\\ {\psi = {v_s},}\\ {2s{\psi ^4} - {\psi _r}{\psi ^3} + \phi {\psi ^2}{\psi _s} - 4s{\psi ^2}{\psi _s} + 3\psi {\psi _r}{\psi _s}} - {\psi ^2}{\psi _{sr}} - 3\phi \psi _s^2 + 6s\psi _s^2 \\ \quad +\, \phi \psi \psi _{ss}^2- 2s\psi {\psi _{ss}}+ 4{\psi ^3} - 4\psi {\psi _s} = 0. \end{array} \end{aligned}$$
(35)

Eliminating v from the system (35), one obtains an inverse potential system (\(\textit{IP}_3\)) of Eq. (1)

$$\begin{aligned} \begin{array}{l} {{\phi _s} = {\psi _r},}\\ {2s{\psi ^4} - {\psi _r}{\psi ^3} + \phi {\psi ^2}{\psi _s} - 4s{\psi ^2}{\psi _s} + 3\psi {\psi _r}{\psi _s}} - {\psi ^2}{\psi _{sr}} - 3\phi \psi _s^2 + 6s\psi _s^2 \\ \quad +\, \phi \psi \psi _{ss}^2 - 2s\psi {\psi _{ss}}+ 4{\psi ^3} - 4\psi {\psi _s} = 0. \end{array} \end{aligned}$$
(36)

4.4 Inverse potential system from \(X_4\)

For the symmetry \(X_4\), it maps into the canonical form \({P} = \frac{\partial }{{\partial v}}\) by introducing canonical coordinates

$$\begin{aligned} \begin{array}{l} r = \dfrac{t}{x},\\ s = u,\\ v\left( {r,s} \right) = \ln \left( x \right) . \end{array} \end{aligned}$$
(37)

Equation (1) is mapped to an invertibly equivalent equation

$$\begin{aligned} \begin{array}{l} - 6s{r^3}{v_r}v_s^3{v_{srr}} + 6s{r^3}v_s^2v_r^2{v_{ssr}} - 2s{r^3}{v_s}v_r^3{v_{sss}} - 24s{r^2}{v_r}v_s^3{v_{sr}} \\ \quad + 12s{r^2}v_s^2v_r^2{v_{ss}} - 3{r^2}{v_r}v_s^2{v_{rr}}{v_{ss}}+\, 6s{r^2}v_s^2{v_{rr}}{v_{ss}} \\ \quad +\, 9{r^2}{v_s}v_r^2{v_{ss}}{v_{sr}} + 12s{r^2}{v_r}v_s^2{v_{ssr}} - 6s{r^2}{v_s}v_r^2{v_{sss}}\\ \quad + 18rs{v_r}v_s^2{v_{ss}} + 12r{v_s}{v_r}{v_{ss}}{v_{sr}}\\ \quad -\, 18rs{v_s}{v_{ss}}{v_{sr}} - 6rs{v_s}{v_r}{v_{sss}} - 6s{r^3}v_s^3{v_{sr}}{v_{rr}} + 12s{r^3}{v_r}v_s^2v_{sr}^2\\ \quad - 2rv_s^2{v_{ss}}{v_{rr}} + 18rs{v_r}v_{ss}^2 - 4r{v_r}v_s^2{v_{ssr}}\\ \quad +\, 6srv_s^2{v_{ssr}} + 2r{v_s}v_r^2{v_{sss}} - 12r{v_s}{v_r}{v_{ss}} + 2s{r^3}v_s^4{v_{rrr}} + 6s{r^3}v_r^3v_{ss}^2 \\ \quad - 4{r^3}{v_r}v_s^3{v_{rr}} + 12s{r^2}v_s^4{v_{rr}}+\, 8{r^3}v_s^2v_r^2{v_{sr}}\\ \quad - 4{r^3}{v_s}v_r^3{v_{ss}} + 3{r^2}v_s^3{v_{sr}}{v_{rr}} - 6{r^2}{v_r}v_s^2v_{sr}^2 \\ \quad + 12s{r^2}v_s^2v_{sr}^2 + 3{r^2}{v_r}v_s^3{v_{srr}} - 6s{r^2}v_s^3{v_{srr}}\\ \quad +\, 18s{r^2}v_r^2v_{ss}^2 - 3{r^2}v_s^2v_r^2{v_{ssr}} + {r^2}{v_s}v_r^3{v_{sss}} + 12rs{v_r}v_s^4 + 8r{v_r}v_s^3{v_{sr}}\\ \quad + 16{r^2}{v_r}v_s^2{v_{sr}} - 18srv_s^3{v_{sr}}-\, 4rv_s^2v_r^2{v_{ss}} \\ \quad - 12{r^2}{v_s}v_r^2{v_{ss}} + 6s{r^3}{v_r}v_s^2{v_{ss}}{v_{rr}} - 18s{r^3}{v_s}v_r^2{v_{ss}}{v_{sr}} \\ \quad - 36s{r^2}{v_s}{v_r}{v_{ss}}{v_{sr}} - 4v_s^3 - v_s^2{v_{ssr}}\\ \quad -\, 4{v_s}{v_{ss}} - 3{v_r}v_{ss}^2 - 2{v_r}v_s^4 + 4sv_s^4 + 3v_s^3{v_{sr}} \\ \quad + 6sv_{ss}^2 + {v_s}{v_r}{v_{sss}} - 2s{v_s}{v_{sss}} + 2rv_s^3{v_{srr}} - 6rv_r^2v_{ss}^2\\ \quad -\, 12r{v_r}v_s^3 + 8rv_s^2{v_{sr}} - 3{v_r}v_s^2{v_{ss}} + 6sv_s^2{v_{ss}} + 3{v_s}{v_{sr}}{v_{ss}}\\ \quad - {r^2}v_s^4{v_{rrr}} - 3{r^2}v_r^3v_{ss}^2 - 8{r^2}v_r^2v_s^3 - 4rv_s^4{v_{rr}}\\ \quad -\, 4{r^2}v_s^3{v_{rr}} - 4rv_s^2v_{sr}^2 = 0. \end{array} \end{aligned}$$
(38)

Introducing the new variable \(\phi = {v_r}\) and \(\psi = {v_s}\) and eliminating v from the locally related intermediate system, one obtains the inverse potential system (\(\textit{IP}_4\))

$$\begin{aligned} \begin{array}{l} {\phi _s} = {\psi _r},\\ - 6s{r^3}\phi {\psi ^3}{\psi _{rr}} + 6s{r^3}{\psi ^2}{\phi ^2}{\psi _{sr}} - 2s{r^3}\psi {\phi ^3}{\psi _{ss}} \\ \quad - 24s{r^2}\phi {\psi ^3}{\psi _r} + 12s{r^2}{\psi ^2}{\phi ^2}{\psi _s} - 3{r^2}\phi {\psi ^2}{\phi _r}{\psi _s}\\ \quad + 6s{r^2}{\psi ^2}{\phi _r}{\psi _s} + 9{r^2}\psi {\phi ^2}{\psi _s}{\psi _r} + 12s{r^2}\phi {\psi ^2}{\psi _{sr}} \\ \quad - 6s{r^2}\psi {\phi ^2}{\psi _{ss}} + 18rs\phi {\psi ^2}{\psi _s} + 12r\psi \phi {\psi _s}{\psi _r}\\ \quad - 18rs\psi {\psi _s}{\psi _r} - 6rs\psi \phi {\psi _{ss}} - 6s{r^3}{\psi ^3}{\psi _r}{\phi _r} \\ \quad + 12s{r^3}\phi {\psi ^2}\psi _r^2 - 2r{\psi ^2}{\psi _s}{\phi _r} + 18rs\phi \psi _s^2 - 4r\phi {\psi ^2}{\psi _{sr}}\\ \quad + 6sr{\psi ^2}{\psi _{sr}} + 2r\psi {\phi ^2}{\psi _{ss}} - 12r\psi \phi {\psi _s} + 2s{r^3}{\psi ^4}{\phi _{rr}}\\ \quad + 6s{r^3}{\phi ^3}\psi _s^2 - 4{r^3}\phi {\psi ^3}{\phi _r} + 12s{r^2}{\psi ^4}{\phi _r}\\ \quad + 8{r^3}{\psi ^2}{\phi ^2}{\psi _r} - 4{r^3}\psi {\phi ^3}{\psi _s} + 3{r^2}{\psi ^3}{\psi _r}{\phi _r} - 6{r^2}\phi {\psi ^2}\psi _r^2 \\ \quad + 12s{r^2}{\psi ^2}\psi _r^2 + 3{r^2}\phi {\psi ^3}{\psi _{rr}} - 6s{r^2}{\psi ^3}{\psi _{rr}}\\ \quad + 18s{r^2}{\phi ^2}\psi _s^2 - 3{r^2}{\psi ^2}{\phi ^2}{\psi _r} + {r^2}\psi {\phi ^3}{\psi _{ss}} + 12rs\phi {\psi ^4} \\ \quad + 8r\phi {\psi ^3}{\psi _r} + 16{r^2}\phi {\psi ^2}{\psi _r} - 18sr{\psi ^3}{\psi _r}\\ \quad - 4r{\psi ^2}{\phi ^2}{\psi _s} - 12{r^2}\psi {\phi ^2}{\psi _s} + 6s{r^3}\phi {\psi ^2}{\psi _s}{\phi _r} \\ \quad - 18s{r^3}\psi {\phi ^2}{\psi _s}{\psi _r} - 36s{r^2}\psi \phi {\psi _s}{\psi _r} - 4{\psi ^3} - {\psi ^2}{\psi _{sr}}\\ \quad - 4\psi {\psi _s} - 3\phi \psi _s^2 - 2\phi {\psi ^4} + 4s{\psi ^4} + 3{\psi ^3}{\psi _r} + 6s\psi _s^2 \\ \quad + \psi \phi {\psi _{ss}} - 2s\psi {\psi _{ss}} + 2r{\psi ^3}{\psi _{rr}} - 6r{\phi ^2}\psi _s^2\\ \quad - 12r\phi {\psi ^3} + 8r{\psi ^2}{\psi _r} - 3\phi {\psi ^2}{\psi _s} + 6s{\psi ^2}{\psi _s} \\ \quad + 3\psi {\psi _r}{\psi _s} - {r^2}{\psi ^4}{\phi _{rr}} - 3{r^2}{\phi ^3}\psi _s^2 - 8{r^2}{\phi ^2}{\psi ^3} - 4r{\psi ^4}{\phi _r}\\ \quad - 4{r^2}{\psi ^3}{\phi _r} - 4r{\psi ^2}\psi _r^2 = 0. \end{array} \end{aligned}$$
(39)

4.5 Inverse potential system from \(X_5\)

When \(f\left( t \right) = t\) for \(X_5\), canonical coordinates induced by \(X_5\) are given by

$$\begin{aligned} \begin{array}{l} r = t,\\ s = u - \dfrac{1}{2}\dfrac{x}{t},\\ v\left( {r,s} \right) = \dfrac{1}{2}\dfrac{x}{t}. \end{array} \end{aligned}$$
(40)

Transformation (40) maps Eq. (1) to the equation

$$\begin{aligned} \begin{array}{l} rv_s^2{v_{ssr}} - r{v_r}{v_s}{v_{sss}} + 3r{v_r}v_{ss}^2 - 3r{v_s}{v_{sr}}{v_{ss}} + s{v_s}{v_{sss}} - 3sv_{ss}^2 + 2{v_s}{v_{ss}} = 0. \end{array}\nonumber \\ \end{aligned}$$
(41)

which is invertibly related to Eq. (1). Introducing the new variable \(\phi = {v_r}\) and \(\psi = {v_s}\) and eliminating v from the locally related intermediate system, one obtains the inverse potential system (\(\textit{IP}_5\))

$$\begin{aligned} \begin{array}{l} {\phi _s} = {\psi _r},\\ r{\psi ^2}{\psi _{sr}} - r\phi \psi {\psi _{ss}} + 3r\phi \psi _s^2 - 3r\psi {\psi _r}{\psi _s} + s\psi {\psi _{ss}} - 3s\psi _s^2 + 2\psi {\psi _s} = 0. \end{array} \end{aligned}$$
(42)

4.6 Inverse potential system from \(X_6\)

When \(g\left( t \right) = {e^t}\) for \(X_6\), canonical coordinates induced by \(X_6\) are given by

$$\begin{aligned} \begin{array}{l} r = - \ln \left( x \right) +t,\\ s = - \dfrac{1}{2}x + u,\\ v\left( {r,s} \right) = - \dfrac{1}{2}{e^{ - t}}. \end{array} \end{aligned}$$
(43)

Transformation (43) maps Eq. (1) to the equation

$$\begin{aligned} \begin{array}{l} {\begin{array}{l} \left( {12sv_r^3v_{ss}^2 - 36s{v_s}v_r^2{v_{sr}}{v_{ss}} + 12sv_s^2v_r^2{v_{ss}}} \right. + 12s{v_r}v_s^2{v_{ss}}{v_{rr}} \\ \quad + 24s{v_r}v_s^2v_{sr}^2 - 24sv_s^3{v_r}{v_{sr}} \end{array}}\\ {\begin{array}{l} \quad -\, 12sv_s^3{v_{sr}}{v_{rr}} - 4s{v_s}v_r^3{v_{sss}} + 12sv_s^2v_r^2{v_{ssr}} + 8sv_s^4{v_r}\\ \quad - 12s{v_r}v_s^3{v_{srr}} + 12s{v_{rr}}v_s^4 + 4sv_s^4{v_{rrr}}s\\ {\begin{array}{l} \left. { \quad -\, 8{v_s}v_r^3{v_{ss}} + 16v_s^2v_r^2{v_{sr}} - 8v_s^3v_r^2 - 8{v_r}v_s^3{v_{rr}}} \right) {e^r} \\ \quad + 3v_r^2v_{ss}^2 - 6{v_s}{v_r}{v_{ss}}{v_{sr}} + {v_r}v_s^2{v_{ss}} + v_s^2{v_{ss}}{v_{rr}}\\ {\quad + \,2v_s^2v_{sr}^2 - v_s^3{v_{sr}} - {v_s}v_r^2{v_{sss}} + 2{v_r}v_s^2{v_{ssr}} - v_s^3{v_{srr}} = 0.}\\ \end{array}} \end{array}}\\ \end{array} \end{aligned}$$
(44)

Introducing the new variable \(\phi = {v_r}\) and \(\psi = {v_s}\) and eliminating v from the locally related intermediate system, one obtains the inverse potential system (\(\textit{IP}_6\))

$$\begin{aligned} \begin{array}{l} {{\phi _s} = {\psi _r},}\\ {\begin{array}{l} {\left( {12s{\phi ^3}\psi _s^2 {-} 36s\psi {\phi ^2}{\psi _r}{\psi _s} {+} 12s{\psi ^2}{\phi ^2}{\psi _s} {+} 12s\phi {\psi ^2}{\psi _s}{\phi _r} {+} 24s\phi {\psi ^2}\psi _r^2 {-} 24s{\psi ^3}\phi {\psi _r}} \right. }\\ \quad - \,12s{\psi ^3}{\psi _r}{\phi _r} - 4s\psi {\phi ^3}{\psi _{ss}} + 12s{\psi ^2}{\phi ^2}{\psi _{sr}} \\ \quad + 8s{\psi ^4}\phi - 12s\phi {\psi ^3}{\psi _{rr}} + 12s{\phi _r}{\psi ^4} + 4s{\psi ^4}{\phi _{rr}} \end{array}}\\ {\begin{array}{l} {\left. {\quad - \,8\psi {\phi ^3}{\psi _s} {+} 16{\psi ^2}{\phi ^2}{\psi _r} {-} 8{\psi ^3}{\phi ^2} {-} 8\phi {\psi ^3}{\phi _{rr}}} \right) {e^r} {+} 3{\phi ^2}\psi _s^2 {-} 6\psi \phi {\psi _s}{\psi _r} {+} \phi {\psi ^2}{\psi _s} }\\ {\quad +\, {\psi ^2}{\psi _s}{\phi _r}+2{\psi ^2}\psi _r^2 - {\psi ^3}{\psi _r} - \psi {\phi ^2}{\psi _{ss}} + 2\phi {\psi ^2}{\psi _{sr}} - {\psi ^3}{\psi _{rr}} = 0.} \end{array}}\\ \end{array} \end{aligned}$$
(45)

Remark 4

The inverse potential system \(\textit{IP}_1\), \(\textit{IP}_2\), \(\textit{IP}_3\), \(\textit{IP}_4\), \(\textit{IP}_5\) and \(\textit{IP}_6\) play an important role in analyzing HS equation, which are all equivalent to the HS equation. The relationship between the solutions of inverse potential system and HS equation is one-to-one. All the inverse potential systems are nonlocally related to HS equation. Figure 2 presents a tree of inverse potential systems arising from Lie point symmetries, which further extend the tree of nonlocally related systems (see Fig. 1) form conservation law-based method.

Fig. 2
figure 2

A tree of inverse potential systems for the Hunter–Saxton equation

Full size image

5 Conclusions

In this paper Lie symmetry analysis method is performed on the HS equation. The direct method is used to derive local conservation laws of the HS equation. The nonlocally related PDE systems of HS equation are constructed with the aid of conservation law-based method. Based on the symmetry classification of the potential systems, we obtain many new local and nonlocal symmetries. A tree of nonlocally related PDE system for HS equation is presented in Fig. 1. Two equivalence transformation theorems of the potential systems are established. In order to extend the tree we established a tree of the inverse potential systems (Fig. 2) by using a symmetry-based method. The results of this paper are helpful for further analysis of the properties of the HS equation.