First-Order Multipliers, Noether Symmetries and Variational Reduction of the Hunter–Saxton Equation

Abstract

We revisit the Hunter–Saxton equation through its classical first-order Lagrangian. The determining system for all first-order multipliers is reduced to one linear equation in two variables. Its solutions include both point-symmetry characteristics and genuinely generalised variational characteristics; in the analytic category, the remaining freedom is locally parameterised by two arbitrary analytic functions. Every member of the known infinite-dimensional point-symmetry ideal is shown to preserve the action up to a total divergence and produces an arbitrary-function family of conserved currents, whereas one finite point symmetry is excluded from the Noether point-symmetry algebra of the Lagrangian. We also correct an omission in the previously reported associations between finite currents and Lie point symmetries. For the scaling symmetry, the radial component of an associated current vanishes identically after transformation. Alignment and multiplier criteria explain this degeneracy and show when it is a property of the conservation-law class. Thus association guarantees invariance of the transformed component, but not that the component retains differential content. Reducing the Lagrangian instead recovers the known similarity equations and their first integrals for two representative symmetries. In the scaling reduction, the reduced Noether symmetry is induced by a commuting member of the infinite-dimensional ideal.

1. Introduction

Symmetries, conservation laws and variational principles provide complementary ways of reducing nonlinear partial differential equations. In classical double reduction [1,2], a conserved current associated with a Lie point symmetry in the sense of Kara and Mahomed [3] is transformed to canonical coordinates. When the transformed current is inherited nontrivially, its invariant component gives a first integral of the symmetry-reduced equation. For an Euler–Lagrange equation, there is a second, intrinsically variational route: one may reduce the action by a variational symmetry and then use the structure of the reduced Lagrangian to obtain first integrals.

A limitation of classical double reduction is that the reduced equation need not inherit the conserved vector associated with the chosen symmetry. The Kara–Mahomed association is necessary, but does not by itself guarantee that the transformed conserved vector retains nontrivial differential content in the canonical variables. In this paper, we exhibit a sharp instance of this non-inheritance, in which the r-component of the transformed conserved vector collapses identically to zero. The reduced divergence relation is then a trivially satisfied identity: no ODE is produced and no first integral is obtained. We refer to this complete and fully explicit form of cancellation as strong degeneracy. Beyond the explicit example, we derive geometric and multiplier criteria that identify when degeneracy occurs and establish conditions under which it is invariant under a change in the current representative. The phenomenon was observed, but not explained, in our earlier paper [4]; one of the contributions below is an explicit algebraic diagnosis.

We study the Hunter–Saxton (HS) equation:

$${\left(u_t+u{u}_x\right)}_x=\frac{1}{2}{u}_x^2,$$

(1)

which arises in the theoretical study of nematic liquid crystals and models orientation waves in the director field [5]. The established literature reveals several complementary, but only partly connected, structures. At the geometric and variational level, the equation is related to geodesic motion on diffeomorphism groups and admits bi-Hamiltonian and variational formulations [6,7]. In particular, Baxter, Van Gorder and Vajravelu [8] recorded the action $\int \int (u_t{u}_x+u{u}_x^2)\mathrm{d}x\mathrm{d}t$. Up to an overall constant, its integrand is the first-order Lagrangian used below. These results establish the variational foundation of (1); they do not, however, classify the full set of point transformations that preserve this action up to a divergence.

Equation (1) also belongs to a broader integrable-wave setting in which multiplier, Lie symmetry and variational methods are often used together. The closest comparison is the Camassa–Holm (CH) equation, from which the HS equation arises as a high-frequency, or short-wave, limit [6]. For a related CH-type equation, Bodibe and Khalique combined Lie symmetry analysis, closed-form solutions and conservation laws [9]. This provides a useful methodological comparison with the present study, although the equations and the resulting conservation-law classifications are different.

The connection with the Korteweg–de Vries (KdV) equation is also structural. Khesin and Misiołek showed that KdV, CH and HS have the same underlying Virasoro symmetry group and can be realised as Euler or geodesic equations for different right-invariant metrics on the Virasoro group or appropriate homogeneous spaces; their bi-Hamiltonian structures consequently belong to a common geometric classification [10]. At the hierarchy level, Lenells constructed a nonlinear change of variables between the bi-Poisson structures of KdV and a modified HS equation containing a linear term, thereby relating their hierarchies of commuting flows and conservation laws [11]. Thus, the explicit nonlinear hierarchy correspondence concerns the modified HS equation, while the standard HS Equation (1) is connected to KdV through the broader bi-Hamiltonian and Virasoro geometric framework.

Related developments for Korteweg–de Vries (KdV)-type equations further illustrate the range of these methods. Bruzón et al. used multipliers to classify conservation laws for a generalised seventh-order KdV equation [12]. Márquez et al. studied symmetries, conservation laws and line-soliton solutions for a two-dimensional generalised KdV equation with power nonlinearity [13], while Márquez, Gandarias and Anco related symmetries, conservation laws and line solitons for a Kawahara–KP equation [14]. These works are cited to place the multiplier and symmetry methodology used below in the wider integrable-systems literature. They complement the structural relationship described above without identifying the standard HS equation directly with the standard KdV equation.

A second strand exposes a much richer integrability structure. Wang constructed infinite hierarchies of higher symmetries, recursion operators and conserved densities [15]. Tian and Liu subsequently traced arbitrary-function conserved densities and commuting hierarchies to reciprocal transformations [16]. Morozov’s contact-symmetry classification for the generalised Hunter–Saxton equation gives the semidirect product ${\mathfrak{gl}}_2\left(\mathbb{R}\right)⋉{\mathfrak{a}}_{\infty }$; the arbitrary-function point symmetries considered below form a distinguished subideal of ${\mathfrak{a}}_{\infty }$ [17]. Collectively, these studies show that the relevant gap is not a shortage of symmetries or conservation laws. Rather, they do not classify all multipliers depending on $(t,x,u,u_t,u_x)$, determine which members of the point algebra preserve the classical first-order action up to a divergence, or explain how the resulting currents behave under symmetry reduction.

Lie point analyses have addressed reductions, optimal systems, conservation laws, and exact and similarity solutions [4,8,18,19]. Closer in method to the present work, Jafari and Mahdion used the multiplier method together with a two-dimensional homotopy operator to construct higher-order HS conservation laws and examined their invariance under classical Lie symmetries [20]. Their construction produces higher-order examples, whereas Theorem 1 below classifies the complete multiplier family under the stated first-order dependence. The two results are therefore complementary in differential order and scope; neither family is claimed here to contain the other. These studies demonstrate the effectiveness of symmetry reduction, but they do not settle the complete first-order multiplier problem. In particular, the known point-symmetry ideal $Y_f=f(t){\partial }_x+f^{\prime }(t){\partial }_u$ supplies the characteristics $f^{\prime }-f{u}_x$, whereas the multiplier $u_x^{-2}$ arises from a generalised rather than a point variational symmetry. A complete classification must accommodate both types and determine their relation to the known first-order action and to equivalence classes of local currents.

Parallel advances in solution theory have clarified the regularity and long-time behaviour of conservative solutions [21,22], stability under $\alpha$-dependent dissipation [23], and a sharp threshold for energy conservation and uniqueness [24]. These developments reinforce the continuing analytical relevance of the equation, while addressing questions distinct from the local Noether–reduction problem considered here.

Against this background, the contribution of the paper is fourfold. First, we reduce the determining system for all multipliers $\Lambda (t,x,u,u_t,u_x)$ to a single linear equation, characterise its local analytic solution space, and exhibit the solutions required below. Second, within the known point-symmetry algebra, we determine the full algebra of Noether point symmetries of the first-order Lagrangian: the entire ideal $Y_f$ consists of Noether symmetries up to a total divergence, whereas $X_1$ does not. Third, we place the previously reported finite multiplier currents in their correct equivalence classes and establish alignment and multiplier criteria for current degeneracy. For $X_3$ the transformed current vanishes on invariant graphs: association guarantees invariance, but not nontrivial inheritance. Fourth, we transform the action in canonical coordinates for $X_3$ and $X_4$. The resulting reduced Euler–Lagrange equations and first integrals clarify the variational mechanism; in the $X_3$ reduction, the first integral is generated by the commuting ideal element $Y_t$. The known similarity solutions are not presented as new.

The paper is organised as follows. Section 2 distinguishes classical double reduction from variational reduction. Section 3 records the Lie point symmetries, gives the complete reduction of the first-order multiplier determining system, and presents canonical currents and their association structure. Section 4 develops the first-order Lagrangian formulation, the Noether determining system, and the Noether conservation laws, including the arbitrary-function family. Section 5 develops general degeneracy criteria and applies them to the $X_3$ non-inheritance. Section 6 reduces the Lagrangian for representative variational symmetries. Section 7 discusses the implications and summarises the conclusions.

2. Classical Double Reduction and Variational Reduction

Consider a scalar $(1+1)$-dimensional PDE of order $q\ge 1$,

$$F\left(t,x,u,\dots ,u_{\left(q\right)}\right)=0,$$

(2)

where $u_{\left(q\right)}$ denotes the collection of all partial derivatives of order q of $u(t,x)$. Suppose (2) admits a Lie point symmetry

$$X={\xi }^t(t,x,u){\partial }_t+{\xi }^x(t,x,u){\partial }_x+\eta (t,x,u){\partial }_u.$$

(3)

A conservation law $D_t{T}^t+D_x{T}^x=0$ is associated with X in the sense of Kara and Mahomed [3] if

$$X^{\left(k\right)}\left(\begin{array}{c}{T}^t\\ T^x\end{array}\right)-\left(\begin{array}{cc}{D}_t{\xi }^t& D_x{\xi }^t\\ D_t{\xi }^x& D_x{\xi }^x\end{array}\right)\left(\begin{array}{c}{T}^t\\ T^x\end{array}\right)+\left(D_t{\xi }^t+D_x{\xi }^x\right)\left(\begin{array}{c}{T}^t\\ T^x\end{array}\right)=0.$$

(4)

Here $X^{\left(k\right)}$ denotes the kth prolongation, where k is the highest derivative order occurring in the conserved current. In the classical double reduction algorithm [1,2], both the conservation law and the symmetry are used in the reduction step. The conserved vector is transformed to canonical variables arising from X, and the association condition (4) ensures that the resulting r-component is independent of the symmetry variable, thereby yielding a first integral. As shown in Section 5, this algorithm can fail to reduce the order when that component is constant or zero. Association supplies an invariance statement; it does not supply a nondegeneracy theorem.

For an Euler–Lagrange equation, let L be a Lagrangian density, with action $\mathcal{S}\left[u\right]=\int \int L\mathrm{d}t\mathrm{d}x$, and let X be a variational symmetry, possibly up to a divergence. Choose canonical coordinates satisfying $X\left(r\right)=0$, $X\left(s\right)=1$ and $X\left(w\right)=0$. After transforming the density, a divergence term may be removed so that the reduced density is independent of s. Restricting to invariant functions $w=w\left(r\right)$ gives a one-dimensional Lagrangian $\mathcal{l}(r,w,w_r)$. Its Euler–Lagrange equation is the usual symmetry-reduced equation. A cyclic coordinate or the absence of explicit r-dependence then yields a first integral through the ordinary one-dimensional Noether theorem. This interchange of restriction and variation is the content of the principle of symmetric criticality [25]. In the two reductions below, its validity is also checked directly by verifying that the reduced Euler–Lagrange equation agrees with the invariant restriction of the field equation. This is a two-stage variational reduction, rather than a classical double reduction through a transformed conserved current. We use this distinction throughout the remainder of the paper.

3. Point Symmetries and First-Order Multipliers

Because (1) is an Euler–Lagrange equation, its local conservation-law multipliers are, under the usual regularity assumptions, the characteristics of generalised variational symmetries. This correspondence is broader than the point-symmetry setting: a multiplier may correspond to a generalised variational symmetry without arising from a Noether point symmetry. We use this distinction to organise the analysis. This section first records the Lie point-symmetry algebra and classifies all multipliers with the stated first-order dependence. Section 4 then determines which members of the point algebra preserve the first-order action, possibly up to a total divergence, and derives their Noether currents. The finite symmetry–multiplier–current correspondence is summarised in

Table 1. Symmetry–multiplier–current correspondence for the five canonical currents.

Current	Multiplier	Noether Origin	Kara–Mahomed Associated Symmetry
$T_1$	${\Lambda }_1=\frac{1}{2}{Q}_{X_4}$	$X_4$	$2X_3-X_1$
$T_2$	${\Lambda }_2=-Q_{X_2}$	$X_2$	${\kappa }_1(X_1+2X_3)+{\kappa }_2{X}_2$
$T_3$	${\Lambda }_3=-Q_{X_3}$	$X_3$	$X_3$
$T_4$	${\Lambda }_4=-Q_{Y_1}$	$Y_f{|}_{f\equiv 1}=Y_1={\partial }_x$	${\kappa }_1(X_1+X_3)+{\kappa }_2{X}_2$
$T_5$	${\Lambda }_5=u_x^{-2}$	None	${\kappa }_1(X_1-\frac{1}{2}{X}_3)+{\kappa }_2{X}_2$

.

This section builds on [4] and records corrected canonical current representatives and symmetry–conservation-law associations. The HS equation, Equation (1), admits the finite-dimensional generators:

$$\begin{array}{cccc}\hfill X_1& =x{\displaystyle \frac{\partial }{\partial x}}+u{\displaystyle \frac{\partial }{\partial u}},\hfill & \hfill X_2& ={\displaystyle \frac{\partial }{\partial t}},\hfill \\ \hfill X_3& =t{\displaystyle \frac{\partial }{\partial t}}+x{\displaystyle \frac{\partial }{\partial x}},\hfill & \hfill X_4& =t^2{\displaystyle \frac{\partial }{\partial t}}+2tx{\displaystyle \frac{\partial }{\partial x}}+2x{\displaystyle \frac{\partial }{\partial u}},\hfill \end{array}$$

(5)

together with the infinite-dimensional ideal

$$Y_f=f(t){\displaystyle \frac{\partial }{\partial x}}+f^{\prime }(t){\displaystyle \frac{\partial }{\partial u}},f\in C^{\infty }.$$

(6)

The finite generators (5) were reported in [18]. Retaining the arbitrary-function branch in the point-symmetry determining system yields (6); its evolutionary characteristics form a point-symmetry subideal within Morozov’s wider contact algebra ${\mathfrak{gl}}_2\left(\mathbb{R}\right)⋉{\mathfrak{a}}_{\infty }$ [17]. The multiplier classification below includes both point and non-point variational characteristics; the Noether point-symmetry classification is given separately in Section 4.

Conservation Laws Arising from Multipliers

The multiplier approach [26,27,28] seeks first-order functions $\Lambda =\Lambda (t,x,u,u_t,u_x)$ satisfying the determining equation

$${\displaystyle \frac{\delta }{\delta u}}\left[\Lambda \left({(u_t+u{u}_x)}_x-\frac{1}{2}{u}_x^2\right)\right]=0,$$

(7)

where $\delta /\delta u$ is the Euler operator

$${\displaystyle \frac{\delta }{\delta u}}={\displaystyle \frac{\partial }{\partial u}}-D_t{\displaystyle \frac{\partial }{\partial u_t}}-D_x{\displaystyle \frac{\partial }{\partial u_x}}+D_t^2{\displaystyle \frac{\partial }{\partial u_{tt}}}+D_x^2{\displaystyle \frac{\partial }{\partial u_{xx}}}+D_x{D}_t{\displaystyle \frac{\partial }{\partial u_{tx}}}-\cdots$$

Here $D_t$ and $D_x$ are the total derivative operators in t and x, respectively. Splitting (7) with respect to the independent jet variables gives a complete answer.

Theorem 1.

On a connected open jet-space region, every smooth multiplier of the form $\Lambda (t,x,u,u_t,u_x)$ is

$$\Lambda =A(t)u_t+x\left(A^{\prime }(t)u_x-A^{″}(t)\right)+\mathsf{\Phi }(t,u_x),$$

(8)

where

$$A^{‴}\left(t\right)=0,2{\mathsf{\Phi }}_{t{u}_x}=u_x^2{\mathsf{\Phi }}_{u_x{u}_x}+2u_x{\mathsf{\Phi }}_{u_x}-2\mathsf{\Phi }.$$

(9)

Conversely, every function (8) satisfying conditions (9) is a first-order multiplier. If the region intersects $u_x=0$, smoothness is understood across that hypersurface; singular branches such as $u_x^{-2}$ are then excluded. On a component where $u_x\ne 0$, such singular branches are admissible.

Proof. 

Write $p=u_t$, $q=u_x$ and

$$F={(u_t+u{u}_x)}_x-\frac{1}{2}{u}_x^2=u_{tx}+u{u}_{xx}+\frac{1}{2}{q}^2.$$

The determining equation is $\delta (\Lambda F)/\delta u=0$. Since $\Lambda$ is of the first order and F is of the second order, direct application of the Euler operator gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{\delta (\Lambda F)}{\delta u}}=& \left({\Lambda }_u-D_t{\Lambda }_p-D_x{\Lambda }_q\right)F-{\Lambda }_p{D}_{t}F-{\Lambda }_q{D}_{x}F\hfill \\ \hfill & +\Lambda u_{xx}+q{D}_x\Lambda +u{D}_x^2\Lambda +D_t{D}_x\Lambda .\hfill \end{array}$$

(10)

The third-order variables $u_{ttx}$, $u_{txx}$ and $u_{xxx}$ have coefficients

$$-{\Lambda }_p+{\Lambda }_p,-u{\Lambda }_p-{\Lambda }_q+u{\Lambda }_p+{\Lambda }_q,-u{\Lambda }_q+u{\Lambda }_q,$$

respectively, and therefore cancel identically. Thus expression (10) is a polynomial of degree two in $u_{tt}$, $u_{tx}$ and $u_{xx}$.

The coefficients of $u_{tt}{u}_{tx}$, $u_{tx}{u}_{xx}$ and $u_{xx}^2$ vanish identically. The remaining nonzero quadratic coefficients, those of $u_{tt}{u}_{xx}$ and $u_{tx}^2$, are, respectively, ${\Lambda }_{pq}-u{\Lambda }_{pp}$ and its negative. Hence

$${\Lambda }_{pq}=u{\Lambda }_{pp}.$$

(11)

The coefficient of $u_{tt}$ gives

$$q^2{\Lambda }_{pp}-2q{\Lambda }_{pu}-2{\Lambda }_{px}=0,$$

(12)

whereas the coefficient of $u_{tx}$ is

$$2{\Lambda }_u-q^2{\Lambda }_{pq}+2u{\Lambda }_{px}+2uq{\Lambda }_{pu}=0.$$

(13)

Using (11) and (12) in (13) yields ${\Lambda }_u=0$. Equation (11) then holds identically in u with both derivatives independent of u, and consequently

$${\Lambda }_{pp}={\Lambda }_{pq}=0.$$

Equation (12) now gives ${\Lambda }_{px}=0$. Thus

$$\Lambda =A\left(t\right)p+B(t,x,q).$$

(14)

Substitution of (14) into Equation (10) shows that the coefficient of $u_{xx}$ is

$$u(B_{xq}-A^{\prime })+B+B_{tq}-q{B}_q-\frac{1}{2}{q}^2{B}_{qq}.$$

Splitting with respect to u gives

$$B_{xq}=A^{\prime },2B_{tq}=q^2{B}_{qq}+2q{B}_q-2B.$$

(15)

The terms free of second-order derivatives give, after using $B_{xq}=A^{\prime }$,

$$u{B}_{xx}+B_{tx}+q{B}_x-q^2{A}^{\prime }=0,$$

and hence

$$B_{xx}=0,q^2{A}^{\prime }-q{B}_x-B_{tx}=0.$$

(16)

From $B_{xx}=0$, write $B_x=\beta (t,q)$. The relation $B_{xq}=A^{\prime }$ gives $\beta =A^{\prime }q+\gamma \left(t\right)$, and therefore

$$B=x\left(A^{\prime }q+\gamma \left(t\right)\right)+\mathsf{\Phi }(t,q).$$

The second equation in (16) becomes $-q(\gamma +A^{″})-{\gamma }^{\prime }=0$. Since this holds identically in q, $\gamma =-A^{″}$ and $A^{‴}=0$. Therefore

$$B=x(A^{\prime }q-A^{″})+\mathsf{\Phi }(t,q).$$

Finally, substitution into the second equation in (15) cancels all terms proportional to x and leaves

$$2{\mathsf{\Phi }}_{tq}=q^2{\mathsf{\Phi }}_{qq}+2q{\mathsf{\Phi }}_q-2\mathsf{\Phi },$$

which is the second condition in (9) after $q=u_x$ is restored.

Conversely, if (8) and (9) hold, then Equations (11)–(16) hold by construction. These are all coefficients obtained by splitting (10), so $\delta (\Lambda F)/\delta u=0$. □

Corollary 1

(Local analytic form). Fix $(t_0,q_0)$ with $q_0\ne 0$. For arbitrary analytic functions $g_0\left(t\right)$ and $g_1\left(t\right)$ near $t_0$, there is a unique local analytic solution of (9) satisfying

$$\mathsf{\Phi }(t,q_0)=g_0\left(t\right),{\mathsf{\Phi }}_q(t,q_0)=g_1\left(t\right).$$

Thus, on a component with $u_x\ne 0$, the general local analytic solution of the remaining multiplier equation is parameterised by two arbitrary analytic functions of one variable.

Proof. 

For $q\ne 0$, the second equation in (9) can be solved for its highest q-derivative:

$${\mathsf{\Phi }}_{qq}={\displaystyle \frac{2}{q^2}}{\mathsf{\Phi }}_{tq}-{\displaystyle \frac{2}{q}}{\mathsf{\Phi }}_q+{\displaystyle \frac{2}{q^2}}\mathsf{\Phi }.$$

The Cauchy–Kowalevski theorem, applied with q as the transverse variable, gives local existence and uniqueness for the stated analytic Cauchy data. □

Remark 1.

Let $\vartheta =u_x{\partial }_{u_x}$. Since $u_x^2{\partial }_{u_x}^2={\vartheta }^2-\vartheta$, the equation for Φ in (9) has the factored form

$$2{\partial }_t{\partial }_{u_x}\mathsf{\Phi }=(\vartheta +2)(\vartheta -1)\mathsf{\Phi }.$$

(17)

The t-independent functions $u_x$ and $u_x^{-2}$ are immediate kernel elements of the operator on the right-hand side.

The arbitrary-function point characteristics and the non-point multiplier used below occur as immediate examples in this theorem:

$$\mathsf{\Phi }=f^{\prime }\left(t\right)-f\left(t\right)u_x,\mathsf{\Phi }=u_x^{-2}$$

(18)

which both solve (9). The first choice is the characteristic of $Y_f$, while the second is the characteristic of a generalised non-point variational symmetry. Expressions involving $u_x^{-1}$ or $u_x^{-2}$ are understood locally on a jet-space domain where $u_x\ne 0$.

No division by $q=u_x$ is used in deriving Theorem 1; hence its determining equations also include multipliers that are smooth across $u_x=0$. The restriction $u_x\ne 0$ applies only to singular examples such as $u_x^{-2}$ and to the analytic Cauchy parameterisation in Corollary 1. Claims of completeness are therefore relative to the stated first-order dependence and the chosen smooth or analytic jet-space region.

Our earlier calculation [4] reported the following five-parameter subfamily of (8):

$$\Lambda =u_t\left({\delta }_2+{\delta }_{3}t-\frac{{\delta }_1{t}^2}{2}\right)+x{u}_x({\delta }_3-{\delta }_{1}t)+{\delta }_{1}x+{\delta }_4{u}_x+{\displaystyle \frac{{\delta }_5}{u_x^2}},$$

(19)

with ${\delta }_i$, $i=1,\dots ,5$, arbitrary constants. It corresponds to a quadratic A and $\mathsf{\Phi }={\delta }_4{u}_x+{\delta }_5{u}_x^{-2}$. Setting one parameter at a time to one and the others to zero produces the five conservation laws $D_t{T}_i^t+D_x{T}_i^x=0$ on solutions of (1).

The conserved vectors corresponding to (19) may include additive trivial currents involving arbitrary functions ${\phi }_1\left(u\right)$, ${\phi }_2\left(x\right)$ and ${\phi }_3\left(t\right)$, as well as constants. These trivial pieces contribute zero divergence identically and play no role in the symmetry–conservation-law classification. For clarity of exposition, we record canonical representatives of the corresponding equivalence classes, obtained by setting all trivial contributions to zero:

$$\begin{array}{cc}\hfill T_1^t& =u_x^2\left(\frac{t^{2}u}{4}-\frac{tx}{2}\right)+u_x(x-tu),\hfill \\ \hfill T_1^x& =u_x\left(xu-\frac{1}{2}{t}^{2}u{u}_t\right)-\frac{t^2{u}_t^2}{4}-\frac{1}{2}xtu{u}_x^2+tu{u}_t,\hfill \end{array}$$

(20a)

$$\begin{array}{cc}\hfill T_2^t& =-\frac{u{u}_x^2}{2},\hfill \\ \hfill T_2^x& =u{u}_x{u}_t+\frac{u_t^2}{2},\hfill \end{array}$$

(20b)

$$\begin{array}{cc}\hfill T_3^t& =u_x^2\left(\frac{x}{2}-\frac{tu}{2}\right),\hfill \\ \hfill T_3^x& =tu{u}_x{u}_t+\frac{t{u}_t^2}{2}+\frac{1}{2}xu{u}_x^2,\hfill \end{array}$$

(20c)

$$\begin{array}{cc}\hfill T_4^t& =\frac{u_x^2}{2},\hfill \\ \hfill T_4^x& =\frac{u{u}_x^2}{2},\hfill \end{array}$$

(20d)

$$\begin{array}{cc}\hfill T_5^t& =-\frac{1}{u_x},\hfill \\ \hfill T_5^x& =-\frac{u}{u_x}+\frac{3x}{2}.\hfill \end{array}$$

(20e)

The final column of Table 1 restates the Kara–Mahomed association classification obtained in our earlier paper [4] for the five canonical currents in (20). The present classification includes an amendment to the $T_1$ entry: although the earlier paper reported no associated Lie point symmetry for this current, re-evaluation of condition (4) yields $2X_3-X_1\to T_1$, as recorded in the table. The multiplier and Noether-origin columns place this classification within the broader symmetry–multiplier–conservation-law correspondence. They record information distinct from the Kara–Mahomed association; in particular, a Noether generator need not coincide with an associated symmetry. In the $T_3$ row, the occurrence of $X_3$ in both columns records two distinct facts: $T_3$ arises from $X_3$ through Noether’s theorem, and the pair $(T_3,X_3)$ is also associated in the Kara–Mahomed sense. Here ${\kappa }_1$ and ${\kappa }_2$ are arbitrary constants, not both zero. The current $T_5$ has no Noether point-symmetry origin because its multiplier is a generalised non-point variational characteristic, although it admits the associated point symmetries shown in the final column.

Table 1 concerns only the five canonical currents in (20) and is not intended as an exhaustive classification of the conservation laws of (1).

The reductions associated with the three parameterised symmetry families paired with $T_2$, $T_4$ and $T_5$ in Table 1 were carried out in detail in [4] and are not repeated here. The present analysis instead concentrates on $X_3$, for which current inheritance fails; $X_4$, for which reduction of the action is especially transparent; and the ideal $Y_f$. This focus avoids duplicating the earlier reduction catalogue and highlights the additional variational structure.

The multiplier method and Noether symmetry analysis have different scopes. A multiplier is the characteristic of a generalised variational symmetry, so the classification above includes the non-point characteristic $u_x^{-2}$. The next section addresses the narrower question of which members of the Lie point algebra preserve the chosen first-order action up to a total divergence. The correspondence above links these two levels of the analysis.

4. Variational Structure, Noether Symmetries and Conservation Laws

Equation (1) is the Euler–Lagrange equation associated with the first-order Lagrangian density

$$L(t,x,u,u_t,u_x)=-{\displaystyle \frac{1}{2}}{u}_x{u}_t-{\displaystyle \frac{1}{2}}u{u}_x^2,$$

(21)

since

$${\displaystyle \frac{\delta L}{\delta u}}=u_{tx}+u{u}_{xx}+{\displaystyle \frac{1}{2}}{u}_x^2.$$

(22)

Both the variational character and this first-order Lagrangian are known [6,8], and conservation laws obtained through Boyer’s extension of Noether’s theorem were studied in [18]. The purpose here is to determine all Noether symmetries arising from point transformations for this particular density and place them inside the multiplier classification of Theorem 1.

A Noether point symmetry (up to a total divergence) of the Lagrangian (21) is a generator $X=\tau {\partial }_t+\xi {\partial }_x+\eta {\partial }_u$ for which there exist gauge functions $B^t(t,x,u)$ and $B^x(t,x,u)$ satisfying the Noether invariance condition [26,29,30]:

$$X^{\left(1\right)}\left(L\right)+L(D_t\tau +D_x\xi )=D_t{B}^t+D_x{B}^x.$$

(23)

Substituting the general infinitesimals of the full Lie point-symmetry algebra,

$$\tau =k_2+k_{3}t+k_4{t}^2,\xi =(k_1+k_3)x+2k_{4}tx+f\left(t\right),\eta =k_{1}u+2k_{4}x+f^{\prime }\left(t\right),$$

(24)

into (23) and solving the resulting determining system gives

$$\tau =k_2+k_{3}t+k_4{t}^2,\xi =k_{3}x+2k_{4}tx+f\left(t\right),\eta =2k_{4}x+f^{\prime }\left(t\right),$$

(25)

together with the gauge functions

$$B^t=-k_{4}u+H^t(t,x),B^x=-k_4{u}^2-{\displaystyle \frac{1}{2}}{f}^{″}\left(t\right)u+H^x(t,x),$$

(26)

where $D_t{H}^t+D_x{H}^x=0$; we use the canonical choice $H^t=H^x=0$. The coefficient $k_1$ is forced to vanish, whereas the function $f\left(t\right)$ remains arbitrary. We have therefore proved the following classification.

Theorem 2.

The Lagrangian (21) admits the following algebra of Noether point symmetries:

$$\mathrm{span}\{X_2,X_3,X_4\}⋉\{Y_f:f\in C^{\infty }\}.$$

The generator $X_1$ is a Lie point symmetry of the field equation but is not a Noether symmetry, even up to a total divergence, of this Lagrangian.

For a first-order Lagrangian, the conserved current associated with a Noether symmetry $(\tau ,\xi ,\eta ;B^t,B^x)$ is

$$\begin{array}{cc}\hfill T^t& =\tau L+Q{\displaystyle \frac{\partial L}{\partial u_t}}-B^t,\hfill \\ \hfill T^x& =\xi L+Q{\displaystyle \frac{\partial L}{\partial u_x}}-B^x,\hfill \end{array}$$

(27)

where $Q=\eta -\tau u_t-\xi u_x$. On solutions of (1), the resulting current satisfies $D_t{T}^t+D_x{T}^x=0$. From (21),

$${\displaystyle \frac{\partial L}{\partial u_t}}=-{\displaystyle \frac{1}{2}}{u}_x,{\displaystyle \frac{\partial L}{\partial u_x}}=-{\displaystyle \frac{1}{2}}{u}_t-u{u}_x.$$

(28)

The three finite-dimensional generators give the following currents, followed by the arbitrary-function family associated with $Y_f$.

(i)

${\mathit{X}}_{\mathbf{2}}={\partial }_{\mathit{t}}$.

Taking $\tau =1$, $\xi =0$, $\eta =0$, $B^t=B^x=0$ yields

$$T_{N_2}^t=-{\displaystyle \frac{1}{2}}u{u}_x^2,T_{N_2}^x={\displaystyle \frac{1}{2}}{u}_t^2+u{u}_t{u}_x.$$

(29)

(ii)

${\mathit{X}}_{\mathbf{3}}=\mathit{t}{\partial }_{\mathit{t}}+\mathit{x}{\partial }_{\mathit{x}}$.

Taking $\tau =t$, $\xi =x$, $\eta =0$, $B^t=B^x=0$ yields

$$T_{N_3}^t={\displaystyle \frac{1}{2}}{u}_x^2(x-tu),T_{N_3}^x={\displaystyle \frac{1}{2}}t{u}_t^2+tu{u}_t{u}_x+{\displaystyle \frac{1}{2}}xu{u}_x^2.$$

(30)

(iii)

${\mathit{X}}_{\mathbf{4}}={\mathit{t}}^{\mathbf{2}}{\partial }_{\mathit{t}}+\mathbf{2}\mathit{tx}{\partial }_{\mathit{x}}+\mathbf{2}\mathit{x}{\partial }_{\mathit{u}}$.

Taking $\tau =t^2$, $\xi =2tx$, $\eta =2x$ and

$$B^t=-u,B^x=-u^2$$

yields

$$\begin{array}{cc}\hfill T_{N_4}^t& =u-x{u}_x+u_x^2\left(tx-\frac{1}{2}{t}^{2}u\right),\hfill \\ \hfill T_{N_4}^x& =u^2-x{u}_t-2xu{u}_x+\frac{1}{2}{t}^2{u}_t^2+t^{2}u{u}_t{u}_x+txu{u}_x^2.\hfill \end{array}$$

(31)

(iv)

The arbitrary-function ideal ${\mathit{Y}}_{\mathit{f}}$.

For $\tau =0$, $\xi =f\left(t\right)$, $\eta =f^{\prime }\left(t\right)$, $B^t=0$ and $B^x=-\frac{1}{2}{f}^{″}\left(t\right)u$, Noether’s formula gives

$$\begin{array}{cc}\hfill T_f^t=& {\displaystyle \frac{1}{2}}f{u}_x^2-{\displaystyle \frac{1}{2}}{f}^{\prime }{u}_x,\hfill \\ \hfill T_f^x=& {\displaystyle \frac{1}{2}}fu{u}_x^2-{\displaystyle \frac{1}{2}}{f}^{\prime }{u}_t-f^{\prime }u{u}_x+{\displaystyle \frac{1}{2}}{f}^{″}u.\hfill \end{array}$$

(32)

A direct differentiation gives the identity

$$D_t{T}_f^t+D_x{T}_f^x=-\left(f^{\prime }-f{u}_x\right)\left(u_{tx}+u{u}_{xx}+{\displaystyle \frac{1}{2}}{u}_x^2\right).$$

(33)

Consequently $Q_f=f^{\prime }-f{u}_x$ is an arbitrary-function family of first-order multipliers. Since the determining equation is linear, both signs are multipliers; identity (33) fixes the sign appearing in this particular divergence identity. This family is the point subfamily $\mathsf{\Phi }=f^{\prime }-f{u}_x$ of Theorem 1 and is not contained in the finite subfamily (19).

The finite currents (29) and (30) coincide with $T_2$ in (20b) and $T_3$ in (20c), respectively. The $X_4$ current is equivalent to $-2T_1$. Indeed, if $\mathsf{\Theta }=xu-t{u}^2$, then

$$(T_{N_4}^t,T_{N_4}^x)=-2(T_1^t,T_1^x)+(D_x\mathsf{\Theta },-D_t\mathsf{\Theta }).$$

(34)

The last pair is a trivial current. Thus the three finite Noether currents correspond, respectively, to $T_2$, $T_3$ and $T_1$. The remaining two currents in (20) are also accounted for. The current $T_4$ in (20d) is exactly (32) with $f\equiv 1$, whereas $T_5$ has multiplier $u_x^{-2}$ and therefore comes from a generalised non-point variational symmetry.

The arbitrary-function densities constructed by Tian and Liu arise from a reciprocal transformation and are expressed in terms of $u_x$ and its higher x-derivatives, without explicit t-dependence [16]. At first differential order, the common member visible in both constructions is the constant-f current $T_4$ (up to current equivalence); nonconstant f in (32) instead gives explicitly time-dependent first-order currents. Thus the two constructions are complementary rather than identical. This comparison is restricted to first differential order and is understood up to current equivalence; no complete classification of their overlap is claimed here.

5. Degeneracy of Associated Currents Under Double Reduction

Classical double reduction requires more than the Kara–Mahomed association. Association guarantees that the transformed conserved vector has no explicit dependence on the symmetry variable; it does not guarantee that its invariant component carries differential information. We now isolate the additional nondegeneracy requirement, give geometric and multiplier criteria for its failure, and apply them to $(T_3,X_3)$.

Let

$$X=\tau {\partial }_t+\xi {\partial }_x+\eta {\partial }_u,Q_X=\eta -\tau u_t-\xi u_x,$$

and choose canonical coordinates $(r,s,w)$ such that $X\left(r\right)=0$, $X\left(s\right)=1$, $X\left(w\right)=0$ and hence $X={\partial }_s$. We assume $(\tau ,\xi )\ne (0,0)$, so that X has a nontrivial horizontal part. An X-invariant graph satisfies the invariant-surface condition $Q_X=0$, or equivalently $w=w\left(r\right)$. This horizontal assumption restricts the criteria in this section to symmetries with a nontrivial action on the independent variables; invariant graphs for the ideal $Y_f$ are treated separately in Section 6.

A conserved vector transforms as follows [1]:

$$T^r={\displaystyle \frac{T^t{D}_{t}r+T^x{D}_{x}r}{\Delta }},T^s={\displaystyle \frac{T^t{D}_{t}s+T^x{D}_{x}s}{\Delta }},\Delta =\left(D_{t}r\right)\left(D_{x}s\right)-\left(D_{x}r\right)\left(D_{t}s\right).$$

(35)

The transformation carries the divergence according to

$$D_r{T}^r+D_s{T}^s={\Delta }^{-1}\left(D_t{T}^t+D_x{T}^x\right).$$

(36)

For a current strictly associated with X, the transformed components have no explicit s-dependence. On invariant graphs, $D_s{T}^s=0$, and the transformed conservation law becomes $D_r{T}^r=0$. It reduces the order only when $T^r$ is a nonconstant differential function of $r,w,w_r,\dots$.

Definition 1

(Current degeneracy). An associated pair $(T,X)$ is degenerate if $T^r$, restricted to X-invariant graphs, is constant with respect to $(r,w,w_r,\dots )$ on the reduced jet space, rather than merely constant along a particular reduced solution. It is strongly degenerate if $T^r\equiv 0$. In either case $D_r{T}^r=0$ is an identity and supplies neither a reduced equation nor a first integral.

5.1. An Alignment Criterion

Theorem 3

(Alignment criterion). Assume $(D_{t}r,D_{x}r)\ne (0,0)$ and $\Delta \ne 0$ on the local canonical coordinate domain. Then the transformed component $T^r$ vanishes on X-invariant graphs if and only if the conserved vector is pointwise parallel to the horizontal part of the generator there, equivalently

$$\tau T^x-\xi T^t=0.$$

(37)

Proof. 

Expanding the total derivatives of $r=r(t,x,u)$ gives

$$\tau D_{t}r+\xi D_{x}r=X\left(r\right)-Q_X{r}_u.$$

On an invariant graph, $X\left(r\right)=Q_X=0$, so $(D_{t}r,D_{x}r)$ is orthogonal to $(\tau ,\xi )$. Since both vectors are nonzero, there is a nonzero function $\lambda$ such that $(D_{t}r,D_{x}r)=\lambda (-\xi ,\tau )$. The numerator of $T^r$ is therefore $\lambda (\tau T^x-\xi T^t)$, which proves the result because $\Delta \ne 0$. □

The theorem has a simple geometric interpretation: a current tangent to the group orbits has no component transverse to them and therefore transports nothing in the reduced variable. The tangency condition (37) is coordinate-free; consequently strong degeneracy is independent of the choice of canonical coordinates.

5.2. A Multiplier Criterion and Current Equivalence

The multiplier is an invariant of a conservation-law equivalence class: equivalent currents have the same multiplier, whereas a trivial current has a zero multiplier. This yields a class-level degeneracy test.

Theorem 4

(Multiplier criterion). Suppose T is a smooth finite-order local current on a connected open region of jet space and is associated with X. Assume that the canonical transformation has $\Delta \ne 0$ and that the off-shell identity

$$D_t{T}^t+D_x{T}^x=\Lambda \left(u_{tx}+u{u}_{xx}+\frac{1}{2}{u}_x^2\right)$$

(38)

holds. Assume also that restriction to the invariant jet submanifold is regular and that the restricted component $T^r$ remains smooth and of finite differential order. If Λ vanishes on the invariant jet submanifold determined by $Q_X=0$, then $(T,X)$ is degenerate. Moreover, every representative of the same conservation-law class that is associated with X is also degenerate.

Proof. 

Restrict (38) to invariant graphs and use the transformed identity (36). Association and $w=w\left(r\right)$ give $D_s{T}^s=0$, whereas the hypothesis gives ${\Lambda |}_{\mathrm{inv}}=0$. Consequently $D_r{T}^r\equiv 0$ as an identity on the reduced jet space. If $T^r$ has differential order n, the coefficient of $w_{(n+1)}$ in $D_r{T}^r$ is $\partial T^r/\partial w_{\left(n\right)}$; descending by differential order shows that all relevant partial derivatives of $T^r$ vanish. Hence $T^r$ is constant.

Any equivalent current differs by a trivial current and therefore has the same multiplier. If that representative is also associated with X, the preceding argument applies unchanged. □

The HS equation is normal on the local jet-space region considered here because it is solved for the leading derivative $u_{tx}$. In this setting, the multiplier is an invariant of a local conservation-law class: a trivial current has a zero multiplier and equivalent currents have the same multiplier. This normality assumption is the reason the final class-level statement of the theorem is valid.

Remark 2

(Scope of the equivalence-class statement). The final assertion of Theorem 4 is conditional on representative-level association. Adding a trivial current preserves the multiplier but need not preserve the strict Kara–Mahomed association with a given symmetry. The theorem therefore states that degeneracy is shared by those representatives of a fixed conservation-law class that are each associated with the same symmetry; it does not assert that every representative of the class is automatically associated or degenerate.

Whether the condition ${\Lambda |}_{\mathrm{inv}}=0$ is also necessary under the regularity assumptions of this section, and whether weak degeneracy ($T^r$ constant but nonzero) occurs in this family, remain natural questions. The analysis of $T_1$ below is consistent with an inconclusive multiplier criterion and a successful reduction.

Corollary 2

(Generating-symmetry obstruction). Let X be a variational symmetry, and suppose that a representative of its Noether current class is associated with X. Then every representative in that class that is associated with X is degenerate under reduction by X.

Proof. 

The multiplier of the Noether current is $Q_X$, up to sign. It vanishes on the invariant surface $Q_X=0$, so Theorem 4 applies. □

This corollary does not contradict the usual successful applications of double reduction. A conservation law may be associated with several symmetries; the criterion excludes reduction by its generating variational symmetry. It does not exclude other associated symmetries for which the multiplier remains nonzero on the corresponding invariant surface.

Generalisations and multi-reduction extensions of double reduction provide a broader context for such trivial reduced identities [31,32]. The present criteria are complementary: the alignment theorem identifies the geometric mechanism, the multiplier theorem formulates an equivalence-class test, and $(T_3,X_3)$ below gives an explicit strong-degeneracy example whose dynamics are recovered variationally.

For example, $T_1$ has multiplier ${\Lambda }_1=-\frac{1}{2}{t}^2{u}_t-tx{u}_x+x$ and is associated with $2X_3-X_1=2t{\partial }_t+x{\partial }_x-u{\partial }_u$, whose characteristic is $Q=-u-2t{u}_t-x{u}_x$. On the invariant surface $Q=0$,

$${\Lambda }_1=x+{\displaystyle \frac{1}{4}}tu-{\displaystyle \frac{3}{4}}tx{u}_x,$$

which is not identically zero. Theorem 4 is a sufficient degeneracy test and is therefore inconclusive in this case; the successful reduction reported in [4] is consistent with the absence of the multiplier obstruction.

Equation (34) also shows why Noether generation should not be conflated with strict association of a chosen representative. The current $T_{N_4}$ is generated by $X_4$, while the equivalent canonical current $T_1$ satisfies the strict criterion for $2X_3-X_1$. Adding a trivial current can change a representative-level association statement, but it cannot change the multiplier test in Theorem 4.

5.3. The Pair $(T_3,X_3)$

The symmetry $X_3=t{\partial }_t+x{\partial }_x$ has characteristic $Q_3=-t{u}_t-x{u}_x$ and canonical variables

$$r={\displaystyle \frac{x}{t}},s=\ln x,w=u,$$

(39)

on a local domain where $t\ne 0$ and $x>0$. Here $X_3={\partial }_s$, and

$$t={\displaystyle \frac{e^s}{r}},x=e^s,u=w.$$

On invariant graphs $w=w\left(r\right)$,

$$u_t=-r^2{e}^{-s}{w}_r,u_x=r{e}^{-s}{w}_r,$$

(40)

so $Q_3{|}_{\mathrm{inv}}=0$. Since $T_3$ is the Noether current of $X_3$ and is associated with it, its multiplier is $-Q_3$ and Theorem 4 predicts degeneracy for every associated representative of its equivalence class.

The explicit calculation gives the stronger conclusion $T_3^r=0$. From (39),

$$D_{t}r=-r^2{e}^{-s},D_{x}r=r{e}^{-s},D_{t}s=0,D_{x}s=e^{-s},\Delta =-r^2{e}^{-2s}.$$

For the canonical representative in (20c), substitution of (39)–(40) yields

$$T_3^t={\displaystyle \frac{1}{2}}r(r-w)e^{-s}{w}_r^2,T_3^x={\displaystyle \frac{1}{2}}{r}^2(r-w)e^{-s}{w}_r^2.$$

(41)

Thus $T_3^x=r{T}_3^t$, and hence

$$(T_3^t,T_3^x)={\displaystyle \frac{T_3^t}{t}}(t,x).$$

The current is pointwise parallel to the horizontal part of $X_3$, as predicted by Theorem 3. Directly,

$$T_3^r={\displaystyle \frac{T_3^t(-r^2{e}^{-s})+r{T}_3^t\left(r{e}^{-s}\right)}{\Delta }}=0.$$

(42)

Therefore $D_r{T}_3^r\equiv 0$ is an identity: this current produces neither an ODE for $w\left(r\right)$ nor a first integral through the classical formula.

Within the association classification summarised in Table 1, $T_3$ is paired only with the span of $X_3$, which is precisely its generating symmetry. The obstruction is consequently structural rather than an artefact of the canonical representative. It is not a defect of the Kara–Mahomed association: association ensures independence from s, but it does not ensure nontrivial differential content. Section 6 shows that the dynamics are recovered by reducing the action rather than this current.

6. Variational Reduction in Canonical Coordinates

For a variational symmetry, the natural object to transform is the Lagrangian density. We carry out this transformation for $X_3$ and $X_4$. The resulting similarity equations and solutions are known; their role here is to show how the first integrals arise directly from the reduced variational structure. For the checks below, write

$$\mathcal{F}\left[u\right]=u_{tx}+u{u}_{xx}+\frac{1}{2}{u}_x^2,$$

so that the HS equation is $\mathcal{F}\left[u\right]=0$.

6.1. The Scaling Symmetry $X_3$

Use the canonical variables (39). On invariant graphs $w=w\left(r\right)$, the oriented Jacobian and transformed densities are

$${\displaystyle \frac{\partial (t,x)}{\partial (r,s)}}=-{\displaystyle \frac{e^{2s}}{r^2}},L{\displaystyle \frac{\partial (t,x)}{\partial (r,s)}}={\displaystyle \frac{1}{2}}(w-r)w_r^2.$$

Hence the reduced Lagrangian is

$${\mathcal{l}}_3(r,w,w_r)={\displaystyle \frac{1}{2}}(w-r)w_r^2.$$

(43)

Its Euler–Lagrange equation is

$$2(w-r)w_{rr}+w_r^2-2w_r=0,$$

(44)

which is the $b=-1$ member of the similarity family obtained by Baxter, Van Gorder and Vajravelu [8]. Direct restriction of the field equation gives

$$\mathcal{F}{|}_{\mathrm{inv}}={\displaystyle \frac{1}{2}}{r}^2{e}^{-2s}\left[2(w-r)w_{rr}+w_r^2-2w_r\right],$$

(45)

which confirms directly that varying the reduced Lagrangian and restricting the field equation give the same reduced equation.

Set $z=w-r$. Then ${\mathcal{l}}_3=\frac{1}{2}z{(z_r+1)}^2$ has no explicit r-dependence. Its energy integral is

$$z_r{\displaystyle \frac{\partial {\mathcal{l}}_3}{\partial z_r}}-{\mathcal{l}}_3={\displaystyle \frac{1}{2}}z(z_r^2-1)=C,$$

or, equivalently,

$$(w-r)(w_r^2-2w_r)=C_1.$$

(46)

Translation in r in the $(r,z)$ coordinates is induced by the ideal element $Y_t=t{\partial }_x+{\partial }_u$. Indeed,

$$[X_3,Y_t]=0,Y_t\left(r\right)=1,Y_t\left(w\right)=1,$$

(47)

so $Y_t$ descends to ${\partial }_r+{\partial }_w$ on the reduced space and leaves $z=w-r$ invariant. In the coordinates $(r,z)$ it becomes translation in r; the energy integral (46) is therefore precisely the reduced Noether integral generated by a member of the infinite-dimensional ideal. Thus the nontrivial reduced dynamics are recovered from the action even though the particular transformed current in (42) is identically zero.

6.2. The Projective Symmetry $X_4$

For $t\ne 0$, set

$$r={\displaystyle \frac{x}{t^2}},s=-{\displaystyle \frac{1}{t}},w=u-{\displaystyle \frac{2x}{t}}.$$

Then $X_4={\partial }_s$, and the oriented Jacobian is $\partial (t,x)/\partial (r,s)=-s^{-4}$. On invariant graphs $w=w\left(r\right)$, direct simplification gives

$$\begin{array}{cc}\hfill L{\displaystyle \frac{\partial (t,x)}{\partial (r,s)}}=& {\displaystyle \frac{1}{2}}w{w}_r^2+D_r\left({\displaystyle \frac{rw}{s^2}}-{\displaystyle \frac{w^2}{s}}\right)+D_s\left(-{\displaystyle \frac{w}{s}}+{\displaystyle \frac{r}{s^2}}\right).\hfill \end{array}$$

(48)

This calculation is local on a domain with $t\ne 0$ (equivalently $s\ne 0$), where the coordinate map has a nonzero Jacobian. The displayed $D_r$- and $D_s$-terms contribute only boundary terms to the transformed action. They may therefore be removed under compactly supported variations, or under boundary conditions for which the corresponding variations vanish, before restricting to invariant graphs. After removal of the displayed divergence, the reduced Lagrangian is

$${\mathcal{l}}_4(w,w_r)={\displaystyle \frac{1}{2}}w{w}_r^2.$$

(49)

The Euler–Lagrange equation and energy integral are therefore

$$2w{w}_{rr}+w_r^2=0,{\displaystyle \frac{1}{2}}w{w}_r^2=C.$$

(50)

Here the direct restriction of the field equation is

$$\mathcal{F}{|}_{\mathrm{inv}}={\displaystyle \frac{1}{2}}{s}^4\left(2w{w}_{rr}+w_r^2\right),$$

(51)

which provides the corresponding symmetric-criticality check. On an interval where $w>0$, the real energy relation requires $C\ge 0$. The case $C=0$ gives the constant branch, while for $C>0$ either sign of the square root gives a nonconstant local branch. Writing $C_1=\pm \sqrt{2C}$, the energy integral becomes $\sqrt{w}{w}_r=C_1$, whence

$$u(t,x)={\displaystyle \frac{2x}{t}}+{\left({\displaystyle \frac{3C_{1}x}{2t^2}}+C_2\right)}^{2/3}.$$

(52)

Equation (52) is included as a check on the reduction and is not claimed as a new Hunter–Saxton solution.

6.3. Invariant Graphs for the Ideal $Y_f$

Let f be smooth and nonvanishing on the interval under consideration. Canonical variables for (6) are

$$r=t,s={\displaystyle \frac{x}{f\left(t\right)}},w=u-{\displaystyle \frac{f^{\prime }\left(t\right)}{f\left(t\right)}}x.$$

(53)

An invariant graph consequently has the form

$$u(t,x)={\displaystyle \frac{f^{\prime }\left(t\right)}{f\left(t\right)}}x+w\left(t\right).$$

(54)

Substitution into (1) yields the compatibility condition

$$2f{f}^{″}-{\left(f^{\prime }\right)}^2=0.$$

(55)

Thus the arbitrary-function Noether symmetry exists for every smooth f, but an invariant solution of (1) of the form (54) exists only when (55) holds. Locally, on an interval where $f\ne 0$, the nonzero solutions of (55) are $f\left(t\right)=c{(at+b)}^2$, where $c\ne 0$ and $a,b$ are not both zero. The corresponding invariant solutions are

$$u(t,x)={\displaystyle \frac{2a}{at+b}}x+w\left(t\right),$$

(56)

where $w\left(t\right)$ is arbitrary. This compatibility distinction is easy to miss if the infinite-dimensional ideal is omitted from the point-symmetry algebra. The case $a=0$ recovers spatial translations. For $f\left(t\right)=t^2$, the ansatz contains the same affine term $2x/t$ as the $X_4$ reduction, although the invariant variable is different.

7. Discussion and Conclusions

Because (1) is an Euler–Lagrange equation, the multiplier and Noether approaches are linked but do not have identical scope. Multipliers correspond to characteristics of generalised variational symmetries, whereas the Noether point-symmetry classification identifies the point-transformation subclass that preserves the chosen first-order action. With this distinction in view, four conclusions are important. First, the infinite-dimensional point ideal $Y_f$ is known within the symmetry algebra, and Morozov’s contact classification places its characteristics inside an even larger abelian ideal. The new issue settled by Theorem 2 is variational: the whole point ideal consists of Noether symmetries of the classical first-order action, allowing a total-divergence boundary term, whereas the finite generator $X_1$ is excluded.

Second, Theorem 1 completes the multiplier calculation rather than merely enlarging the earlier finite ansatz. The solution space for $\mathsf{\Phi }(t,u_x)$ contains the point characteristics $f^{\prime }-f{u}_x$, the generalised characteristic $u_x^{-2}$, and further solutions of the linear determining Equation (9). The five currents in (20) therefore form a useful finite subfamily, not the general solution.

More precisely, the theorem is complete within the stated first-order jet dependence. Smooth multipliers crossing $u_x=0$ remain included in the determining equation, while singular multipliers and the two-function analytic parameterisation are understood locally on components where $u_x\ne 0$. This separates the functional classification from the five explicit representatives used in the reduction analysis.

Infinite hierarchies of higher symmetries and conserved densities, including arbitrary-function constructions obtained through reciprocal transformations, are already known for the Hunter–Saxton equation [15,16] and its generalisations [17]. The family (32) is different in scope: it is an explicitly time-dependent first-order local Noether family generated directly by the point-symmetry ideal $Y_f$. At first differential order, the common member visible in comparison with the reciprocal families of Tian and Liu is the constant-f current $T_4$. Thus the contribution here concerns low-order multiplier and Noether point-symmetry structure, not the existence of infinite conservation hierarchies in general.

Third, association and inheritance are different statements. The alignment criterion, Theorem 3, identifies the geometric mechanism: inheritance fails when the current is tangent to the symmetry orbits. The multiplier criterion, Theorem 4, makes the constant-current obstruction a statement about all associated representatives of the same conservation-law class. For the pair $(T_3,X_3)$, the Kara–Mahomed condition is valid, but the transformed component is $T_3^r=0$. The reduced Lagrangian (43), by contrast, retains the dynamics and gives both the similarity Equation (44) and its first integral (46). This comparison shows why variational symmetries are naturally handled at the level of the action when a preserved variational structure is available.

Fourth, the classification and reduction are linked by (47): $Y_t$ commutes with $X_3$, descends to the reduced variational problem and generates its first integral. The infinite ideal therefore has an operational role in reduction, not only a classificatory one.

The reductions in Section 6 serve as methodological checks rather than claims of new similarity solutions. The structural content lies in the complete reduction of the first-order multiplier determining system, the classification of Noether point symmetries, the placement of all five finite currents, and the general degeneracy criterion.

Taken together, these results show that the multiplier, Noether and reduction viewpoints are complementary. The multiplier calculation captures both point and generalised variational characteristics; the Noether classification isolates the variational part of the point algebra; and the degeneracy criteria explain when an associated current fails to survive reduction. A natural next step is to solve the factored linear Equation (17) for $\mathsf{\Phi }(t,u_x)$ in useful functional classes and to study the action of the finite algebra on the resulting currents. The analytic parameterisation in Corollary 1 resolves the local analytic problem; this future direction concerns broader smooth or otherwise structured function classes.