Integrability structures of the generalized Hunter--Saxton equation

We consider integrability structures of the generalized Hunter--Saxton equation. In particular, we obtain the Lax representation with nonremovable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate an infinite-dimensional Lie algebra of higher symmetries, and prove existence of infinite number of cosymmetries of higher order. Further, we give an example of employing the higher order symmetry to constructing exact globally defined solutions for the generalized Hunter--Saxton equation.


Introduction
The Hunter-Saxton equation was introduced in [12] to describe the nonlinear instability of the director field in the nematic liquid crystal and then has been a subject of thorough investigation. As it was shown in [12], equation (1) admits a Lagrangian formulation with Lagrangian L = (u t − u u x ) u x . In [13] a bi-Hamiltonian structure, a Lax representation, a nonlocal recursion operator, and a series of conservation laws have been found. A tri-Hamiltonian formulation for (1) was proposed in [27]. Inverse scattering solutions for (1) were constructed in [2]. In [15] it has been proven that equation (1) can be understood as a geodesic equation associated to a rightinvariant metric on an appropriate homogeneous space related to the Virasoro group. The pseudo-spherical formulation for equation (1) and quadratic pseudopotentials were proposed and used to find nonlocal symmetries and conservation laws in [31]. In [11], the nonlocal symmetries were used to construct exact solutions and a nonlocal recursion operator for  (1). Nonlocal recursion operators, a fourth order local recursion operator, series of higher symmetries and conservation laws for equation (1) have been constructed in [36], see also [33]. The further discussion of the physical interpretation of equation (1) can be found in [4].
In this paper we consider the generalization of the Hunter-Saxton equation (1). This equation with β = 1 2 has applications in geometry of Einstein-Weil structures [34,7], and in hydrodynamics [10]. In [5,29] a nonlocal transformation was used to construct a general solution for (2). In [25] we have shown that equation (2) is linearizable via the contact transformation (t, x, u, u t , u x ) → (t,x,ũ,ũt,ũx) given by the formulae (3) This transformation maps (2) to the Euler-Poisson equatioñ In its turn equation (4) in integrable by quadratures via Laplace's method, [28, § 9.3]. The general solution to (4) combined with the inverse transformation to (3) provides the parametric formula for the general solution to equation (2), see details in [25]. This formula is locally defined and does not give global solutions to (2).
In the present paper we study integrability properties of equation (2). In Section 3 we find the Lax representation for (2) with arbitrary β . We show that this Lax representation includes the non-removable spectral parameter. We study contact symmetries of this equation in Section 4. We show that the Lie algebra of contact symmetries of equation (2) is the semi-direct sum s 4 ⋉ a ∞ of the four-dimensional Lie algebra s 4 ∼ = gl 2 (R) and the infinitedimensional Abelian ideal a ∞ . Then in Section 5 we apply the approach of [17,18,19,22] to find local and nonlocal recursion operators for symmetries of (2). In Section 6 we study the action of local recursion operators to the subalgebra s 4 . This action generates a Lie subalgebra s ∞ of the algebra of higher symmetries of equation (2). We show that s ∞ has an interesting structure of the so-called Lie algebra of matrices of complex size, [9,30]. Cosymmetries of (2) and recursion operators for cosymmetries are discussed in Section 7. Finally, in Section 8 we use a higher symmetry from s ∞ to construct globally defined invariant solutions of equation (2).
The evolutionary vector field associated to an arbitrary vector-valued smooth function ϕ: J ∞ (π) → R m is the vector field for any φ and i.
A function ϕ: J ∞ (π) → R m is called a (generator of an infinitesimal) symmetry of equation E when E ϕ (F) = 0 on E. The symmetry ϕ is a solution to the defining system where ℓ E = ℓ F | E with the matrix differential operator The symmetry algebra Sym(E) of equation E is the linear space of solutions to (8) endowed with the structure of a Lie algebra over R by the Jacobi bracket {ϕ, ψ} = E ϕ (ψ) − E ψ (ϕ).
The algebra of contact symmetries Sym 0 (E) is the Lie subalgebra of Sym(E) defined as Sym(E) ∩C ∞ (J 1 (π)). Let the linear space W be either R N for some N ≥ 1 or R ∞ endowed with local coordinates w s , s ∈ {1, . . . , N} or s ∈ N, respectively. Locally, a differential covering of E is a trivial bundle τ: J ∞ (π) × W → J ∞ (π) equipped with extended total derivatives Define the partial derivatives of w s by w s (9) is referred to as the covering equations or the Lax representation of equation E. EXAMPLE 1. A differential covering for the Hunter-Saxton equation (1) has been presented in [31]. In a slightly different notation this is defined on J ∞ (π) × R with π: or by the system of the covering equations   The compatibility condition (w t ) x = (w x ) t of this system coincides with equation (1). The τ-vertical parts of the right-hand sides of (10) are linear combinations of the vector fields ∂ w , w ∂ w , and w 2 ∂ w . These vector fields generate the Lie algebra sl 2 (R) referred to as the universal algebra of the covering, [23]. ⋄ Consider operator E φ obtained by replacing D x k to D x k in (6). Solutions φ = φ (x i , u α I , w j ) to equationẼ φ (F) = 0 are referred to as shadows of symmetries in the covering τ, or just shadows.
A PDE E has two important coverings: the tangent covering TE and the cotangent covering T * E. Their covering equations are given by systems ℓ E (q) = 0 and ℓ * E (p) = 0, respectively, where ℓ * E is the adjoint operator to ℓ E . The local sections of the tangent covering are (generators of) symmetries, while the local sections of the cotangent covering are referred to as cosymmetries. The cosymmetries generate conservation laws for equation E, see discussion in [22,Ch. 1]. and Example 4 below. EXAMPLE 2. The covering equations for the tangent and cotangent coverings of equation (5) have the form and For α = 0 equations (12) and (13) for some operator S. Likewise, a recursion operator for cosymmetries of a PDE E is a Bäcklund autotransformation in the cotangent covering T * E. Taking adjoint operators to both sides of (14) we get Therefore operator S * is a recursion operator for cosymmetries.

Lax representation
Based on Example 1 we conjecture that the generalized Hunter-Saxton equation (5) admits a Lax representation with the same universal algebra sl 2 (R). We assume also that the coefficients of the covering equations are functions of u, u x , and u xx , that is, there exists system (5) coincides with the integrability conditions of (16). Direct computations give such a system: When α = 0, this system coincides with (11). The parameter λ = 0 in both systems (11) and (17) is non-removable. In accordance with [24, § §3.2, 3.6], [16,14], to prove this assertion it is sufficient to note that symmetry V = x ∂ x + u ∂ u of equation (5) does not admit a lift to a symmetry of system (17). Therefore the action e εV : The map System (17) can be written in the form of the pseudospherical type surface equations see discussion of the pseudospherical type equations in [32,6,31,11] and references therein.

Contact symmetries
The Lie algebra Sym 0 (E) of contact symmetries of equation (5) is generated by functions 2 and the family of solutions U = U(t, u x ) to the linear PDE The commutator table of Sym 0 (E) is given by equations This table implies that the contact symmetry algebra of (5) is the semi-direct sum Sym 0 (E) = s 4 ⋉ a ∞ of the four-dimensional subalgebra s 4 = φ 0,0 , φ 1,0 , φ 1,1 , φ 1,0 that is isomorphic to gl 2 (R), and the infinite-dimensional Abelian ideal a ∞ spanned by solutions to (19). Equation (19) has solutions of the form ψ(A) = A u x + A ′ . These functions generate a sub-ideal b ∞ a ∞ . The action of s 4 on b ∞ is given by equations This action has the following reformulation: the vector space A of smooth functions A = A(t) has a s 4 -module structure defined by formulae REMARK 1. The problem to find all the local symmetries of the form U(t, u x ) is as hard as the problem to find all solutions to equation (5), since equation (19) is contact-equivalent to (5). The proof of this statement mimics the proof of contact equivalence of equations (5) and (4) presented in [25]. ⋄

Recursion operators
In this section we use the methods of [17,18,19,22] to find local and nonlocal recursion operators for symmetries of equation (5).
To construct local recursion operators of first order we search for shadows of symmetries of the form where q is a solution to (12). Direct computations then give the following shadows: where E is the right hand side of equation (5). Therefore we have PROPOSITION 1. Differential operators provide local recursion operators for symmetries of E.
Proof follows from the general results of [22], or from identities Notice that the local recursion operators have the following commutator table: in other words, they constitute the Lie algebra sl 2 (R).
To find nonlocal recursion operators for symmetries we consider the Whitney product of the tangent covering (12) and the cotangent covering given by equation (13). Then Green's formula provides the canonical conservation law [22, p. 22 In particular, substituting for the solution p = u −2 x of (13) into (26) defines nonlocality s 1 by equations Likewise, the cosymmetry p = u α+1 x defines the nonlocality s 2 by system We obtain four shadows of symmetries in the tangent covering of the form σ = Q 1 q t + Q 2 q x + Q 3 q + Q 4 s 1 + Q 5 s 2 with nontrivial functions Q 4 and Q 5 : From the second equation in (27) we have therefore the nonlocal recursion operators are associated with shadows σ 4 and σ 5 . In the same way from the second equation in (28) we have therefore shadows σ 6 and σ 7 produce the nonlocal recursion operators respectively.

Higher symmetries
The action of the local recursion operators (22) -(24) on the contact symmetries (18) produces the Lie subalgebra s ∞ ⊂ Sym(E). In this section we study the structure of s ∞ . We for r times due to (25).
In particular, we have (recall notation of (20), (21)) Combining this with (29) we obtain PROPOSITION 2. Representation (20), (21) admits a prolongation to the Lie algebra s ∞ given by formula To show that s ∞ has the structure of the so-called Lie algebra of matrices of the complex size introduced in [9] and studied in [30], we recall the constructions of the last paper.
Let d denote the Lie algebra of differential operators of the form p n (t) ∂ n t + p n−1 (t) ∂ n−1 t + . . .+ p 1 (t) ∂ t + p 0 (t) where p k ∈ C[t] for k ∈ {0, . . . , n} and n ∈ N ∪ {0}, with the Lie bracket defined by the commutator. For fixed λ ∈ C consider the subalgebra gl(λ ) ⊂ d generated by the differential operators 1, The Lie algebra gl(λ ) is isomorphic to U(sl 2 (C))/I λ , where U(sl 2 (C)) is the universal enveloping algebra of sl 2 (C) and I λ is the ideal in U(sl 2 (C)) generated by the differential Comparing (21), (30), and (31), we obtain the following statement: THEOREM. The Lie algebra s ∞ ⊂ Sym(E) is isomorphic to gl(α + 3).
EXAMPLE 3. Symmetries φ 2,0 , ... , φ 2,4 are given by equations ⋄ REMARK 2. When α = 0, the Lie algebra s ∞ is a proper subalgebra of Sym(E). The family of symmetries of third order was found in [36]. We have η m ∈ s ∞ . The action of the local recursion operators R i on η m provides a family of higher symmetries of increasing order. This family is not included in s ∞ . We have no examples of higher symmetries that are not included in s ∞ when α = 0. ⋄ REMARK 3. The local recursion operators R i preserve the ideal a ∞ , since R i map solutions of equation (19) to solutions of the same equation, ⋄ REMARK 4. When α = 0, the family of local recursion operators of fourth order P m = P m,1 • P m,2 • P m,3 • D x with was constructed in [36]. We have P m (ψ(A)) = 0, hence P m is not a linear combination of the recursion operators of the form R (p,q,r) . We have no examples of local recursion operators that do not belong to the span of R (p,q,r) when α = 0. ⋄

Cosymmetries
Equations (12) and (13) coincide when α = 0, hence in this case cosymmetries are the same as the generators of symmetries. For other values of α we have PROPOSITION 3. All the cosymmetries of equation (5) with α = 0 that belong to C ∞ (J 1 (π)) have the form ψ = V (t, u x ), where functions V are solutions to the PDE which is the adjoint equation for (19).
Equation (33) is equivalent to equation (5) w.r.t. the pseudogroup of contact transformations on J 2 (π). The proof of this assertion is similar to the proof of contact equivalence of equations (5) and (4) given in [25]. Therefore the problem to find all solutions to equation (33) is as hard as the problem to find all solutions to equation (5). Nevertheless, we can find some particular solutions of (33). For example, when V t = 0, this equation get the form of a linear ordinary differential equation of second order. The general solution of this ODE is a linear combination with constant coefficients of two fundamental solutions ψ 1 = u −2 x and ψ 2 = u α+1 x . Equation (5) has higher cosymmetries. E.g., a family of cosymmetries of third order is defined by formulae when α = −4 and when α = −4, where H and G are arbitrary functions of their arguments.
xx is a cosymmetry of equation (2) for each α = −2. The related conservation law is The simple induction shows that for certain functions W k . Therefore for each α = −2 equation (5) has cosymmetries of arbitrary higher order. ⋄

Invariant solutions
In this section we give an example of implementing higher symmetries for finding globallydefined solutions of the generalized Hunter-Saxton equation. Namely, we construct φ 2,0invariant solutions to (5). These solutions satisfy the over-determined system that contains (5) and equation u tt = (u u xx + (α + 2) −1 u 2 x ) 2 u xx .
The compatibility conditions for this system get the form u t = u u x − u 3 x (α + 2) 2 u 3 xx (u x u xxx − 2 (α + 3) u 2 xx ), whence to obtain φ 2,0 -invariant solutions of (5) we proceed as follows: first, we find the general solution to ODE (38) in the form u = S(x, c 1 , c 2 , c 3 , c 4 ). This solution depends on four arbitrary constants c 1 , ... , c 4 . Second, we replace these constants by unknown functions s 1 (t), ... , s 4 (t) and substitute the obtained function u = S(x, s 1 (t), s 2 (t), s 3 (t), s 4 (t)) into equation (37). This yields a system of ODES that defines functions s i (t). Equation (38) has four-dimensional solvable algebra of point symmetries generated by vectors ∂ x , ∂ u , x ∂ x , u ∂ u , therefore this equation is integrable by quadratures in accordance Lie algebra of higher symmetries and then study the structure thereof, in particular we have found the basis of this Lie algebra. We have shown that the higher symmetries from the obtained Lie algebra can be used to construct global exact solutions for the generalized Hunter-Saxton equation. Furthermore, we have employed recursion operators to prove existence of an infinite number of cosymmetries of higher order, which indicates that the space of nontrivial conservation laws of higher order is infinite-dimensional as well.
We hope that the methods used in this paper are applicable to study other properties of the generalized Hunter-Saxton equation related to integrability such as variational symplectic and Poisson structures. We intend to address these issues in our future work.