The Diffiety Institute Preprint Series

Preprint DIPS–9/2001 June 21, 2001

Available via INTERNET: http://diffiety.ac.ru/

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

On geometry of the Liouville equation

by

A. V. KISELEV

On geometry of the Liouville equation

A. V. KISELEV

Abstract. Various geometrical structures related to the Liouville equation, are considered. Properties of the symmetry algebra are discussed, and local conserved currents are constructed. Bäcklund transformations for the Liouville equation are integrated, and nontrivial generalizations of the latter are studied, as well as the structures on them.

Introduction

The 2-dimensional elliptic Liouville equation

E = {u_{xx} + u_{yy} − exp(2u) = 0}, x ≡ x^{1}, y ≡ x^{2}, (1)

appears in numerous models of mathematical physics. In Riemannian geometry it represents the Gauss equation expressed in isothermic coordinates, for the Lobatchevsky plane [9]. Equation (1) may be treated as an example of the continuous elliptic Toda system [12, 15], see also [18] associated with the simple Lie algebra 𝔰𝔩_{2}(C); the coefficient 2 in the exponent stands for the Cartan 1×1 matrix of the algebra. Finding N-instanton solutions, minimizing the free Yang–Mills equations’ action, for self-duality equations F_{μν} = ∗F_{μν}, where F_{μν} is the stress tensor, also leads [20] to the Liouville equation.

Equation (1) was considered by Poincaré in [16], where uniformization of algebraic curves was studied. It is known [22, 1] that for surfaces of genus 0 the specially regularized action, computed on a classical solution of the Liouville equation, is a generating function for accessor parameters that characterize uniformization of a Riemannian surface; moreover, the action for equation (1) is a potential for the Weil–Peterson metric on the Teichmüller space of marked Riemannian surfaces of (0, N) type.

The elliptic Liouville equation plays an important role in the modern field theory, namely, in the strings theory, where the quantum Liouville field appears as a conformal anomaly [1, 17]. Also equation (1) relates to self-similar solutions of the Kadomtsev– Pogutse equations [6].

Equation (1) may be obtained formally from the hyperbolic equation [13]

E′ = {F ≡ u_{ξη} − exp(λu) = 0}, ξ ≡ y^{1}, η ≡ y^{2}, (2)

with λ = 2 by the substitution ξ = (x+iy)/2, η = (x−iy)/2. In many cases investigation of equation (1) in coordinates ξ ∈ ℂ, η ∈ ℂ, ξ̄ = η is convenient, because the results of computations in the internal coordinates ξ, η, u, u_{k} ≡ ∂^{k}u/∂ξ^{k}, _{l}u ≡ ∂^{l}u/∂η^{l} are simpler than similar expressions in real internal coordinates considered on (1).

Key words and phrases. Liouville equation, symmetries, conservation laws, Bäcklund transformations, the Toda systems.

2000 Mathematics Subject Classification. 35J60, 53A30, 58A25, 58J70, 58J72, 58Z05.

Received December 14, 2000; revised March 14, 2001.

In Section 1 of this paper a nontrivial n-dimensional generalization of (1) is proposed. In Section 2 generating sections of symmetries and conservation laws are studied and the Lie algebra of classical symmetries of the Liouville equation is built. Invariant solutions are considered and a special property of the energy-momentum tensor is discussed. Classical and higher conservation laws for the hyperbolic Liouville equation are constructed in Section 3. For a special case of higher generating sections an infinite series of rational (nonpolynomial) higher conserved currents is obtained. In Section 4 shadows of classical nonlocal symmetries corresponding to Abelian coverings, based on an arbitrary conserved current for the Liouville equation, are described. Bäcklund autotransformation for the Liouville equation u_{ξη} = exp(2u) and transformations between the latter, the wave equation v_{ξη} = 0, and the scal^{+}-Liouville equation w_{ξη} = exp(−2w) are integrated. Classical symmetries and conserved currents for the elliptic Toda systems associated with the semisimple Lie algebras of rank 2 are obtained in Section 5.

1. The n-dimensional elliptic Liouville equation

1.1. Conformal equivalence of 2-dimensional metrics. Consider two pointwise conformally equivalent Riemannian metrics ds_{j}^{2} = f_{j}(x, y) (dx^{2} + dy^{2}), f_{j} > 0, j = 1, 2, on an open 2-dimensional Riemannian manifold of constant Gauss curvature

K_{j} = −(2f_{j})^{−1}Δ ln f_{j} = const_{j} .

Let f_{2} = f_{1} exp(2u), then u(x, y) satisfies the equation [9]

Δu = −K_{2}f_{1} exp(2u) − K_{1}f_{1}.

It is easy to see that the 2-dimensional elliptic Liouville equation (1) corresponds to the pair consisting of the flat metric with f_{1} ≡ 1, K_{1} ≡ 0 and the metric on the Lobatchevsky plane (K_{2} ≡ −1).

Corollary 1. Every solution of equation (1) implies existence of the Lobatchevsky plane model conformally equivalent to Euclidean space with the diagonal metric g_{ij} = δ_{ij}.

Definition 1. We call the equation corresponding to f_{1} ≡ 1, K_{1} ≡ 0, K_{2} ≡ +1, the scal^{+}-Liouville equation.

1.2. General case: n ∈ ℕ. Let the number n of independent variables x^{1}, …, x^{n} be greater than 2. Then constancy of the open n-dimensional Riemannian manifold’s scalar curvature

(3) scal ≡ R = const

is a natural condition appearing in a general case of pointwise conformal equivalence between the Euclidean metric and the metric on the n-dimensional Riemannian manifold. In order to adjust this construction to the 2-dimensional case (1) let us fix R = −2 (i.e., the Gauss curvature K = −1 for n = 2) and let

ds^{2} = exp(2u)dx^{k}dx^{k}. (4)

Condition (3) is a nonlinear PDE for u(x^{1}, …, x^{n}).

Theorem 1. Condition (3) is

(5) (n − 1)Δu + \frac{(n − 1)(n − 2)}{2}(grad u)^{2} = exp(2u),

where Δ is the Laplacian in the Euclidean space 𝔼^{n}.

Proof. Scalar curvature R of metric (4) is defined by the formula R = exp(−2u)R_{qqi}^{i} (summation on repeated indices is assumed). One has

R_{qqi}^{i} = ∂_{i} Γ_{qq}^{i} − ∂_{q} Γ_{qi}^{i} + Γ_{pi}^{i}Γ_{qq}^{p} − Γ_{pq}^{i}Γ_{qi}^{p} (no summation),

where the Christoffel symbols are

Γ_{ij}^{k} = ∂_{i}u δ_{j}^{k} + ∂_{j}u δ_{i}^{k} − ∂_{l}u δ_{ij}δ^{kl}.

Computing sums on q, i, p = 1, … n, one obtains (5).

Corollary 2. On a Riemannian manifold with constant negative scalar curvature −2, there exists a class of conformal metrics parametrized by solutions of equation (5).

2. Classical symmetries of the Liouville equation

The determining equation on generating sections φ of symmetries and conservation laws [2] for the elliptic Liouville equation (1) is

D_{1}^{2}φ + D_{2}^{2}φ − 2 exp(2u)φ = 0.

Let z, z̄ be the complex coordinates on the plane (x, y):

z = x + iy, z̄ = x − iy, (6)

and let w(z) be an arbitrary analytic function.

Generating sections of classical symmetries and conservation laws depend on z and z̄, function u and its first derivatives u_{z}, uz̄: φ = φ(z, z̄, u, u_{z}, uz̄).

Proposition 1. Generating sections of classical symmetries and conservation laws over ℝ for equation (1) are:

φ = w(z)u_{z} + \overline{w(z)}uz̄ + \frac{1}{2}(w′(z) + \overline{w′(z)}). (7)

The real Lie algebra of classical infinitesimal symmetries of the 2-dimensional elliptic Liouville equation consists of vector fields

X_{w} = −w(z)∂ − \overline{w(z)} \overline{∂} + \frac{1}{2} (∂w + \overline{∂w}) \frac{∂}{∂u} + ⋯, (8)

where ∂ and \overline{∂} correspond to differentiating with respect to z and z̄.

Proposition 1 and formula (8) imply

Theorem 2. The algebra sym E of classical infinitesimal symmetries of the elliptic Liouville equation is isomorphic to the Lie algebra Vect S^{1} of analytic vector fields on circumference S^{1} [8].

The isomorphism is a projection −dz(X_{w}) for one turn and for another turn it is a reconstruction of the Lie field (8) by generating section (7) corresponding to the analytic w(z).

Remark 1. The Lie fields X_{w} possess an obvious complex structure J: X_{w} ↦ X_{iw} such that J(JX_{w}) = −X_{w}.

Remark 2. The Liouville equation has a discrete symmetry x^{1} → x^{2} invoking change of orientation of the plane (x^{1}, x^{2}); under its action analytic functions of complex argument z are mapped to conjugate ones, i.e., to antianalytic ones.

2.1. Invariant solutions. Obtaining solutions of equation (1), invariant under the action of a certain point symmetry X_{w} of the form (8), is described in [6]. Denote by W (z) = α ∫ dz/w(z) + iαβ, α, β ∈ ℝ; then the answer is

(9)  u_{1}(z, z̄) = \frac{1}{2} ln \frac{W′\overline{W′}}{sinh^{2}(Im W)},

(10)  u_{2}(z, z̄) = \frac{1}{2} ln \frac{W′\overline{W′}}{(Im W)^{2}} ,
(11)  u_{3}(z, z̄) = \frac{1}{2} ln \frac{W′\overline{W′}}{sin^{2}(Im W)}.

Remark 3. Functions (9)–(11) map successively one into another:

W_{1}(z) = i ln H(z),  W_{2}(z) = tan \frac{W_{1}(4z)}{2},  W_{3}(z) = ln W_{2}(z),

where H(z) is an arbitrary analytic function [13], and finally they provide general solution of the Liouville:

u[H] = \frac{1}{2} ln \frac{4H′\overline{H′}}{(1 − H\overline{H})^{2}} . (12)

Remark 4. One easily checks that the analytic function w(z) is expressed via harmonic one v(x, y) ≡ Im W (x + iy) by

w(z) = \frac{const}{v_{y} + iv_{x}} , const ∈ ℝ \ \{0\}.

Remark 5. General solution (12) may be also obtained by integrating the elliptic Toda system associated with the simple Lie algebra A_{1} [12, 15].

Remark 6. According to Bianchi, 1879 (cf. [7]), one can construct a solution of equation (1) using an arbitrary harmonic function v(x, y) in a way similar to (9)–(11):

u_{4}(z, z̄) = \frac{1}{2} ln \frac{v_{x}^{2} + v_{y}^{2}}{cos^{2} v} .

2.1.1. The energy-momentum tensor on u = ∞ curves. Let caustic [5] be a connected component of the curve

{ z | |H(z)| = 1, ∂^{l}H(z) ≠ 0, l ≥ 1 },

where solution (12) becomes ∞. The traceless energy-momentum tensor for equation (1) may be considered as T (z) = ∂^{2}u − (∂u)^{2}, \overline{∂}T = 0.

Proposition 2. The energy-momentum tensor T (z) is continuous on the caustics.

Proof. It is easy to check that for arbitrary solution (12)

T (z) = \frac{1}{2} (\frac{H′′′}{H′} − \frac{3}{2}(\frac{H′′}{H′} )^{2}) ≡ \frac{1}{2}\{H, z\},

where {H(z), z} is the Shwarzian derivative. The fact that T (z) does not depend on values H(z) with |H| = 1, implies the result.

2.1.2. Nonlinear superposition for the Liouville equation. The Cauchy–Riemann equation ∂H/∂z̄ = 0 is linear, and superposition principle is valid for it. Thus one obtains nonlinear superposition principle for equation (1) by means of formula (12): if H an everywhere nongenerating function and a linear combination of arbitrary analytic functions H_{1}(z) and H_{2}(z), then u[H], u′[H_{1}], and u′′[H_{2}] are independent solutions of the elliptic Liouville equation (see (12) for the definition of u[H]).

Remark 7. The constants H = z_{0} ∈ ℂ may be used for constructing new solutions though none is defined by formula (12) for them solely; the trivial function H = 0 (or v = 0) must be regarded as the neutral element (matching no solution (12)!)

Corollary 3. Solution u[H] and “antisolution” u[−H] vanish in presence of an arbitrary solution u[H_{1}].

Really, u[(H_{1} + H) + (−H)] = u[H_{1}].

Remark 8. The Lie algebra structure [H_{1}, H_{2}] = Wronsk(H_{1}, H_{2}) in the ring A of analytic functions provides another mapping A × A → Sol E into the set of solutions to the Liouville equation, i.e., H_{1} ⊗ H_{2} ↦ u [ [H_{1}, H_{2}] ].

3. Conservation laws and higher symmetries for the Liouville equation

3.1. Classical conservation laws for the hyperbolic Liouville equation. The results obtained in [10] may be carried over literally to the hyperbolic case (2), which is a simplification in some sense. Namely, the following statement is valid:

Proposition 3. For the hyperbolic Liouville equation one has

1. Generating sections γ(ξ, η, u, u_{ξ}, u_{η}) of classical symmetries and conservation laws are

γ^{(ξ)}(ξ, u_{ξ}) = (\frac{∂}{∂ξ} + λu_{ξ})Φ(ξ) or γ^{(η)}(η, u_{η}) = (\frac{∂}{∂η} + λu_{η})Ψ(η), (13)

where Φ(ξ), Ψ(η) are arbitrary functions.

2. Each of the generating sections corresponds a nontrivial conservation law.

3. The symmetries above are Noetherian symmetries of the action with the Lagrangian density L = u_{ξ}u_{η}/2 + exp(λu)/λ.

4. The generating section γ^{(ξ)} corresponds to the conserved current

(14) h = (uΦ′′ + \frac{λ}{2}uu_{ξ}Φ′ + \frac{λ}{2}uu_{ξξ}Φ) dξ

+ (\frac{λ}{2}uΦ exp(λu) − Φ exp(λu) + u_{η}Φ′ + \frac{λ}{2}u_{ξ}u_{η}Φ) dη,

the current for γ^{(η)} may be obtained by the interchange ξ ↔ η, Φ ↔ Ψ.

Proof of the proposition is quite similar to the one given in [10] and is omitted.

Remark 9. Conserved currents h for both equations (1) and (2) form a Lie algebra with the bracket induced by the Lie algebra of analytic functions [a, b] = Wronsk(a, b).

3.2. Generating sections of higher symmetries and conservation laws. Generating sections γ of higher symmetries and conservation for equation (2) are [7, 21]:

γ_{n} = (D_{ξ} + λu_{1})Φ(ξ, w, w_{1}, …, w_{n−3}), (15)

where u_{k} ≡ ∂^{k}u/∂ξ^{k}, w ≡ u_{2} − \frac{λ}{2}u_{1}^{2}, w_{k} ≡ D_{ξ}^{k}w, D_{ξ} is the total derivative with respect to ξ, Φ is an arbitrary function (there exists a similar class of higher generating sections with ξ and η interchanged; second order sections are absent).

Remark 10. The noncommutative Lie algebra of higher symmetries of equation (2) contains an infinite commutative subalgebra defined by the higher Korteweg–de Vries equations [21].

The complex substitution u(ξ, η)|_{(z/2,z̄/2)} = u(z, z̄) with λ = 2 transforms (2) to equation (1) in coordinates (6): 4u_{zz̄} = exp(2u), and also gives generating sections of symmetries and conservation laws, defined over ℂ, for the latter. In order to define these structures over ℝ and apply them to equation (1), one has to compensate imaginary part with adding the conjugate structure:

φ = (D_{z} + 2u_{z})Φ(z, w, w_{1}, …) + (Dz̄ + 2uz̄)Φ̄(z̄, w̄, w̄_{1}, …).

Example 1. The function φ = 2u_{xxx} −3 exp(2u)u_{x} −u_{x}^{3} +3u_{x}u_{y}^{2} is a generating section of higher symmetry and higher conservation law for equation (1). The corresponding conserved current h = S_{1} dy − S_{2} dx may be reconstructed with the same method [19] as in the case of classical conserved currents [10]; the components of h are

4S_{1} = 3u_{x}^{2}u_{y}^{2} − u_{x}^{4} − 6uu_{x}u_{y}u_{xy} − 6 exp(2u)u_{x}^{2} + 2 exp(2u)u_{xx}

+ 4u_{x}u_{xxx} − 3 exp(4u) + 3u_{y}^{2}u exp(2u) − 3u_{y}^{2}uu_{xx} + 8u exp(2u)u_{xx}

+ 3u_{x}^{2}uu_{xx} − 6u_{y}^{2} exp(2u) − 4uu_{xxxx} + \frac{6u_{y}^{2}}{u} exp(2u)

+ \frac{3}{u} exp(4u) − \frac{3u_{xx}}{u} exp(2u) − \frac{3u_{y}^{2}}{u^{2}} exp(2u) + 13u exp(2u)u_{x}^{2},

4S_{2} = 6u_{x}u_{y}uu_{xx} + 4u_{y}u_{xxx} − u_{y}u_{x}^{3} + 3u_{x}u_{y}^{3} − 4uu_{xxxy} − 3uu_{xy}u_{y}^{2}

+ 3uu_{x}^{2}u_{xy} + 6u_{xy} exp(2u) − \frac{6u_{x}u_{y}}{u} exp(2u)

− \frac{3u_{xy}}{u} exp(2u) + \frac{3u_{x}u_{y}}{u^{2}} exp(2u).

It is interesting that in the simplest case the conserved current is not polynomial, but a rational function.

3.2.1. Higher conservation laws. An infinite series of polynomial conserved currents for equation (2) was described in [4], but the formulae given there do not describe all local conservation laws. Below higher conservation laws for equation (2) are obtained. They are rational in u(ξ, η). We obtain higher conserved current h corresponding to generating section γ_{n} defined in (15).

Let the divergence ω be

ω = (u_{ξη} − exp(λu))(D_{ξ}Φ + λu_{1}Φ) dξ ∧ dη, ω|_{E} = d̄h = 0.

Literally repeating the reasoning for classical case, we have

G(ℓ_{ω} ∘ u) ≡ s^{ξ} dξ + s^{η} dη = [uD_{ξ}(Φ_{ξ} + ∑_{r=0}^{n−3} Φ_{w_{r}}w_{r+1} + λu_{1}Φ)] dη

+ [_{1}u(Φ_{ξ} + ∑_{r=0}^{n−3} Φ_{w_{r}}w_{r+1} + λu_{1}Φ) + λ(u_{ξη} − e^{λu})Φ

+ ∑_{s=0}^{n−3} ∑_{t=1}^{s+2} ∑_{p=1}^{t}(−1)^{p+1}\binom{t}{p}D_{ξ}^{p−1} ((u_{ξη} − e^{λu})Φ_{ξw_{s}} \frac{∂w_{s}}{∂u_{t}} u_{t−p})

+ ∑_{s=0}^{n−3} ∑_{r=0}^{n−3} ∑_{t=1}^{s+2} ∑_{p=1}^{t}(−1)^{p+1}\binom{t}{p}D_{ξ}^{p−1} ((u_{ξη} − e^{λu})w_{r+1}Φ_{w_{s}w_{r}} \frac{∂w_{s}}{∂u_{t}} u_{t−p})

+ ∑_{r=1}^{n−2} ∑_{t=1}^{r+2} ∑_{p=1}^{t}(−1)^{p+1}\binom{t}{p}D_{ξ}^{p−1} ((u_{ξη} − e^{λu})Φ_{w_{r−1}} \frac{∂w_{r}}{∂u_{t}} u_{t−p})

+ λ ∑_{r=0}^{n−3} ∑_{t=1}^{r+2} ∑_{p=1}^{t}(−1)^{p+1}\binom{t}{p}D_{ξ}^{p−1} ((u_{ξη} − e^{λu})u_{1}Φ_{w_{r}} \frac{∂w_{r}}{∂u_{t}} u_{t−p})] dξ,

where G is the Green operator and ℓ_{ω} = ∑_{σ} ∂ω/∂u_{σ} · D_{σ} is the universal linearization operator [2, 19]. The required conserved current h = S^{ξ} dξ + S^{η} dη corresponding to G(ℓ_{ω} ∘ u) is

h = ∫^{0} dτ A_{τ}^{∗} (s^{ξ} dξ + s^{η} dη) ,  d̄h = 0,

where A_{τ}: (y^{i}, u_{σ}) ↦ (y^{i}, u_{σ} exp(τ)).

3.2.2. Infinite series of rational higher conserved currents. Let us obtain higher conserved currents h for equation (2) in the case Φ = w_{k}, k ≥ 0. The function Φ supplies a higher generating section by means of (15).

Let \left[{• \atop •, …, •}\right] be the coefficient in the expansion of the q-th total derivative with

respect to ξ for the exponent exp(λu(ξ, η)):

D_{ξ}^{q} exp(λu) = ∑_{m=0}^{q} ∑_{j_{1}+···+j_{q}=m} exp(λu)λ^{m}\left[{q \atop j_{1}, …, j_{q}}\right]u_{1}^{j_{1}} ⋯ u_{q}^{j_{q}}, q ≥ 0.

The coefficients \left[{• \atop •, …, •}\right] satisfy the following recursion relation:

\left[{q \atop j_{1}, …, j_{q}}\right] = \left[{q \atop j_{1} − 1, …, j_{q}}\right] + \left[{q \atop j_{1}, …, j_{q−1} + 1}\right] · δ_{j_{q},1} · (j_{q−1} + 1)

+ ∑_{k=1}^{q−2} \left[{q \atop j_{1}, …, j_{k} + 1, j_{k+1} − 1, …, j_{q}}\right] · (j_{k} + 1),

where ∑_{k} j_{k} = m ∈ ℕ, and \left[{… \atop …, j_{k} − 1, …}\right] ≡ 0, if ther exists k such that j_{k} = 0. The

relation follows from the Leibniz rule.

Remark 11. The following two integrals appear in computation of the conserved current h with arbitrary σ = (σ_{1}, …, σ_{k}), k > 0:

∫^{0} u_{σ_{1}} ⋯ u_{σ_{k}} exp(kτ) dτ = \frac{1}{k}u_{σ_{1}} ⋯ u_{σ_{k}};

if the integrand contains exponent of u(ξ, η), one has to integrate by parts:

∫^{0} u_{σ_{1}} ⋯ u_{σ_{k}} exp(kτ) exp (λu exp(τ)) dτ = \frac{u_{σ_{1}} ⋯ u_{σ_{k}}}{λ^{k}u^{k}} ·

∑_{j=1}^{k−1}(−1)^{j+1}(\frac{(k − 1)!}{(k − j)!}(λu)^{k−j} exp(λu) − (−1)^{k}(k − 1)! exp(λu)).

Proposition 4. The higher conserved current h = S_{1} dη − S_{2} dξ for equation (2) reconstructed from generating section (15) with Φ = w_{k}, k ≥ 0, has the components

S_{1} = \frac{1}{2}_{1}uu_{k+3} − \frac{λ}{6}_{1}u ∑_{l=0}^{k+1} \binom{k + 1}{l}u_{l+1}u_{k−l+2} + \frac{λ}{3}_{1}uu_{1}u_{k+2} − \frac{λ}{8}_{1}uu_{1} ·

· ∑_{l=0}^{k} \binom{k}{l}u_{l+1}u_{k−l+1} + ∑_{p=0}^{k+2} \binom{k + 2}{p}(−1)^{k−p} ∑_{m=0}^{k−p+2} \binom{k − p + 2}{m} ·

· u_{k−m+2} ∑_{n=0}^{m} ∑_{j_{1}+···+j_{m}=n} e^{λu}\left[{m \atop j_{1}, …, j_{m}}\right]u_{1}^{j_{1}} … u_{m}^{j_{m}} ·

· (\frac{λ^{n}}{2} + \frac{1}{λu^{n+1}} · \{ ∑_{j=1}^{n}(−1)^{j}(λu)^{n−j+1} · \frac{n!}{(n − j + 1)!} − (−1)^{n}n!\})

− \frac{λ}{2} ∑_{l=0}^{k+1} \binom{k + 1}{l} ∑_{p=0}^{l} \binom{l + 1}{p}(−1)^{l−p} ∑_{i_{1}=0}^{l−p} \binom{l − p}{i_{1}} ∑_{i_{2}=0}^{i_{1}} \binom{i_{1}}{i_{2}} ·

· u_{k−p−i_{1}+2}u_{p+i_{1}−i_{2}} ∑_{m=0}^{i_{2}} ∑_{j_{1}+···+j_{i_{2}}=m} e^{λu}\left[{i_{2} \atop j_{1}, …, j_{i_{2}}}\right] · u_{1}^{j_{1}} … u_{i_{2}}^{j_{i_{2}}} ·

· (\frac{λ^{m}}{3} + \frac{1}{λ^{2}u^{m+2}} \{ ∑_{j=1}^{m+1}(−1)^{j}(λu)^{m−j+2} \frac{(m + 1)!}{(m − j + 2)!}

+ (−1)^{m}(m + 1)!\}) + \frac{λ}{2} ∑_{l=0}^{k+1} \binom{k + 1}{l} ∑_{p=0}^{k−l+1} \binom{k − l + 2}{p}(−1)^{k−l−p} ·

· ∑_{i_{1}=0}^{k−l−p+1} \binom{k − l − p + 1}{i_{1}} ∑_{i_{2}=0}^{i_{1}} \binom{i_{1}}{i_{2}}u_{k−p+2}u_{p+i_{1}−i_{2}} ∑_{m=0}^{i_{2}} ·

· ∑_{j_{1}+···+j_{i_{2}}=m} e^{λu} · \left[{i_{2} \atop j_{1}, …, j_{i_{2}}}\right]u_{1}^{j_{1}} … u_{i_{2}}^{j_{i_{2}}} · (\frac{λ^{m}}{3}

+ \frac{1}{λ^{2}u^{m+2}} · \{ ∑_{j=1}^{m+1}(−1)^{j}(λu)^{m−j+2} \frac{(m + 1)!}{(m − j + 2)!}

+ (−1)^{m}(m + 1)!\}) − \frac{λ}{3}uu_{ξη}u_{k+2} + \frac{λ^{2}}{8} uu_{ξη} ∑_{l=0}^{k} \binom{k}{l}u_{l+1} ·

· u_{k−l+1} + u_{k+2}e^{λu} − \frac{u_{k+2}}{λu} e^{λu} − \frac{1}{2} ∑_{l=0}^{k} \binom{k}{l} ·

· \frac{u_{l+1}u_{k−l+1}}{λu^{2}} \{λ^{2}u^{2} − 2λu + 2\}e^{λu}

+ λ ∑_{p=0}^{k+1} \binom{k + 2}{p}(−1)^{k−p} ∑_{i_{1}=0}^{k−p+1} \binom{k − p + 1}{i_{1}} ∑_{i_{2}=0}^{i_{1}} \binom{i_{1}}{i_{2}} ·

· u_{k−p−i_{1}+2}u_{p+i_{1}−i_{2}} ∑_{m=0}^{i_{2}} ∑_{j_{1}+···+j_{i_{2}}=m} e^{λu}\left[{i_{2} \atop j_{1}, …, j_{i_{2}}}\right] ·

· u_{1}^{j_{1}} … u_{i_{2}}^{j_{i_{2}}}(\frac{λ^{m}}{3} + \frac{1}{λ^{2}u^{m+2}} ·

· \{ ∑_{j=1}^{m+1}(−1)^{j}(λu)^{m−j+2} \frac{(m + 1)!}{(m − j + 2)!} + (−1)^{m}(m + 1)!\})

− \frac{λ^{2}}{2} ∑_{l=0}^{k} \binom{k}{l} ∑_{p=0}^{l} \binom{l + 1}{p}(−1)^{l−p} ∑_{i_{1}=0}^{l−p} \binom{l − p}{i_{1}} ∑_{i_{2}=0}^{i_{1}} \binom{i_{1}}{i_{2}} ·

· ∑_{i_{3}=0}^{i_{2}} u_{l−p−i_{1}+1} · u_{k−l+i_{1}−i_{2}+1}u_{p+i_{2}−i_{3}} ∑_{m=0}^{i_{3}} ∑_{j_{1}+···+j_{i_{3}}=m} e^{λu} ·

· \left[{i_{3} \atop j_{1}, …, j_{i_{3}}}\right] · u_{1}^{j_{1}} … u_{i_{3}}^{j_{i_{3}}} · (\frac{λ^{m}}{4} + \frac{1}{λ^{3}u^{m+3}} ·

· \{ ∑_{j=1}^{m+2}(−1)^{j}(λu)^{m−j+3} \frac{(m + 2)!}{(m − j + 3)!} − (−1)^{m}(m + 2)!\})

− \frac{λ^{2}}{2} ∑_{l=0}^{k} \binom{k}{l} ∑_{p=0}^{k−l} \binom{k − l + 1}{p}(−1)^{k−l} ∑_{i_{1}=0}^{k−l} \binom{k − l}{i_{1}} ∑_{i_{2}=0}^{i_{1}} \binom{i_{1}}{i_{2}} ∑_{i_{3}=0}^{i_{2}} ·

· \binom{i_{2}}{i_{3}}u_{k−l−i_{1}+1}u_{l+i_{1}−i_{2}+1}u_{p+i_{2}−i_{3}} ∑_{m=0}^{i_{3}} ∑_{j_{1}+···+j_{i_{3}}=m} e^{λu} ·

· \left[{i_{3} \atop j_{1}, …, j_{i_{3}}}\right]u_{1}^{j_{1}} … u_{i_{3}}^{j_{i_{3}}} · (\frac{λ^{m}}{4} + \frac{1}{λ^{3}u^{m+3}} ·

· \{ ∑_{j=1}^{m+2}(−1)^{j}(λu)^{m−j+3} \frac{(m + 2)!}{(m − j + 3)!} − (−1)^{m}(m + 2)!\}),

S_{2} = −\frac{λ}{3}uu_{1}u_{k+3} + \frac{λ^{2}}{8} uu_{1} ∑_{l=0}^{k} \binom{k + 1}{l}u_{l+1}u_{k−l+2}

+ \frac{λ}{6}u ∑_{l=0}^{k+2} \binom{k + 2}{l}u_{l+1}u_{k−l+3} − \frac{λ}{3}uu_{2}u_{k+2}

+ \frac{λ^{2}}{8} uu_{2} ∑_{l=0}^{k} \binom{k}{l}u_{l+1}u_{k−l+1} − \frac{1}{2}uu_{k+4}.

4. Coverings of the Liouville equation, nonlocal symmetries, and Bäcklund transformations

4.1. On nonlocal symmetries. One can construct an Abelian covering c̃^{h} structure [2] in the bundle c^{h}: Ẽ → E^{∞} using a conserved current h = S^{ξ} dξ + S^{η} dη for an equation E with 2 independent variables. Then the fiber coordinate w is differentiated with respect to ξ and η in the following way: w_{ξ} = S^{ξ}, w_{η} = S^{η}; the total derivatives are extended to the form D̃_{i} = D̄_{i} + S^{y^{i}} ∂/∂w.

Consider the covering c̃^{h}, provided by classical conserved current (14) corresponding to generating section γ^{ξ} of the form (13). The search for shadows of nonlocal c̃^{h}- symmetries of equation (2), i.e., solving the determining equation ℓ̃_{F}^{E′}(γ̃) = 0, where the universal linearization operator ℓ̃_{F} contains extended total derivatives D̃_{i} leads to

Proposition 5. The shadows γ̃(ξ, η, w, u, u_{ξ}, u_{η}) of the classical nonlocal c̃^{h}-symmetries are

γ̃^{(ξ)}(ξ, u_{ξ}) = (\frac{∂}{∂ξ} + λu_{ξ})Φ̃(ξ) or γ̃^{(η)}(η, u_{η}) = (\frac{∂}{∂η} + λu_{η})Ψ̃(η), (16)

where Φ̃(ξ), Ψ̃(η) are arbitrary functions.

In other words, the set of shadows of classical nonlocal c̃^{h}-symmetries of equation (2) is isomorphic to the Lie algebra of classical symmetries (13), and no new symmetries of the latter equation are obtained.

4.2. Bäcklund transformations. Bäcklund autotransformation for the Liouville equation and Bäcklund transformation between equation (2), the wave equation v_{ξη} = 0 and the equation w_{ξη} = exp(−2w) corresponding to the Gaussian curvature K_{2} ≡ +1 in the sense of Section 1.1, were found in [3]. Let us integrate all the these cases.

Remark 12. These transformations depend on formal parameter k, but the latter can be eliminated with by a scale transformation.

4.2.1. Bäcklund autotransformation. Equation (2) possesses the Bäcklund autotransformation

(u′ − u)_{ξ} = \frac{1}{k} · exp(u′ + u),
(u′ + u)_{η} = k · (exp(u′ − u) − exp(u − u′)) ,

k ≠ 0 being a formal parameter. Set k = 1 and denote τ^{−} = u′ − u, τ^{+} = u′ + u. It is easy to see that the functions τ^{−} and τ^{+} satisfy the system

(17)  τ_{ξ}^{−} = exp(τ^{+}),
(18)  τ_{η}^{+} = 2 sinh τ^{−}.

From (17) it follows that τ^{+} = ln τ_{ξ}^{−}; then equation (18) implies τ_{ξη}^{−} = 2τ_{ξ}^{−} sinh τ^{−}, and

τ_{η}^{−} = 2 cosh τ^{−} + δ(η), (19)

where δ(η) is the integration constant. Resolving (19), one obtains the function τ^{−}(ξ, η) depending now on another integration constant ϵ(ξ), and thus one reconstructs τ^{+}(ξ, η). Knowing the functions τ^{−} and τ^{+}, one gets a pair of solutions u, u′ of equation (2), and these solutions are bound by Bäcklund autotransformation.

There exists a similar transformation with ξ ↔ η.

4.2.2. Transformation between the Liouville equation and the wave equation. Bäcklund transformation between equation (2) and the equation v_{ξη} = 0 is [18, 3]:

(v − u)_{ξ} = \frac{1}{k} · exp(v + u), (v + u)_{η} = −k · exp(u − v).

As before, denote τ^{−} = u − v, τ^{+} = v + u; these functions satisfy the system

τ_{ξ}^{−} = − exp(τ^{+}), τ_{η}^{+} = − exp(τ^{−}),

i.e., each of the functions τ^{−}, τ^{+} is a potential for the other, e.g. τ^{+} = ln(−τ_{ξ}^{−}). Solving an easily obtained equation τ_{ξη}^{−} = exp(τ^{−})τ_{ξ}^{−} for τ^{−}(ξ, η), one gets an equation with separating variables,

\frac{dθ}{exp(θ)} = \frac{dη}{exp(δ(η))}, (20)

where θ = (τ^{−} + δ)(ξ, η) and δ(η) is the integration constant. The variable ξ is a parameter in this equation, and thus another integration constant ϵ(ξ) appears. The function

θ = − ln ( − ∫_{ϵ(ξ)}^{η} exp(−δ(κ)) dκ)

is a solution of equation (20), and thus one successively reconstructs functions the τ^{−} and τ^{+} and obtains a pair u, v, where u(ξ, η) is a solution of and v(ξ, η) is a solution of the wave equation.

Remark 13. Assuming ξ = η̄ = z/2, one has Δ_{2}v(x, y) = 0, i.e., the above-mentioned transformation bounds equation (1) and the 2-dimensional Laplace equation in complex coordinates (6).

4.2.3. Transformation between the Liouville equation and the scal^{+}-Liouville equation. This transformation may be written down in the following way [3]:

(w − u)_{ξ} = \frac{1}{k} · (exp(w + u) + exp(−w − u)),
(w + u)_{η} = −k · exp(u − w),

where w_{ξη} = exp(−2w). Then −w is a solution of the scal^{+}-Liouville equation (see Section 1.1). Denote τ^{+} = u + w, τ^{−} = u − w, then

τ_{η}^{+} = − exp(τ^{−}), τ_{ξ}^{−} = −2 cosh τ^{+},

implying τ^{−} = ln(−τ_{η}^{+}) and τ_{ξη}^{+} = 2τ_{η}^{+} cosh τ^{+}. Solving the last equation gives τ_{ξ}^{+} = 2sinh τ^{+} + δ(ξ), where δ(ξ) is the integration constant; further resolving the equation on τ^{+}(ξ, η) provides another integration constant ϵ(η). By τ^{+}(ξ, η) one reconstructs τ^{−}(ξ, η) and finally obtains a pair of solutions u = (τ^{+} + τ^{−})/2 and w = (τ^{+} − τ^{−})/2.

Remark 14. The function w(ξ, η) = \frac{1}{2} ln \frac{v_{ξ}v_{η}}{cosh^{2} v} is a solution of the equation w_{ξη} =

exp(−2w), where v(ξ, η) is an arbitrary solution of the wave equation v_{ξη} = 0.

Some more results concerning nonlocal structures (zero curvature representation and recursion operator for the Liouville equation) can be found in [7].

5. On the Toda systems associated with the semisimple Lie algebras of rank 2

The 2-dimensional elliptic Toda systems [12, 15] associated with the semisimple Lie algebras A_{2}, B_{2} ≃ C_{2}, D_{2} ≃ A_{1} ⊕ A_{1}, and G_{2}, of rank 2, represented in the coordinates z, z̄, u^{k}(z, z̄) are:

u_{zz̄}^{k} = \frac{1}{4} exp(∑_{l=1}^{2} K_{l}^{k}u^{l}), k = 1, 2, (21)

where K_{l}^{k} is the Cartan matrix of the corresponding Lie algebra. Denote u^{1} ≡ u and u^{2} ≡ v, and let w(z) be an arbitrary analytic function. The following results take place [11]:

Proposition 6. The following facts hold:

1. Generating sections φ = (φ^{1}, φ^{2}) for classical infinitesimal symmetries of Toda system (21) associated to the simple Lie algebra of rank 2 with the Cartan matrix

K = ( \begin{matrix}2 & −a \\ −1 & 2 \end{matrix} ) , a = const, (22)

are

φ^{1} = w(z)u_{z}^{1} + \overline{w(z)}u1 ; z̄ + \frac{a + 2}{4 − a} (w_{z} + w̄z̄) ,
φ^{2} = w(z)u_{z}^{2} + \overline{w(z)}u2 ; z̄ + \frac{3}{4 − a} (w_{z} + w̄z̄) .

2. Classical conserved currents H = S^{z} dz̄−Sz̄ dz for this system have the components

S^{z} = −avw̄_{z̄z̄} − uw̄_{z̄z̄} + w̄u2 ; z̄ + w_{z}uz̄ − \frac{1}{2} exp(2u − av)w + awv_{z}vz̄

− uw̄z̄uz̄ + aw̄z̄vz̄ − \frac{1}{8}auw exp(2v − u) + \frac{1}{2}auw̄v_{z̄z̄} + w̄z̄uz̄

+ \frac{1}{2}avw̄z̄uz̄ − avw̄z̄vz̄ + \frac{1}{2}auw̄z̄vz̄ − \frac{1}{8}avw exp(2u − av)

− \frac{1}{2}awv_{z}uz̄ − \frac{1}{2}awvz̄u_{z} − aw̄vz̄uz̄ + aw_{z}vz̄ + wu_{z}uz̄

+ aw̄v2 ; z̄ + \frac{1}{4}uw exp(2u − av) + \frac{1}{4}avw exp(2v − u)

− uw̄u_{z̄z̄} + \frac{1}{2}avw̄u_{z̄z̄} − avw̄v_{z̄z̄} − \frac{1}{2}a exp(2v − u)w,

Sz̄ = −avw_{zz} − uw_{zz} + aw̄v_{z}vz̄ + w_{z}u_{z} + w̄z̄u_{z} + wu_{z}^{2}

− awu_{z}v_{z} − \frac{1}{2}aw̄u_{z}vz̄ − \frac{1}{2}aw̄uz̄v_{z} − avw_{z}v_{z} + \frac{1}{2}avw_{z}u_{z}

+ \frac{1}{2}auwv_{zz} + \frac{1}{4}avw̄ exp(2v − u) − avwv_{zz} + \frac{1}{2}auw_{z}v_{z}

− \frac{1}{8}avw̄ exp(2u − av) + \frac{1}{2}avwu_{zz} − \frac{1}{8}auw̄ exp(2v − u)

− uwu_{zz} + \frac{1}{4}uw̄ exp(2u − av) − uw_{z}u_{z} − \frac{1}{2}a exp(2v − u)w̄

− \frac{1}{2} exp(2u − av)w̄ + aw̄z̄v_{z} + aw_{z}v_{z} + w̄u_{z}uz̄ + awv_{z}^{2}.

For the Lie algebra A_{2} the constant a = 1, for B_{2}, a = 2, and for G_{2}, a = 3. The Cartan matrix for C_{2} is transpose one for B_{2}, so the change u ↔ v is needed for this case. The Lie algebra D_{2} is not simple, and the conserved current for this algebra is a sum of two independent conserved currents (14).

Computations of conserved currents H for system (21) were rather tedious and Jet [14] software was used.

Acknowledgements. The author is thankful to the participants of the seminar “The Lie algebras, differential geometry, and mathematical physics”, IUM, for their remarks and constructive criticism. Also the author expresses his profound gratitude to I.S. Krasil’shchik, A.V. Ovchinnikov, and A.M. Verbovetsky, for stimulating discussions and advice. Special thanks are addressed to M. Marvan for his Jet software.

References

[1] A.A. Belavin, A.N. Polyakov, and A.B. Zamolodchikov, Infinite conformal symmetry group in two-dimensional quantum field theory, Nucl. Phys. 241 (1984), n. 2, 333–380.

[2] A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Khor’kova, I.S. Krasil’shchik, A.V. Samokhin, Yu.N. Torkhov, A.M. Verbovetsky, and A.M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Amer. Math. Soc., Providence, RI, 1999. Edited and with a preface by Krasil’shchik and Vinogradov.

[3] R.K. Dodd and R.K. Bullough, Bäcklund transformations for the sine–Gordon equations, Proc. Roy. Soc., London, A351 (1976), 499–523.

[4] R.K. Dodd and R.K. Bullough, Polynomial conserved densities for the sine–Gordon equations, Proc. Roy. Soc., London, A352 (1977), 481–503.

[5] G.P. Georgadze, A.K. Pogrebkov, and M.K. Polivanov, On global solutions of the Cauchy problem for the Liouville equation φ_{tt}(t, x)−φ_{xx}(t, x) = \frac{1}{2}m exp(φ(t, x)), Dokl. AN SSSR 243 (1978), n. 2, 318–320 (in Russian).

[6] V.N. Gusyatnikova, A.V. Samokhin, V.S. Titov, et al., Symmetries and conservation laws of Kadomtsev–Pogutse equations, Acta Appl. Math. 15 (1989) n. 1, 23–64.

[7] N.H. Ibragimov (ed.), CRC Handbook of Lie Group to Differential Equations. Vol. II. Applications in engineering and physical sciences, CRC Press, Boca Raton, Fl., 1995.

[8] V.G. Kac and A.K. Raina, Bombai Lectures on Highest Weight Representation of Infinite Dimensional Lie Algebras, World Scientific, Singapore et al., 1987.

[9] J.L. Kazdan and F.W. Warner, Curvature functions for open 2-manifolds, Ann. Math., (2) 99 (1974), n. 2, 203–219.

[10] A.V. Kiselev, Classical conservation laws for the elliptic Liouville equation, Moscow Univ. Phys. Bull. (2000), n. 6, 11–13.

[11] A.V. Kiselev, Conservation laws for the elliptic Toda systems associated with the algebras A_{2}, B_{2}, C_{2}, D_{2}, G_{2}, Proc. XXII Conf. of the young sci. at MSU mech.–math. dept. (2000) MSU, Moscow, Russia.

[12] A.N. Leznov and M.V. Saveliev, Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems. Birkhäuser Verlag, Basel, 1992.

[13] J. Liouville, Sur l’equation aux différences partielles d^{2} log λ/du dv ± λ/(2a^{2}) = 0, J. de math. pure et appliquée 18 (1853), n. 1, 71–72.

[14] M. Marvan, Jet. A software for differential calculus on jet spaces and diffieties, ver. 3.9 (August 1997) for Maple V Release 4. Opava, Czhech Republic, 1997; http://diffiety.ac.ru/

[15] A.V. Ovchnnikov, Toda Systems Associated with the Lie Algebras and the W -algebras in some Problems of Mathematical Physics. PhD thesis, Physics dept., MSU, Moscow, Russia, 1996 (in Russian).

[16] H. Poincaré, Les fonctions fuchsiennes et l’equation Δu = exp(u), J. math. pures et appl., 5^{e} ser. (1898), n. 4, 157–230.

[17] A.M. Polyakov, Gauge Fields and Strings. Harwood Acad. Publs., NY, 1987.

[18] W.F. Shadwick, The Bäcklund problem for the equation ∂^{2}z/∂x^{1}∂x^{2} = f(z), J. Math. Phys. 19 (1978), n. 11, 2312–2317.

[19] A.M. Vinogradov, The C-spectral sequence, Lagrangian formalism, and conservation laws, I. The linear theory. II. The nonlinear theory, J. Math. Anal. Appl. 100 (1984), 1–129.

[20] E. Witten, Some exact multipseudoparticle solutions of classical Yang–Mills theory, Phys. Rev. Lett. 38 (1977), n. 3, 121-124.

[21] A.V. Zhiber and A.B. Shabat, The Klein–Gordon equations admitting nontrivial group, Dokl. AN SSSR 247 (1979), n. 5, 1103–1107 (in Russian).

[22] P.G. Zograf and L.A. Tachtajan, On the Liouville equation, accessor parameters and geometry of the Teichmüller space for the Riemannian surfaces of genus 0, Math. sbornik 132(174) (1987), n. 2, 147–166 (in Russian).

Lomonosov Moscow State University, Physics Department, Chair of mathematics, 119899, Vorobjevy Gory, Moscow, Russia.

E-mail address: arthemy@mccme.ru