The Diffiety Institute Preprint Series

Preprint DIPS–15 December 20, 2001

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

The Diffiety Institute

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

On one-parametric family of Bäcklund autotransformations for the Liouville equation

by

Arthemy V. Kiselev

On one-parametric family of Bäcklund autotransformations for the Liouville equation

Arthemy V. Kiselev

Abstract. Scaling symmetry of the Liouville equation is proved to provide a oneparametric family of one-dimensional non-abelian coverings determining a Bäcklund autotransformation. Deformation of the structural element is shown to be equal to the Frölicher-Nijenhuis bracket of the symmetry lifting and the structural element of the coverings. An intriguing by-formula in total derivatives is obtained. Mathematics Subject Classification (2000): 35Q53, 58J72, 32G08, 13N15.

This paper illustrates new cohomological concepts [1, 2] in the theory of Bäcklund transformations between PDE. Notation and all definitions follow [1, 3]. First, let us introduce the notion of Bäcklund (auto)transformation in terms of coverings over PDE.

1. Let E_{i} ⊂ J^{k_{i}}(π_{i}), i = 1, 2, be two differential equations and τ_{i}: Ẽ → E_{i} be coverings with the same total space Ẽ. Then the diagram

E_{1}^{∞} ⟵^{τ_{1}} Ẽ ⟶^{τ_{2}} E_{2} (1)

is called a Bäcklund transformation B(Ẽ, τ_{i}, E_{i}) between the equations E_{i}. Diagram (1) is called a Bäcklund autotransformation if E_{1}^{∞} = E_{2}^{∞} = E^{∞}.

Remark 1. Let τ_{j}: Ẽ_{j} → E_{j}, j = 1, 2, be two coverings and μ: Ẽ_{1} → Ẽ_{2} be a diffeomorphism that maps the Cartan distribution C_{τ_{1}}D(Ẽ_{1}) into C_{τ_{2}}D(Ẽ_{2}). Then the diagram B(Ẽ_{j}, τ_{1}, τ_{2} ∘ μ, E_{j}) is also a Bäcklund transformation between the equations E_{j}.

Remark 2. Let τ: Ẽ → E^{∞} be a covering and μ: Ẽ → Ẽ be a nontrivial diffeomorphism of manifolds preserving Cartan distribution, e.g., a discrete symmetry that cannot be restricted to E^{∞}. Then the diagram

\begin{matrix}Ẽ ⟶^{μ} Ẽ, \\ _{τ}↘ ↙_{τ} \\ E^{∞} \end{matrix} (2)

is also Bäcklund autotransformation for E. In the sequel, this construction is used for generating Bäcklund autotransformations for the Liouville equation.

2. The hyperbolic Liouville equation E is

E = {F ≡ u_{xy} − exp(2u) = 0}. (3)

Consider the covering structure in τ_{t}: Ẽ_{t} → E^{∞} provided by the extended total derivatives

D̃_{x} = D̄_{x} + ũ_{x} ∂/∂ũ, D̃_{y} = D̄_{y} + ũ_{y} ∂/∂ũ, [D̃_{x}, D̃_{y}] = 0, (4) if the partial derivatives of the nonlocal variable ũ w.r.t. x and y are

(5)  ũ_{x} = u_{x} + exp(−t) · exp(ũ + u),  ũ_{y} = −u_{y} + 2 exp(t) · sinh(ũ − u).

Let the diffeomorphism μ be the swapping u ↔ ũ of the fiber variable u and the nonlocal variable ũ combined with x ↦ −x and y ↦ −y. Then diagram (2) determines Bäcklund autotransformation B(Ẽ_{t}, τ_{t}, τ_{t} ∘ μ, E) for Eq. (3), and for the Liouville equation, the equations Ẽ_{t} of Bäcklund autotransformation [4] are

(6)  (ũ − u)_{x} = exp(−t) · exp(ũ + u),
(7)  (ũ + u)_{y} = 2 exp(t) · sinh(ũ − u).

Proposition 1. Let t_{1} ≠ t_{2}. Then the coverings τ_{t_{1}} and τ_{t_{2}} are non-equivalent.

Proof. Consider the Whitney sum τ_{t_{1}} ⊕ τ_{t_{2}} of the coverings τ_{t_{1}} and τ_{t_{2}} with nonlocal variables ũ and ũ′ respectively. These coverings are equivalent if their Whitney sum is reducible. Following [1], we verify this condition in local coordinates and prove that the system

(8)  (D̄_{x} + (u_{x} + exp(ũ + u − t_{1})) \frac{∂}{∂ũ} + (u_{x} + exp(ũ′ + u − t_{2})) \frac{∂}{∂ũ′} ) ϕ = 0,

(D̄_{y} + (−u_{y} + 2 exp(t_{1}) sinh(ũ − u)) \frac{∂}{∂ũ} + (−u_{y} + 2 exp(t_{2}) sinh(ũ′ − u)) \frac{∂}{∂ũ′} ) ϕ = 0 (9)

has only constant solution ϕ = const for t_{1} ≠ t_{2}. Put u_{k} ≡ ∂^{k}u/∂x^{k} and u\overline{k} ≡ ∂^{k}u/∂y^{k} for any k ∈ ℕ. The characteristics of Eq. (8) are

\frac{dũ}{u_{x} + exp(ũ + u − t_{1})} = \frac{dũ′}{u_{x} + exp(ũ′ + u − t_{2})} = \frac{du}{u_{1}} = \frac{du_{k}}{u_{k+1}} = dq_{1},

\frac{dũ}{−u_{y} + 2 exp(t_{1}) sinh(ũ − u)} = \frac{dũ′}{−u_{y} + 2 exp(t_{2}) sinh(ũ′ − u)} = \frac{du}{u_{1}} = \frac{du\overline{k}}{u\overline{k+1}} = dq_{2},

k ∈ ℕ. We have

dq_{1} = \frac{d(ũ − u)}{exp(ũ + u − t_{1})} = \frac{d(ũ′ − u)}{exp(ũ′ + u − t_{2})},

whence

\frac{d(ũ − ũ′)}{exp(ũ − t_{1}) − exp(ũ′ − t_{2})} = 0.

Consequently, ũ′ − ũ = δ = const and δ depends at most on t_{1} and t_{2}. Then we have ũ_{x} = ũ′_{x} and ũ_{y} = ũ′_{y} as equations for δ. The first equation implies δ = t_{1} − t_{2} and the second one gives

exp(δ) sinh(ũ′ + δ − u) = sinh(ũ′ − u).

Using hyperbolic-trigonometry formulas, we obtain

exp(δ) sinh(ũ′ − u) cosh(δ) + exp(δ) cosh(ũ′ − u) sinh(δ) = sinh(ũ′ − u).

Note that exp(δ) cosh(δ) − 1 = exp(δ) sinh(δ). Finally, we emphasize that the equation sinh(ũ′ − u) = cosh(ũ′ − u) has no roots in ℂ. Therefore, system (8)-(9) has only constant solution ϕ = const for t_{1} ≠ t_{2}.

Proposition 1 means that the parameter t cannot be eliminated by a coordinate transformation if the later does not depend on t itself.

3. Consider the scaling symmetry of base coordinates

X^{0} = −x \frac{∂}{∂x} + y \frac{∂}{∂y} .

It corresponds to the generating function φ = xu_{x} − yu_{y}. The scaling symmetry can be extended to the entire E^{∞}:

X̂ = −x \frac{∂}{∂x} + y \frac{∂}{∂y} + ∑_{k≥1} ku_{k} \frac{∂}{∂u_{k}} − ∑_{k≥1} ku\overline{k} \frac{∂}{∂u\overline{k}} . (10)

Remark 3. The diffeomorphisms A_{t} = exp(tX̂) form an abelian group. Really, we see that x(t) = exp(−t) x(0), y(t) = exp(t) y(0), u(t) = u(0), u_{k}(t) = exp(kt) u_{k}(0), u\overline{k}(t) = exp(−kt) u\overline{k}(0), k ≥ 1. Obviously, A_{0} = id and A_{t_{1}} ∘ A_{t_{2}} = A_{t_{2}} ∘ A_{t_{1}} = A_{t_{1}+t_{2}}.

Proposition 2. The symmetry X̂ cannot be extended to a symmetry of the covering equation Ẽ_{t}.

Proof. Assume the converse. By ˜϶_{φ} we denote the evolutionary vector field ∑_{σ} D̃_{σ}(φ) · ∂/∂u_{σ} on Ẽ_{t}, φ ∈ C^{∞}(Ẽ_{t}), and ℓ̃_{F} (φ) = ˜϶_{φ}(F). Suppose there is a smooth function a ∈ C^{∞}(Ẽ_{t}) such that the system

ℓ̃_{F} (φ) = 0,  D̃_{x^{i}}(a) = ˜϶_{φ,a}(ũ_{x^{i}}) ≡ (˜϶_{φ} + a ∂/∂ũ) (ũ_{x^{i}}),  x^{1} ≡ x, x^{2} ≡ y, (11)

holds. This means that the field ˜϶_{φ,a} is a local symmetry of the covering equation Ẽ_{t} and X̂ is constructively extended onto Ẽ_{t}. Nevertheless, system (11) is not compatible since

D̃_{x} ∘ D̃_{y}(a) ≠ D̃_{y} ∘ D̃_{x}(a).

Really, D̃_{x} ∘ D̃_{y}(a) − D̃_{y} ∘ D̃_{x}(a) does not depend on a at all and equals

xu_{x}^{2} exp(t + u − ũ) + u_{x}yu_{y} exp(t + ũ − u) − xu_{x} exp(2ũ) − u_{x}yu_{y} exp(t + u − ũ)

−2yu_{y} exp(2t + ũ + u) + 2xu_{x} exp(2t + ũ + u) − xu_{x}^{2} exp(t + ũ − u) + x exp(2u)u_{x}

−y exp(2u)u_{y} + 2 exp(t)xu_{x}^{2} + yu_{y} exp(2ũ) − 2 exp(t)yu_{y}u_{x} ≠ 0.

This contradiction concludes the proof.

Thus, the scaling symmetry X̂ is a τ_{t}-shadow only and X̂ provides a family of covering equations Ẽ_{t} over E^{∞}, Ẽ_{t} parametrized by t ∈ ℝ.

4. In local coordinates, the structural element U_{t} of the covering diffiety Ẽ_{t} is

(12) U_{t} = ∑_{σ} d_{C}(u_{σ}) ⊗ \frac{∂}{∂u_{σ}} + (dũ − (u_{x} + exp(ũ + u − t)) dx +

+ (u_{y} − 2 exp(t) sinh(ũ − u)) dy) ⊗ \frac{∂}{∂ũ},

where d_{C} is the Cartan differential, d_{C}u_{σ} = du_{σ} − ∑_{i} u_{σ+1_{i}} dx^{i}.

By [[·, ·]]^{FN} we denote the Frölicher-Nijenhuis bracket [3, 2]: [[Ω, Θ]]^{FN}(f) = L_{Ω}(Θ(f)) −(−1)^{μν} · L_{Θ}(Ω(f)), where Ω, Θ ∈ D(Λ^{∗}(E)), f ∈ C^{∞}(E), degrees μ, ν are μ = deg Ω, ν = deg Θ; L_{Ω} = [i_{Ω}, d]: Λ^{k}(E) → Λ^{k+deg Ω}(E) is the Lie derivative, and i_{Ω} : Λ^{k}(E) → Λ^{k+deg Ω−1}(E) is the inner product.

Theorem ([2]). Let τ: Ẽ → E be a covering and A_{t}: Ẽ → Ẽ be a smooth family of diffeomorphisms such that A_{0} = id and τ_{t} = τ ∘ A_{t}: Ẽ → E is a covering for any t ∈ ℝ. Then the structural element U_{τ_{t}} evolves by

\frac{dU_{τ_{t}}}{dt} = [[X̂_{t}, U_{τ_{t}}]]^{FN}, (13)

where X̂_{t} is a τ_{t}-shadow for any t ∈ ℝ.

If the covering τ_{t} corresponds to Bäcklund autotransformation (6)-(7) for the Liouville equation (3), then we have

(14) \frac{dU_{t}}{dt} = (exp(ũ + u − t) dx − 2 exp(t) sinh(ũ − u) dy) ⊗ \frac{∂}{∂ũ}.

Now we claim that the scaling symmetry X̂ is the τ_{t}-shadow such that Eq. (13) holds, where U_{τ_{t}} is the structural element U_{t} (12) of the covering τ_{t} defined in (4) and U_{t} evolves according to (14). We need Lemmas 1-4 to prove it.

We say that i is the degree of Ω, if Ω ∈ D(Λ^{i}(Ẽ)). If Ẽ is a finite-dimensional manifold, we have an isomorphism D(Λ^{∗}(Ẽ)) ≃ Λ^{∗}(Ẽ) ⊗ D(Ẽ) and thus any derivation Ω ∈ D(Λ^{∗}(Ẽ)) is representable as a finite sum, the summands are Ω = ω ⊗ X, where ω ∈ Λ^{∗}(Ẽ) and X ∈ D(Ẽ). For such elements, the Frölicher-Nijenhuis bracket is

(15) [[ω ⊗ X, θ ⊗ Y]]^{FN} = ω ∧ θ ⊗ [X, Y] + ω ∧ L_{X}(θ) ⊗ (Y) + (−1)^{i} dω ∧ (X θ) ⊗ Y −

− (−1)^{ij} θ ∧ L_{Y} (ω) ⊗ X − (−1)^{(i+1)j} dθ ∧ (Y ω) ⊗ X,

where X, Y ∈ D(Ẽ), ω ∈ Λ^{i}(Ẽ) and θ ∈ Λ^{j}(Ẽ).

Lemma 1. [[X̂, U_{t}]]^{FN} dũ = (dU_{t}/dt) dũ.

Lemma 2. [[X̂, U_{t}]]^{FN} dx = [[X̂, U_{t}]]^{FN} dy = [[X̂, U_{t}]]^{FN} du = 0.

Proof. The proof of Lemmas 1 and 2 is staightforward by multiple using of (15) and thus omitted.

However, the calculation of coefficients of [[X̂, U_{t}]]^{FN} at ∂/∂u_{k} or ∂/∂u\overline{k}, k ≥ 1, is not so transparent.

Remark 4. Let u(x) and f(u) be smooth functions, D_{x} be the total derivative w.r.t. x; take an integer n > 0 and a positive integer l ≤ n − 1. Then

D_{x} (\frac{∂}{∂u_{l}} D_{x}^{n−1}(f(u))) = \frac{∂}{∂u_{l}} D_{x}^{n}(f(u)) − \frac{∂}{∂u_{l−1}} D_{x}^{n−1}(f(u)). (16)

Corollary 1. Moreover,

(n + 1)u_{n+1} \frac{∂}{∂u_{n+1}} D_{x}^{n+1}(f(u)) = (n + 1)u_{n+1} \frac{∂}{∂u_{n}} D_{x}^{n}(f(u)) = (n + 1)u_{n+1} · f′(u). (17)

Lemma 3. Let u(x) and f(u) be smooth functions, D_{x} be the total derivative w.r.t. x. Denote u_{k} ≡ D_{x}^{k}(u(x)), k ≥ 0, u_{0} ≡ u. Then for any integer n ≥ 0 the relation

n · D_{x}^{n}(f(u)) = ∑_{m=1}^{n} m u_{m} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) (18)

holds.

Proof. We prove (18) by induction on n. For n = 0 relation (18) is valid. For n ≥ 0 one has

(n + 1) D_{x}^{n+1}(f(u)) = D_{x}(n D_{x}^{n}(f(u)) + D_{x}^{n}(f(u))) =

by the inductive assumption,

= D_{x} (∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) + D_{x}^{n}(f(u))) =

by the Leibnitz rule,

= ∑_{m=1}^{n} mu_{m+1} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) + ∑_{m=1}^{n} mu_{m} D_{x} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) + D_{x} D_{x}^{n}(f(u)) =

by (16) applied to the second sum,

= ∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n+1}(f(u)) + ∑_{m=1}^{n} mu_{m+1} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) −

− ∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m−1}} D_{x}^{n}(f(u)) + D_{x}D_{x}^{n}(f(u)) =

by the definition of D_{x} and the subscript shift in the latter sum,

= ∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n+1}(f(u)) + ∑_{m=0}^{n}(m + 1)u_{m+1} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) −

− ∑_{m=0}^{n−1}(m + 1)u_{m+1} \frac{∂}{∂u_{m}} D_{x}^{n}(f(u)) =

since almost all summands in the latter two sums coincide,

= ∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n+1}(f(u)) + (n + 1)u_{n+1} \frac{∂}{∂u_{n}} D_{x}^{n}(f(u)) =

by (17),

= ∑_{m=1}^{n} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n+1}(f(u)) + (n + 1)u_{n+1} \frac{∂}{∂u_{n+1}} D_{x}^{n+1}(f(u)) =

= ∑_{m=1}^{n+1} mu_{m} \frac{∂}{∂u_{m}} D_{x}^{n+1}(f(u)). Q. E. D.

Lemma 4. [[X̂, U_{t}]]^{FN} du_{k} = [[X̂, U_{t}]]^{FN} du\overline{k} = 0, k ≥ 1.

Proof. Let k be a positive integer. Taking (15) into account, consider the 1-form

[[X̂, U_{t}]]^{FN} du_{k} = ((k − 1)D̄_{y}u_{k} − ∑_{l=1}^{k−1} lu_{l} \frac{∂}{∂u_{l}} D̄_{x}^{k−1}(exp(2u))) · dy;

we also see that all coefficients of dx, du, du_{l}, du\overline{l} vanish for all l ≥ 1. Finally, note that D̄_{y}u_{k} = D̄_{x}^{k−1}(exp(2u)). By Lemma 3, the coefficient of dy is trivial. Arguing as above, we see that [[X̂, U_{t}]]^{FN} du\overline{k} = 0. This completes the proof.

Theorem 1. The τ_{t}-shadow (10) satisfies the relation

[[X̂, U_{t}]]^{FN} = (exp(ũ + u − t) dx − 2 exp(t) sinh(ũ − u) dy) ⊗ \frac{∂}{∂ũ},

i.e., the group of diffeomorfisms A_{t} = exp(tX̂) provides smooth one-parametric family (5) of non-equivalent one-dimensional non-abelian coverings over Liouville’s equation (3). These coverings correspond to Bäcklund autotransformations for the Liouville equation. Bäcklund transformations are given by diagram (2). Infinitesimal part of the structural element is given by (14).

Final remark. The reader will have no difficulty in showing that quite analogous theorems are valid for Bäcklund transformations between Liouville’s equation (3) and the wave equation v_{xy} = 0:

(v − u)_{x} = exp(−t) exp(u + v), (v + u)_{y} = − exp(t) exp(u − v), t ∈ ℝ,

and Bäcklund transformations between the Liouville equation and the scal^{+}-Liouville equation Υ_{xy} = exp(−2Υ):

(Υ − u)_{x} = 2 exp(−t) cosh(Υ + u), (Υ + u)_{y} = − exp(t) exp(u − Υ), t ∈ ℝ.

Scaling symmetry (10) is the required τ_{t}-shadow in both cases, and Lemma 3 is an efficient tool to prove these theorems.

Problems of direct integration in nonlocal variables of these Bäcklund (auto)transformations and their permutability will be the object of another paper.

Acknowledgement. The author would like to express his gratitude to Prof. I. S. Krasil’shchik for his essential remarks and constructive criticism. The author thanks A. V. Ovchinnikov for improving the readability of this paper.

References

[1] 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.

[2] S.Igonin and I.S.Krasil’shchik, Advanced Studies in Pure Mathematics, Mathematical Society of Japan, (2001) (to appear). See also arXiv:nlin.SI/0010040.

[3] I.S. Krasil’shchik and P.H.M. Kersten Symmetries and recursion operators for classical and supersymmetric differential equations (Dordrecht etc.: Kluwer Acad. Publ., 2000).

[4] R.K.Dodd and R.K.Bullough Proc. Roy. Soc., London. A351 (1976).

Lomonosov Moscow State University, Faculty of Physics, Department of Mathematics, Moscow, Russia., Independent University of Moscow, Math Colledge, Moscow, Russia.

  E-mail address: arthemy@mccme.ru