The Diffiety Institute Preprint Series

Preprint DIPS–1/2003 February 8, 2003

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

The Diffiety Institute

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

Homogeneous spaces associated with Miura transformations

by

Sergei Igonin

Homogeneous spaces associated with Miura transformations

SERGEI IGONIN

Abstract. For scalar evolution equations, we prove that a zero-curvature representation with values in a Lie algebra 𝔤 and a certain type homogeneous space of 𝔤 determine a Miura type transformation. Using this result, we show how to construct and classify Miura type transformations for a given equation.

Recall that an action of a Lie algebra 𝔤 on a manifold W is a homomorphism ρ: 𝔤 → D(W) to the Lie algebra D(W) of vector fields on W. The action is said to be transitive if for each point a ∈ W the mapping

𝔤 → T_{a}W, v ↦ ρ(v)_{a}

is surjective. In this case W is called a homogeneous space of 𝔤. Two actions ρ_{i}: 𝔤 → D(W_{i}), i = 1, 2, are said to be isomorphic if there is a diffeomorphism φ: W_{1} → W_{2} such that ρ_{2} = φ_{∗}ρ_{1}.

Below our considerations are always local. The results are valid in both categories of smooth and analytic manifolds. Depending on the category considered, all functions are supposed to be smooth or analytic.

Consider two scalar evolution equations

u_{t} = P (u, u_{1}, …, u_{p}), u_{k} = \frac{∂^{k}u}{∂x^{k}}, (1)

v_{t} = R(v, v_{1}, …, v_{r}), v_{k} = \frac{∂^{k}v}{∂x^{k}} . (2)

and a transformation

(3)  u = S(v, v_{1}, …, v_{n})

such that for any solution v(x, t) of (2) function (3) satisfies (1). Such transformations are called Miura type transformations (MT in short) by analogy with the famous Miura transformation connecting the KdV and the modified KdV equations. The maximal integer n such that (3) depends nontrivially on v_{n} is called the order of the MT.

In this paper we develop a description of MTs in terms of homogeneous spaces of Lie algebras.

Introduce new variables

(4)  w^{i} = ∂^{i−1}v/∂x^{i−1}, i = 1, …, n,

and rewrite system (2), (3) as follows

(5)  \begin{aligned}\frac{∂w^{i}}{∂x} & = w^{i+1}, i = 1, …, n − 1, \\ \frac{∂w^{n}}{∂x} & = a(w^{1}, …, w^{n}, u), \\ \frac{∂w^{i}}{∂t} & = b^{i}(w^{1}, …, w^{n}, u, …, u_{p−1}), i = 1, …, n. \end{aligned}

where p is the order of (1). And vice versa, it is easily seen that any consistent system of this form with

(6)  \frac{∂}{∂u}a(w^{1}, …, w^{n}, u) ≠ 0

determines a MT of order n for (1) as follows:

• substitute (4) to (5),

• from equation (5) express u = S(v, v_{1}, …, v_{n}),

• let D = ∑_{i≥0} v_{i+1}∂/∂v_{i}, then equation (2) is given by

v_{t} = b^{1}(v, v_{1}, …, v_{n−1}, S, D(S), …, D^{p−1}(S)).

Consider the total derivative operators

D_{x} = \frac{∂}{∂x} + ∑_{j≥0} u_{j+1} \frac{∂}{∂u_{j}} ,

D_{t} = \frac{∂}{∂t} + ∑_{j≥0} D_{x}^{j}(P (u, u_{1}, …, u_{p}))\frac{∂}{∂u_{j}} ,

and more general overdetermined systems

(7)  \begin{aligned}\frac{∂w^{i}}{∂x} & = a^{i}(w^{1}, …, w^{n}, u), i = 1, …, n, \\ \frac{∂w^{i}}{∂t} & = b^{i}(w^{1}, …, w^{n}, u, …, u_{p−1}), i = 1, …, n. \end{aligned}

consistent modulo (1). Clearly, an invertible change of variables

w^{i} ↦ f^{i}(w^{1}, …, w^{n})

leads to a new system of the form (7). Two systems related by such a change of variables are said to be equivalent.

System (7) is completely determined by the vector fields

A = ∑_{i=1}^{n} a^{i}(w^{1}, …, w^{n}, u)\frac{∂}{∂w^{i}} , B = ∑_{i=1}^{n} b^{i}(w^{1}, …, w^{n}, u, …, u_{p−1})\frac{∂}{∂w^{i}} .

Consistency of (7) modulo (1) is equivalent to the equation

(8)  [D_{x} + A, D_{t} + B] = 0.

Remark 1. Coordinate-independent study of such objects is performed in [1, 6] in the framework of the theory of coverings of PDEs. Our results are considerably inspired by this theory. Another source of the present ideas is the paper [3], which announces some remarkable results on MTs for the KdV equation.

Recall that two functions

(9)  M(u), N(u, …, u_{p−1})

with values in a Lie algebra 𝔤 constitute a zero-curvature representation (ZCR in short) for (1) if

(10)  [D_{x} + M, D_{t} + N] = D_{x}N − D_{t}M + [M, N] = 0

Then each action ρ: 𝔤 → D(W) and a choice of local coordinates w^{1}, …, w^{n} in W determine a consistent system of the form (7) with A = ρ(M) and B = ρ(N), since equation (8) follows from (10). Clearly, different choices of coordinates in W or isomorphic actions determine equivalent systems (7).

For each k ∈ ℕ we define the subalgebra 𝔤_{k} of 𝔤 by induction on k as follows: • 𝔤_{0} = 0,

• 𝔤_{1} is the subalgebra generated by all the elements

M(u) − M(u′) ∈ 𝔤,

• 𝔤_{k+1} is generated by the subspaces 𝔤_{k} and [𝔤_{k}, M(u)]. Set also 𝔤̃ = ∪_{k≥0}𝔤_{k}.

Let us present the main result of this paper.

Theorem 1. Consider ZCR (9) with values in 𝔤 and an action ρ: 𝔤 → D(W), where dim W = n. The corresponding system (7) is equivalent to a system of the form (5) with (6) (i.e., determines a MT for (1)) if and only if the following conditions hold.

(1) The subalgebra ρ(𝔤̃) acts on W transitively.

(2) The subalgebra ρ(𝔤_{n−1}) acts on W nontransitively. In this case a nonconstant function w on W invariant under ρ(𝔤_{n−1}) is unique up to a change w ↦ g(w). The functions

(11)  w^{i} = ρ(M(u))^{i−1}(w), i = 1, …, n,

do not depend on u and are local coordinates in which system (7) constructed by ρ takes the desired form (5), (6).

Proof. If the system constructed by ρ is of the form (5), (6) then by the definition of 𝔤_{k} we obtain that the image of 𝔤_{k} in each tangent space is spanned by ∂/∂w^{n−k+1}, …, ∂/∂w^{n}. This obviously implies Conditions 1 and 2.

Conversely, let ρ satisfy Conditions 1, 2. Consider a generic point a ∈ W and the local orbits O_{k} ⊂ W of this point under the action of ρ(𝔤_{k}). Conditions 1, 2 say that

(12)  dim O_{n−1} < n, ∃ c : O_{k} is open in W ∀ k ≥ c.

On the other hand, from the definition of 𝔤_{k} it follows easily that if U ∩O_{m} = U ∩O_{m+1} for some m ≥ 0 and some neighbourhood U of a then U ∩ O_{k} = U ∩ O_{m} for all k ≥ m.

Combining this with (12), we obtain

(13)  a = O_{0} ⊊ O_{1} ⊊ ⋯ ⊊ O_{n}, dim O_{k} = k, k = 0, 1, …, n.

Therefore, there is a nonconstant function w invariant under ρ(𝔤_{n−1}), and it is unique up to a change w ↦ g(w). Consider now functions (11). Using (13) and the definition of 𝔤_{k}, by induction on k one proves

(14)  ρ(𝔤_{n−k})(w^{k}) = 0, ρ(𝔤_{n−k+1})(w^{k}) ≠ 0, k = 1, …, n.

In particular, w^{1}, …, w^{n−1} are invariant under ρ(𝔤_{1}), which implies that w^{1}, …, w^{n} do not depend on u. Combining (13) and (14), we obtain that w^{1}, …, w^{n} form a system of local coordinates in W. It is easily seen that system (7) constructed by ρ is of the form (5), (6) in these coordinates. □

Corollary 1. If 𝔤_{m} = 𝔤_{m+1} for some m ≥ 0 then ZCR (9) cannot produce MTs of order greater than m.

Proof. By Theorem 1, a MT of order n is determined by a transitive action ρ of 𝔤̃ such that ρ(𝔤_{n−1}) is not transitive. Since in our case 𝔤̃ = 𝔤_{k} for any k ≥ m, there are no such actions for n > m. □

It can be shown that for any equation (1) there is a universal ZCR with values in the (possibly infinite-dimensional) Wahlquist-Estabrook Lie algebra 𝔴𝔢 of (1) such that any consistent system (7) arises from an action of 𝔴𝔢 (see [2, 8, 6] and references therein). In particular, by Theorem 1, all MTs of (1) are determined by some transitive actions of 𝔴𝔢.

There is an algorithmic procedure to find the universal ZCR for a given equation (1), see [2, 6]. However, in this way the Lie algebra 𝔴𝔢 is described in terms of generators and relations, which makes it difficult to study transitive actions of 𝔴𝔢. Using heavy computations, for a number of equations the explicit structure of 𝔴𝔢 in terms of wellknown Lie algebras was described (see [8] and references therein). It is shown in [5] how these computations can be simplified.

Example 1. According to [8], for the KdV equation u_{t} = u_{3} + u_{1}u we have

𝔴𝔢 ≅ H ⊕ 𝔰𝔩_{2}(ℂ) ⊗_{ℂ} ℂ[λ]

where H is the 5-dimensional nilpotent Heisenberg algebra with the basis r_{i}, i = −2, −1, 0, 1, 2 and the commutator table

[r_{−1}, r_{1}] = [r_{2}, r_{−2}] = r_{0}, [r_{i}, r_{j}] = 0 ∀ i + j ≠ 0.

The universal ZCR is M(u) = X_{1} + \frac{1}{3}uX_{2} + \frac{1}{6}u^{2}X_{3}, where

X_{1} = r_{1} − \frac{1}{2}y + \frac{1}{2}λz, X_{2} = r_{−1} + z, X_{3} = r_{−2},

and h, y, z is a basis of 𝔰𝔩_{2} with the relations [h, y] = 2y, [h, z] = −2z, [y, z] = h. Here the form of N(u, u_{1}, u_{2}) in (9) is not important for us.

We have

𝔤_{1} = ⟨X_{2}, X_{3}⟩, 𝔤_{2} = ⟨r_{−2}, r_{−1}, z, 2r_{0} − h⟩,

and

𝔤_{3} = 𝔤_{k} = 𝔤̃ = ⟨𝔰𝔩_{2} ⊗ ℂ[λ], r_{−2}, r_{−1}, r_{0}, r_{1}⟩ ∀ k ≥ 3.

By Corollary 1, any MT of the KdV equation is of order not greater than 3. A weaker fact that each MT is reduced to a MT of order not greater than 3 ‘by introduction of a potential’ was announced in [3].

Let us explain how our method of constructing MTs includes the one of [3].

Consider the part of this ZCR with values in 𝔣 = 𝔰𝔩_{2}(ℂ) ⊗_{ℂ} ℂ[λ]. We have

𝔣_{1} = ⟨z⟩, 𝔣_{2} = ⟨z, h⟩, 𝔣_{3} = 𝔣̃ = 𝔣.

By Theorem 1, each homogeneous space of 𝔣 of dimension ≤ 3 determines a MT for the KdV equation. Conditions 2 is satisfied, since dim 𝔣_{k} = k for k = 0, 1, 2.

According to [5], for a transitive action ρ: 𝔣 → D(W) the image ρ(𝔣) is finitedimensional and is of the form

(15)  𝔰𝔩_{2} ⊗ ℂ[λ]/(f(λ)), f(λ) ∈ ℂ[λ].

Let

f(λ) = a ∏_{s=1}^{k}(λ − e_{s})^{k_{s}}, a, e_{s} ∈ ℂ, a ≠ 0, e_{i} ≠ e_{j} ∀ i ≠ j.

Then Lie algebra (15) is isomorphic to

(16)  ⨁_{s=1}^{k} 𝔰𝔩_{2} ⊗ ℂ[λ]/(λ^{k_{s}}).

The Lie groups

∏_{s} SL_{2}(ℂ[λ]/(λ^{k_{s}})).

that appear in [3] have (16) as their Lie algebras. Thus construction of MTs is reduced to local description of homogeneous spaces of dim ≤ 3 of these Lie groups. This description and the corresponding MTs are presented in [3].

Example 2. The equation

(17)  u_{t} = u_{5} + 10uu_{3} + 25u_{1}u_{2} + 20u^{2}u_{1}

admits the following Lax pair [4]

L_{t} = [L, A], (18)

L = D_{x}^{3} + 2uD_{x} + u_{1},
A = 9D_{x}^{5} + 30uD_{x}^{3} + 45u_{1}D_{x}^{2} + (20u^{2} + 35u_{2})D_{x} + (10u_{3} + 20uu_{1}).

Let us show how this Lax pair leads to a ZCR and MTs for (17). As usual, equation (18) is equivalent to consistency of the following system

(19)  Lψ = λψ, ψ_{t} = −Aψ,

where λ is a parameter. Introduce new variables

(20)  q^{1} = ψ, q^{2} = ψ_{x}, q^{3} = ψ_{xx} + uψ

and rewrite (19) as follows

(21)  Q_{x} = −MQ, Q_{t} = −NQ,

where

Q = (\begin{matrix}q^{1} \\ q^{2} \\ q^{3} \end{matrix}) , M = (\begin{matrix}0 & −1 & 0 \\ −u & 0 & −1 \\ λ & −u & 0 \end{matrix}) ,

N = (\begin{matrix}u_{3} + 8uu_{1} + 3uλ & −u_{2} − 4u^{2} & 9λ \\ f + 3u_{1}λ & −6uλ & −u_{2} − 4u^{2} \\ 5u^{2}λ + 2u_{2}λ & f − 3u_{1}λ & 3uλ − u_{3} − 8uu_{1} \end{matrix}) ,

f = u_{4} + 8u_{1}^{2} + 9uu_{2} + 9λ^{2} + 4u^{3}.

Let us treat λ as a formal variable. Then the matrices M, N constitute a ZCR (9) with values in 𝔰𝔩_{3}(ℂ) ⊗_{ℂ} ℂ[λ]. More precisely, M and N take values in the subalgebra 𝔤 ⊂ 𝔰𝔩_{3} ⊗ ℂ[λ] generated by the two matrices

(22)  U = (\begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) , V = (\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ −λ & 0 & 0 \end{matrix}) .

The subalgebra 𝔤_{1} is spanned by the matrix U, the subalgebra 𝔤_{2} has the basis

U, [U, V] = (\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & −1 \end{matrix}) ,

and 𝔤_{3} = 𝔤. By Corollary 1, this ZCR leads to MTs of order not greater than 3. By Theorem 1, every homogeneous space W of 𝔤 with dim W ≤ 3 indeed determines a MT of order dim W. Conditions 2 is satisfied, since dim 𝔤_{k} = k for k = 0, 1, 2.

For example, substituting λ = 0 we obtain an epimorphism φ: 𝔤 → 𝔰𝔩_{2}(ℂ). The canonical transitive action of 𝔰𝔩_{2} on the projective line determines the MT [4]

u = −v_{1} − \frac{1}{2}v^{2}, v_{t} = v_{5} − 5(v_{1}v_{3} + v_{2}^{2} + v_{1}^{3} + 4vv_{1}v_{2} + v^{2}v_{3} − v^{4}v_{1}).

Remark 2. It is easily seen that for any Lax pair L_{t} = [L, A] such that L depends on u, u_{1} only and is linear in u_{1} there is a similar to (20) change of variables that transforms (19) to a ZCR (21) of the form (9). Therefore, each Lax pair of the described type leads to MTs.

Remark 3. A ZCR of the form (9) with values in a Lie algebra 𝔤 is equivalent to a homomorphism from the Wahlquist-Estabrook algebra to 𝔤. Therefore, we have a homomorphism from the Wahlquist-Estabrook algebra of (17) onto the algebra generated by (22). We conjecture that the kernel of this homomorphism is nilpotent. Then any MT of (17) is obtained from the MTs described here by introduction of a potential in the sense of [3].

References

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[5] S. Igonin. Coverings and the fundamental group for KdV type equations. nlin.SI/0301042

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[8] H. N. van Eck. A non-Archimedean approach to prolongation theory. Lett. Math. Phys. 12 (1986), 231–239.

Independent University of Moscow and University of Twente Current address: University of Twente, Department of Mathematics, P.O. Box 217, 7500 AE En-

schede, The Netherlands

 E-mail address: igonin@mccme.ru