Sine-Gordon equation and representations of affine Kac-Moody algebra ŝl_{2}.

Yuly Billig

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, Canada E3B 5A3

We give a representation-theoretic interpretation of the sine-Gordon equation using the action of affine Kac-Moody algebra ŝl_{2} on the Weyl algebra. In this setup τ - functions become functions in non-commuting variables. The skew Casimir operators that we introduce here, give rise to a hierarchy of partial differential equations that includes the sine-Gordon equation and two copies of the Korteweg-de Vries equation.

0. Introduction

One of the most fascinating applications of the Kac-Moody theory is the use of affine Lie algebras and their groups to exhibit hidden symmetries of soliton equations. In 1981 M. Sato [20] and Drinfeld-Sokolov [6] (see also [7]) discovered fundamental links between soliton equations and infinite-dimensional Lie groups. In an important sequence of papers [2], [3], [4], Date, Jimbo, Kashiwara and Miwa gave a construction of the Kadomtsev-Petviashvili and Korteweg-de Vries hierarchies based on the representation theory of affine Kac-Moody algebras. The affine algebra a_{∞} produces the KP hierarchy, while the KdV hierarchy is linked to ŝl_{2}.

Date, Jimbo, Kashiwara and Miwa used the vertex operator realization of the basic highest weight module, which was discovered earlier by Lepowsky and Wilson [18]. In this realization for ŝl_{2}, the space of the basic module is identified with the polynomial algebra in infinitely many variables x = (x_{1}, x_{3}, x_{5}, …). A completion of this space is interpreted as the space of τ-functions. In this framework, the Bäcklund transformation that raises the soliton number of an N-soliton τ-function, is given by the exponential of a vertex operator.

Kac and Wakimoto [16] discovered that the Casimir operator equation

Ω(τ ⊗ τ) = 0  (0.1)

decomposes in a hierarchy of PDEs in Hirota form.

The sine-Gordon equation

u_{xt} = sin(u)  (0.2)

is of great significance to both mathematics and physics. This equation appeared first in geometry and describes the surfaces of constant negative curvature in ℝ^{3}. The physical importance of the sine-Gordon equation is related to the fact that it is a soliton equation which is manifestly Lorentz-invariant.

The sine-Gordon equation played a special role in the development of the soliton theory. The Bäcklund transformation method, which is a precursor for the Lie theory approach, was first developed in the geometric setup for this equation. Before the connection between soliton equations and infinite-dimensional Lie algebras was discovered in 1980’s, Mandelstam wrote a paper [19] on quantized sine-Gordon equation, in which he used operators somewhat similar to the vertex operators.

From the AKNS method [1] it is known that the sine-Gordon equation is related to ŝl_{2}, but the representation-theoretic interpretation of this equation was missing.

The main goal of the present paper is to fill this gap. Our starting point is the following observation. From the original Hirota’s paper [13] we know that the sine-Gordon equation (0.2) can be written as a system of equations in Hirota form on a pair of functions (τ_{0}, τ_{1})

\{ \begin{matrix}D_{x}D_{t}(τ_{0} ∘ τ_{0} − τ_{1} ∘ τ_{1}) = 0 \\ D_{x}D_{t}(τ_{0} ∘ τ_{1}) = τ_{0}τ_{1}, \end{matrix}  (0.3)

where u is related to (τ_{0}, τ_{1}) by u = 4 arctan (\frac{τ_{1}}{τ_{0}} ). To describe the soliton solutions in terms of τ-functions, we to put τ_{0} and τ_{1} together into τ = τ_{0} + iτ_{1}. Then the N-soliton solution for the sine-Gordon is given by (cf. [13]):

τ = τ_{0} + iτ_{1} =

= 1+∑_{k=1}^{N} ∑_{1≤j_{1}<...<j_{k}≤N} α_{j_{1}} … α_{j_{k}}i^{k} ∏_{1≤r<s≤k} (\frac{z_{j_{r}} − z_{j_{s}}}{z_{j_{r}} + z_{j_{s}}} )^{2} exp ((∑_{s=1}^{k} z_{j_{s}}^{−1}) t + (∑_{s=1}^{k} z_{j_{s}}) x)

with real-valued α_{j}’s and z_{j}’s.

If we compare this with the expression for the N-soliton solution for the KdV hierarchy [11], [14]

τ = 1 + ∑_{k=1}^{N} ∑_{1≤j_{1}<...<j_{k}≤N} α_{j_{1}} … α_{j_{k}} ∏_{1≤r<s≤k} (\frac{z_{j_{r}} − z_{j_{s}}}{z_{j_{r}} + z_{j_{s}}} )^{2} exp ( ∑_{m∈ℕ_{odd}} (∑_{s=1}^{k} z_{j_{s}}^{m}) x_{m}) ,

then it becomes clear that time t in the sine-Gordon equation is nothing but variable x_{−1} missing in the KdV picture! The “correct” set of variables should include both positive and negative odd integers: x = (…, x_{−3}, x_{−1}, x_{1}, x_{3}, …). We shall use the symbols ℤ_{odd}, ℤ_{ev}, ℕ_{odd}, ℕ_{ev} to denote odd/even integers/natural numbers.

Our extended hierarchy will be completely symmetric in positive and negative directions. This means that the ŝl_{2}-module that we need to construct should not be of the highest/lowest weight. However, the Casimir operator is a crucial ingredient for the construction of the hierarchy of differential equations (see (0.1) above). Formally, this operator is well-defined only for the highest/lowest weight modules. Nevertheless, we are still able to use the Casimir operator in our situation.

In the vertex operator realization of the basic highest weight module, ŝl_{2} acts by differential operators on the algebra of polynomials ℂ[x_{1}, x_{3}, x_{5}, …]. In this paper we represent ŝl_{2} by the same differential operators, but now acting on the algebra of differential operators in variables x_{1}, x_{3}, x_{5}, …, by left multiplication. The algebra of differential operators is isomorphic to the Weyl algebra W generated by p_{i}, q_{j}, i, j ∈ ℕ_{odd} with relations

p_{i}p_{j} = p_{j}p_{i}, q_{i}q_{j} = q_{j}q_{i}, p_{i}q_{j} − q_{j}p_{i} = c_{i}δ_{ij} · 1, c_{i} ∈ ℂ\{0}.

Though the Weyl algebra is non-commutative, still the partial differentiations \frac{∂}{∂p_{i}}, \frac{∂}{∂q_{j}} are well-defined. This allows us to view the elements of the Weyl algebra as functions on a quantized space.

The Weyl algebra has a Poincaré-Birkhoff-Witt decomposition

W = ℂ[q_{1}, q_{3}, …] ⊗ ℂ[p_{1}, p_{3}, …].

Since both subalgebras ℂ[q_{1}, q_{3}, …] and ℂ[p_{1}, p_{3}, …] are commutative, there is a linear bijection between W and the polynomial algebra ℂ[…, x_{−3}, x_{−1}, x_{1}, x_{3}, …]:

π : W → ℂ[x_{1}, x_{3}, …] ⊗ ℂ[x_{−1}, x_{−3}, …].

The image of a differential operator under map π is called the symbol of a differential operator (see e.g., [17]).

The idea of our approach is to write a Kac-Moody group invariant equation on a Weyl algebra element τ̂ and then convert it into a hierarchy of partial differential equations in Hirota form on the symbol τ = π(τ̂). Then we can use the Kac-Moody group action to generate the soliton solutions from the trivial solution τ̂ = 1.

The function τ̂ = 1 is not anymore a solution of (0.1), since we have changed the action to be the left multiplication and 1 now represents the identity operator. However, τ̂ = 1 is trivially a solution of

Ω(τ̂ ⊗ τ̂) − (τ̂ ⊗ τ̂)Ω = 0.  (0.4)

This algebraic equation transforms into a hierarchy of the non-linear PDEs in the variables x = (…, x_{−3}, x_{−1}, x_{1}, x_{3}, …) which contains two KdV subhierarchies going in positive and negative directions.

The simplest new equation in this double KdV hierarchy is the following (cf. (4.34)):

u_{yt} = \frac{∂}{∂x} (u_{xxy} + u_{x}u_{y} + u_{x})  (0.5)

with x = x_{1}, y = x_{−1}, t = x_{3}. This is a generalization of the KdV with two spatial variables, but different from the Kadomtsev-Petviashvili equation.

The sine-Gordon equation can not possibly occur in the double KdV hierarchy (0.4) since the system of Hirota equations (0.3) can not be expressed in terms of τ = τ_{0} + iτ_{1} alone.

Using the boson-fermion correspondence we construct “skew” Casimir operators that split τ into even and odd parts (Proposition 2.2). It remains a mystery whether the skew Casimir operators are related to the Dirac operator. We construct a hierarchy of equations by considering skew analogs of (0.4) (see (2.7), (2.8)).

As expected, the sine-Gordon equation with respect to the variables x_{1} and x_{−1} appears in it, “linking” the two KdV hierarchies.

The generalization of the present work to the case of ŝl_{n} should yield a hierarchy containing the finite Toda chain. The infinite Toda chain was derived by Kac and van de Leur [15] from the two component vertex operator representation of a_{∞}. It would be interesting to understand the connection between these two approaches.

Here is a brief description of the structure of the paper. In Section 1 we construct a representation of affine Lie algebra a_{∞} on a completion of the Weyl algebra W. We also review the boson-fermion correspondence and the construction of the Casimir operator. In Section 2 we consider the reduction of this representation to ŝl_{2}, introduce the skew Casimir operators and study their properties. In Section 3 we show how to convert the skew Casimir operator equations into a generating series of PDEs in Hirota form and construct N-soliton solutions for these equations. In the final section we derive non-linear partial differential equations from the Hirota equations. We also obtain the Leibnitz formula for the Hirota formalism and use it for the treatment of systems of Hirota equations.

Acknowledgements. This work is supported by the Natural Sciences and Engineering Research Council of Canada. I am grateful to Jack Gegenberg for his suggestion to study this problem.

1. Boson-fermion correspondence and representations of a_{∞}

In the previous works on applications of infinite-dimensional Lie algebras to soliton equations the main idea is to represent Lie algebras by differential operators. In this setting the Casimir operator equation becomes an infinite hierarchy of non-linear PDEs in Hirota form. The fact that the Casimir operator commutes with the Lie algebra allows one to construct soliton solutions of this hierarchy.

The approach that we take here is to use the action not on the space of functions, but rather on the Weyl algebra, or, equivalently, on the space of differential operators themselves.

We consider the Weyl algebra W, an associative algebra with 1 generated by the elements p_{i}, q_{i}, i ∈ ℕ with the defining relations

p_{i}q_{j} − q_{j}p_{i} = −\frac{4}{j}δ_{ij}1, p_{i}p_{j} = p_{j}p_{i}, q_{i}q_{j} = q_{j}q_{i},  i, j ∈ ℕ.  (1.1)

The scaling factor −\frac{4}{j} in the first of these relations is chosen to put the vertex operators considered below, in a more symmetric form. Essentially, W is a factor of the universal enveloping algebra of the Heisenberg algebra in which the central element is identified with 1.

The Weyl algebra has a natural representation on the space of polynomials in infinitely many variables (Fock space) F = ℂ[x_{1}, x_{2}, …] by the operators of differentiation and multiplication:

p_{j} ↦ −\frac{2}{j} \frac{∂}{∂x_{j}} , q_{j} ↦ 2x_{j}, j ∈ ℕ.

Thus the Weyl algebra can be also interpreted as the algebra of differential operators on F.

In spite of the non-commutativity of the Weyl algebra, the partial derivatives \frac{∂}{∂p_{j}} , \frac{∂}{∂q_{j}} are well defined (which follows from the fact that the relations (1.1) survive the partial differentiation). These partial derivatives are inner derivations of W ([5], section 4.6):

\frac{∂}{∂p_{j}} (f(q, p)) = \frac{j}{4} [q_{j}, f(q, p)] , \frac{∂}{∂q_{j}} (f(q, p)) = −\frac{j}{4} [p_{j}, f(q, p)] , f(q, p) ∈ W.

(1.2)

Here q = (q_{1}, q_{2}, …), p = (p_{1}, p_{2}, …).

The exponentials exp(α\frac{∂}{∂p_{j}}), α ∈ ℂ, (resp. exp(α\frac{∂}{∂q_{j}})) are the shift operators p_{i} ↦ p_{i} + δ_{ij}α, q_{i} ↦ q_{i}, (resp. p_{i} ↦ p_{i}, q_{i} ↦ q_{i} + δ_{ij}α), which are well-defined commuting automorphisms of W.

Clearly, an element f(q, p) ∈ W can be written in many ways. We call the normal form of f(q, p) its presentation in which the generators q_{i}’s are grouped to the left of p_{j}’s in all monomials. The normal form can be found using the standard Poincaré-Birkhoff-Witt procedure. It also corresponds to the decomposition W = W^{−} ⊗ W^{+}, where W^{−} (resp. W^{+} ) is an abelian algebra generated by q_{1}, q_{2}, … (resp. p_{1}, p_{2}, …).

The subalgebra W^{−} (resp. W^{+}) is isomorphic to the algebra of polynomials in infinitely many variables ℂ[x_{1}, x_{2}, …] (resp. ℂ[x_{−1}, x_{−2}, …] ). Consider the isomorphism of vector spaces

π : W^{−} ⊗ W^{+} → ℂ[x_{1}, x_{2}, … x_{−1}, x_{−2}, …],
π (f(q_{1}, q_{2}, …)g(p_{1}, p_{2}, …)) = f(x_{1}, x_{2}, …)g(x_{−1}, x_{−2}, …).

The map π coincides with the notion of a symbol of a differential operator expressed in the Weyl algebra language. This map allows us to convert the elements of the Weyl algebra into ordinary functions in commuting variables.

We define the normally ordered product : fg : of f(q, p) and g(q, p) to be

: f(q, p)g(q, p) := π^{−1} (π(f(q, p))π(g(q, p))) .

We will consider a ℤ-grading of W = ⊕_{n∈ℤ} W_{n} by assigning degrees to the generators as follows:

deg(p_{i}) = i, deg(q_{i}) = −i, i ∈ ℕ.

Next we introduce a completion \overline{W} of the Weyl algebra, in which the vertex operators will be later defined. We set the completion of W_{n} to be

\overline{W}_{n} = ∏_{{i≤0,j≥0 \atop i+j=n}} W_{i}^{−} ⊗ W_{j}^{+}  (1.3)

and define \overline{W} as

\overline{W} = ⊕_{n∈ℤ} \overline{W}_{n}.  (1.4)

It follows from the Poincaré-Birkhoff-Witt argument that \overline{W} has a well-defined structure of an associative algebra.

In physics literature the generators of the Weyl algebra p_{i}, q_{i} are called free bosons. The boson-fermion correspondence is a way of constructing free fermions out of free bosons and vice versa ([9],[14]). The free fermions are the generators ψ_{i}, ψ_{i}^{∗}, i ∈ ℤ, of the Clifford algebra C_{∞} satisfying the relations

ψ_{i}ψ_{j}^{∗} + ψ_{j}^{∗}ψ_{i} = δ_{ij}, ψ_{i}ψ_{j} + ψ_{j}ψ_{i} = 0, ψ_{i}^{∗}ψ_{j}^{∗} + ψ_{j}^{∗}ψ_{i}^{∗} = 0, i, j ∈ ℤ.

Form the formal generating series (fermion fields)

ψ(z) = ∑_{k∈ℤ} ψ_{k}z^{−k} and ψ^{∗}(z) = ∑_{k∈ℤ} ψ_{k}^{∗}z^{k}.

Proposition 1.1. (cf. [14], Theorem 14.10) The Clifford algebra C_{∞} can be represented on the space B = ℂ[u, u^{−1}] ⊗ \overline{W} by vertex operators

ψ(z) ↦ z^{u \frac{∂}{∂u}} u exp (\frac{1}{2} ∑_{j∈ℕ} q_{j}z^{j}) exp (\frac{1}{2} ∑_{j∈ℕ} p_{j}z^{−j}) ,

(1.5)

ψ^{∗}(z) ↦ u^{−1}z^{−u \frac{∂}{∂u}} exp (−\frac{1}{2} ∑_{j∈ℕ} q_{j}z^{j}) exp (−\frac{1}{2} ∑_{j∈ℕ} p_{j}z^{−j}) .

Here the correspondence between two formal series is the correspondence between their respective components.

Note that the components of the vertex operators belong to the algebra B = ℂ[u, u^{−1}]⊗ \overline{W} and act on B by left multiplication. The symbol z^{u \frac{∂}{∂u}} is interpreted as z^{u \frac{∂}{∂u}} (u^{m}) = z^{m}u^{m}.

The only difference between the statement above and Theorem 14.10 in [14] is that here we consider the action of these vertex operators by left multiplication on ℂ[u, u^{−1}]⊗\overline{W} and not on ℂ[u, u^{−1}] ⊗ F as differential operators. Proposition 1.1 is thus an immediate corollary.

The obvious relations between the operator of multiplication by u and the fields ψ(z), ψ^{∗}(z)

ψ(z)u = zuψ(z), ψ^{∗}(z)u = z^{−1}uψ^{∗}(z)

can be rewritten for their components as

ψ_{k}u = uψ_{k+1}, ψ_{k}^{∗}u = uψ_{k+1}^{∗}.  (1.6)

The classical matrix Lie algebras can be embedded into Clifford algebras and Weyl algebras. It is well-known that this can be also done for the affine Kac-Moody algebras [8] and, in particular, for the affine algebra a_{∞} of infinite rank.

The algebra a_{∞} is a non-trivial one-dimensional central extension of the Lie algebra \overline{a}_{∞} of infinite matrices with finitely many non-zero diagonals:

\overline{a}_{∞} = \{ ∑_{i,j∈ℤ} a_{ij}E_{ij} | ∃n ∈ ℕ |i − j| > n ⇒ a_{ij} = 0\} ,

a_{∞} = \overline{a}_{∞} ⊕ ℂ1.

The action of a_{∞} on ℂ[u, u^{−1}] ⊗ \overline{W} is determined by the formula:

E_{ij} ↦ \{ \begin{matrix}ψ_{i}ψ_{j}^{∗} if i ≠ j or i = j > 0, \\ −ψ_{j}^{∗}ψ_{i} if i = j ≤ 0. \end{matrix}

Using (1.5) we can write a generating series for this action ([14], 14.10.9):

∑_{i,j∈ℤ} E_{ij}z_{1}^{i}z_{2}^{−j} ↦ \frac{1}{1 − \frac{z_{2}}{z_{1}}} ((\frac{z_{1}}{z_{2}} )^{u \frac{∂}{∂u}} Γ(z_{1}, z_{2}) − 1) ,

where

Γ(z_{1}, z_{2}) = exp (\frac{1}{2} ∑_{j=1}^{∞}(z_{1}^{j} − z_{2}^{j})q_{j}) exp (\frac{1}{2} ∑_{j=1}^{∞}(z_{1}^{−j} − z_{2}^{−j})p_{j}) .

In order to get a connection with the differential equations in Hirota form, we should also consider the tensor square B ⊗ B of the a_{∞}-module B = ℂ[u, u^{−1}] ⊗ \overline{W}.

The key link is the Casimir operator ([14], 14.11.2)

Ω = ∑_{k∈ℤ} ψ_{k} ⊗ ψ_{k}^{∗}

which acts on a completion \overline{B ⊗ B} and commutes with the a_{∞}-action ([14], 14.11). To construct this completion we view W ⊗ W as a Weyl algebra in twice as many generators and take its completion as in (1.3), (1.4). The algebra \overline{B ⊗ B} is the tensor product of \overline{W ⊗ W} with two copies of ℂ[u, u^{−1}].

2. Skew Casimir operators

It is well-known that affine Kac-Moody algebra ŝl_{2} = sl_{2} (ℂ[t, t^{−1}]) ⊕ ℂK may be embedded into a_{∞} by

H_{2j+1} ↦ ∑_{i∈ℤ} E_{i,i+2j+1} , A_{j} ↦ ∑_{i∈ℤ}(−1)^{i+j}E_{i,i+j} , K ↦ 1 , j ∈ ℤ, (2.1)

where

H_{2j+1} = ( \begin{matrix}0 & t^{j} \\ t^{j+1} & 0 \end{matrix} ) , A_{2j} = ( \begin{matrix}−t^{j} & 0 \\ 0 & t^{j} \end{matrix} ) , A_{2j+1} = ( \begin{matrix}0 & t^{j} \\ −t^{j+1} & 0 \end{matrix} ) .

The restriction of the representation of a_{∞} on B to this subalgebra yields (cf. [14], 14.13):

H_{k} ↦ −\frac{k}{2} p_{k} , H_{−k} ↦ \frac{k}{2} q_{k} , k ∈ ℕ_{odd},  (2.2)
∑_{j∈ℤ} A_{j}z^{−j} ↦ \frac{1}{2} ((−1)^{u \frac{∂}{∂u}} Γ(z) − 1) ,  (2.3)

where

Γ(z) = exp ( ∑_{j∈ℕ_{odd}} q_{j}z^{j}) exp ( ∑_{j∈ℕ_{odd}} p_{j}z^{−j}) .  (2.4)

Lie algebra 𝔤 = ŝl_{2} has a ℤ_{2}-grading 𝔤 = 𝔤_{0} ⊕ 𝔤_{1}, where 𝔤_{0} is the principal Heisenberg subalgebra spanned by H_{j} , j ∈ ℤ_{odd} and K, while 𝔤_{1} is spanned by A_{j} + \frac{1}{2}δ_{j,0}K, j ∈ ℤ. The following lemma is an immediate consequence of the formulas (2.2)-(2.4) above:

Lemma 2.1. (i) xu = ux for x ∈ 𝔤_{0},

(ii) xu = −ux for x ∈ 𝔤_{1}.

This result can also be seen from the realization of elements of 𝔤 as infinite matrices. The conjugation by u is a shift operator: u^{−1}E_{ij}u = E_{i+1,j+1} − δ_{i,0}δ_{j,0}1. It follows from (2.1) that the elements in 𝔤_{0} are represented by matrices invariant under this shift, while elements of 𝔤_{1} are represented by anti-invariant matrices.

The subspace 1 ⊗ \overline{W} in B is clearly invariant under a_{∞} and ŝl_{2}-actions. While the Casimir operator Ω does not leave the space \overline{W ⊗ W} invariant, it can be easily modified into an operator Λ_{0}, for which \overline{W ⊗ W} is invariant:

Λ_{0} = (1 ⊗ u)Ω(u^{−1} ⊗ 1).

Moreover using the conjugation by 1 ⊗ u^{k} we obtain a whole family of skew Casimir operators:

Λ_{k} = (1 ⊗ u^{k+1})Ω(u^{−1} ⊗ u^{−k}) , k ∈ ℤ.

Note that the conjugation by elements u^{k}⊗1 produces the same family because Ω commutes with u ⊗ u.

It is easy to see that

∑_{k∈ℤ} Λ_{k}z^{−k} = (ψ(z)u^{−1}) ⊗ (uψ^{∗}(z)).

Restricted to \overline{W ⊗ W}, the generating series Λ(z) = ∑_{k∈ℤ} Λ_{k}z^{−k} can be explicitly writ-

ten as

Λ(z) = exp (\frac{1}{2} ∑_{j∈ℕ} q_{j}z^{j}) exp (\frac{1}{2} ∑_{j∈ℕ} p_{j}z^{−j})⊗exp (−\frac{1}{2} ∑_{j∈ℕ} q_{j}z^{j}) exp (−\frac{1}{2} ∑_{j∈ℕ} p_{j}z^{−j}) .

The skew Casimir operators Λ_{k} do not anymore commute with the action of a_{∞}, but as we see in the following Proposition, behave nicely relatively to the subalgebra ŝl_{2} in a_{∞}.

Proposition 2.2. (a) For x ∈ 𝔤_{0}

(x ⊗ 1 + 1 ⊗ x)Λ_{k} = Λ_{k}(x ⊗ 1 + 1 ⊗ x), k ∈ ℤ.

(b) For x ∈ 𝔤_{1}

(x ⊗ 1 + 1 ⊗ x)Λ_{k} = −Λ_{k}(x ⊗ 1 + 1 ⊗ x), k ∈ ℤ_{odd},
(x ⊗ 1 − 1 ⊗ x)Λ_{k} = −Λ_{k}(x ⊗ 1 − 1 ⊗ x), k ∈ ℤ_{ev}.

Proof. Recall that Λ_{k} = (1⊗u^{k+1})Ω(u^{−1}⊗u^{−k}). Part (a) is obvious because xu = ux for x ∈ 𝔤_{0} by Lemma 2.1 and x ⊗ 1 + 1 ⊗ x commutes with the Casimir operator Ω.

Verification of (b) is also straightforward. Suppose that x ∈ 𝔤_{1} and k is odd. By Lemma 2.1, x commutes with the even powers of u and anticommutes with the odd powers of u. Thus

(x ⊗ 1 + 1 ⊗ x)(1 ⊗ u^{k+1})Ω(u^{−1} ⊗ u^{−k})
= (1 ⊗ u^{k+1})(x ⊗ 1 + 1 ⊗ x)Ω(u^{−1} ⊗ u^{−k})
= −(1 ⊗ u^{k+1})Ω(u^{−1} ⊗ u^{−k})(x ⊗ 1 + 1 ⊗ x).

If k is even then

(x ⊗ 1 − 1 ⊗ x)(1 ⊗ u^{k+1})Ω(u^{−1} ⊗ u^{−k})
= (1 ⊗ u^{k+1})(x ⊗ 1 + 1 ⊗ x)Ω(u^{−1} ⊗ u^{−k})

= −(1 ⊗ u^{k+1})Ω(u^{−1} ⊗ u^{−k})(x ⊗ 1 − 1 ⊗ x).

Let τ̂ = τ̂(z_{1}, z_{2}, …) be a formal Laurent series in finitely many formal real variables z_{1}, z_{2}, … with coefficients in \overline{W}. Let τ̂^{∗}(z_{1}, z_{2}, …) be the complex conjugate of τ̂.

Proposition 2.3. The sets of solutions τ̂(z_{1}, z_{2}, …) of equations

(τ̂ ⊗ τ̂)Λ_{k} = Λ_{k}(τ̂^{∗} ⊗ τ̂^{∗}), k ∈ ℤ_{odd},  (2.5)

and

(τ̂ ⊗ τ̂^{∗})Λ_{k} = Λ_{k}(τ̂^{∗} ⊗ τ̂), k ∈ ℤ_{ev},  (2.6)

are invariant under the transformation

τ̂(z_{1}, z_{2}, …) ↦ τ̂′(z_{1}, z_{2}, …, z) = τ̂(z_{1}, z_{2}, …) exp (αiΓ(z)) ,

where α ∈ ℝ and z is a formal real variable.

The equations (2.5) and (2.6) are understood here as equalities of the corresponding \overline{W ⊗ W}-valued coefficients in the formal Laurent series.

Proof. Since the components of Γ(z) represent the elements of 𝔤_{1}, we get from Proposition 2.2 that for k ∈ ℤ_{odd}

exp (αiΓ(z)) ⊗ exp (αiΓ(z)) Λ_{k} = exp (αi (Γ(z) ⊗ 1 + 1 ⊗ Γ(z))) Λ_{k}
= Λ_{k} exp (−αi (Γ(z) ⊗ 1 + 1 ⊗ Γ(z))) = Λ_{k} exp (−αiΓ(z)) ⊗ exp (−αiΓ(z)) ,

and in a similar way for k ∈ ℤ_{ev}:

exp (αiΓ(z)) ⊗ exp (−αiΓ(z)) Λ_{k} = Λ_{k} exp (−αiΓ(z)) ⊗ exp (αiΓ(z)) .

Suppose that τ̂ is a solution of the equation (2.5)

(τ̂ ⊗ τ̂)Λ_{k} = Λ_{k}(τ̂^{∗} ⊗ τ̂^{∗}), k ∈ ℤ_{odd}.

Then for τ̂′ = τ̂ exp (αiΓ(z)) we have

(τ̂′ ⊗ τ̂′)Λ_{k} = (τ̂ ⊗ τ̂) (exp (αiΓ(z)) ⊗ exp (αiΓ(z))) Λ_{k}
= (τ̂ ⊗ τ̂)Λ_{k} (exp (−αiΓ(z)) ⊗ exp (−αiΓ(z)))
= Λ_{k}(τ̂^{∗} ⊗ τ̂^{∗}) (exp (−αiΓ(z)) ⊗ exp (−αiΓ(z))) = Λ_{k}((τ̂′)^{∗} ⊗ (τ̂′)^{∗}).

Thus τ̂′ is also a solution of (2.5). The case of (2.6) is completely analogous.

Corollary 2.4. The series τ̂ = (1 + α_{1}iΓ(z_{1})) … (1 + α_{N}iΓ(z_{N})) in formal real variables z_{1}, …, z_{N} with α_{1}, …, α_{N} ∈ ℝ is a solution for both (2.5) and (2.6).

Proof. Trivially τ̂ = 1 is a solution for both equations. Thus by the above proposition, τ̂ = exp (α_{1}iΓ(z_{1})) … exp (α_{N}iΓ(z_{N})) satisfies (2.5) and (2.6). Finally, exp (αiΓ(z)) = 1 + αiΓ(z) because Γ^{2}(z) = 0 ([14], 14.11.15), and the claim of the Corollary follows.

Consider the real and the imaginary parts of τ̂: τ̂_{0}(z_{1}, z_{2}, …) = Re(τ̂), τ̂_{1}(z_{1}, z_{2}, …) = Im(τ̂). The equations (2.5) and (2.6) can be rewritten as the systems

\{ \begin{matrix}(τ̂_{0} ⊗ τ̂_{0} − τ̂_{1} ⊗ τ̂_{1})Λ_{k} − Λ_{k}(τ̂_{0} ⊗ τ̂_{0} − τ̂_{1} ⊗ τ̂_{1}) = 0 \\ (τ̂_{0} ⊗ τ̂_{1} + τ̂_{1} ⊗ τ̂_{0})Λ_{k} + Λ_{k}(τ̂_{0} ⊗ τ̂_{1} + τ̂_{1} ⊗ τ̂_{0}) = 0 \end{matrix} k ∈ ℤ_{odd}  (2.7)

and

\{ \begin{matrix}(τ̂_{0} ⊗ τ̂_{0} + τ̂_{1} ⊗ τ̂_{1})Λ_{k} − Λ_{k}(τ̂_{0} ⊗ τ̂_{0} + τ̂_{1} ⊗ τ̂_{1}) = 0 \\ (τ̂_{0} ⊗ τ̂_{1} − τ̂_{1} ⊗ τ̂_{0})Λ_{k} + Λ_{k}(τ̂_{0} ⊗ τ̂_{1} − τ̂_{1} ⊗ τ̂_{0}) = 0 \end{matrix} k ∈ ℤ_{ev}.  (2.8)

The requirement that α and z are real may be dropped if we apropriately adjust the statement of Proposition 2.3:

Proposition 2.5. The sets of solutions (τ̂_{0}, τ̂_{1}) of the systems (2.7) and (2.8) are invariant under the transformation

(τ̂_{0}, τ̂_{1}) ↦ (τ̂′_{0}, τ̂′_{1}) = (τ̂_{0}, τ̂_{1}) ( \begin{matrix}1 & αΓ(z) \\ −αΓ(z) & 1 \end{matrix} )

where z is a formal variable and α ∈ ℂ.

The proof parallels the one of Proposition 2.3.

3. Hirota bilinear equations and their solutions

The differential nature of the Weyl algebra is encoded in its defining relations (1.1). In this section we shall see how to convert purely algebraic equations

(τ̂ ⊗ τ̂)Λ_{k} = Λ_{k}(τ̂^{∗} ⊗ τ̂^{∗}), k ∈ ℤ_{odd},
(τ̂ ⊗ τ̂^{∗})Λ_{k} = Λ_{k}(τ̂^{∗} ⊗ τ̂), k ∈ ℤ_{ev},

into a hierarchy of differential equations in Hirota form on the “dequantized” functions τ_{0} = π(τ̂_{0}), τ_{1} = π(τ̂_{1}).

When a product of two “functions” in W is written in the normally ordered form, expressions involving partial derivatives (1.2) appear. One special case of this procedure is recorded in the following Lemma and will be later used in our calculations.

Lemma 3.1. Let p, q be generators of the Weyl algebra satisfying pq − qp = c · 1. Then for τ̂ = τ̂(q, p) we have

exp(zp)τ̂ = \{exp (cz \frac{∂}{∂q} ) τ̂\} exp(zp)

and

τ̂ exp(zq) = exp(zq) \{exp (cz \frac{∂}{∂p}) τ̂\} .

Here z is a formal variable and the above equalities are interpreted as equalities of formal power series.

Proof. We shall prove the second identity, the first will then follow by applying the automorphism p ↦ q, q ↦ −p. From (1.2) we get that

τ̂(q, p)q = qτ̂(q, p) + c\frac{∂τ̂}{∂p} = (q + c \frac{∂}{∂p}) τ̂(q, p).

By induction we obtain

τ̂(q, p)q^{n} = (q + c \frac{∂}{∂p})^{n} τ̂(q, p).

To complete the proof, we apply the above equality to the Taylor expansion of exp(zq):

τ̂(q, p) exp(zq) = exp (zq + cz \frac{∂}{∂p}) τ̂(q, p)

= exp(zq) \{exp (cz \frac{∂}{∂p}) τ̂(q, p)\} .

At this point we are ready to transform (2.7) and (2.8) into a hierarchy of Hirota bilinear equations. Note that the terms involved in these equations are of the form (f ⊗ g)Λ_{k} ± Λ_{k}(f ⊗ g). We shall work with such expressions using the generating series (f ⊗ g)Λ(z) ± Λ(z)(f ⊗ g), f ⊗ g ∈ \overline{W} ⊗ \overline{W}. Denote the variables in the first copy of \overline{W} by q′_{i}, p′_{i} and in the second copy of \overline{W} by q′′_{i}, p′′_{i}. Then

(f ⊗ g)Λ(z) ± Λ(z)(f ⊗ g) =

= f(q′, p′)g(q′′, p′′) exp (\frac{1}{2} ∑_{j=1}^{∞}(q′_{j} − q′′_{j})z^{j}) exp (\frac{1}{2} ∑_{j=1}^{∞}(p′_{j} − p′′_{j})z^{−j})

± exp (\frac{1}{2} ∑_{j=1}^{∞}(q′_{j} − q′′_{j})z^{j}) exp (\frac{1}{2} ∑_{j=1}^{∞}(p′_{j} − p′′_{j})z^{−j}) f(q′, p′)g(q′′, p′′).

We shall assume that both f(q′, p′) and g(q′′, p′′) are in the normal form. Moreover since p′_{i} commutes with q′′_{j}, we can replace f(q′, p′)g(q′′, p′′) with the normally ordered product : f(q′, p′)g(q′′, p′′) : , i.e. move all q’s to the left of all p’s.

Making the change of variables p_{i} = \frac{1}{2}(p′_{i} + p′′_{i}), p̃_{i} = \frac{1}{2}(p′_{i} − p′′_{i}), q_{i} = \frac{1}{2}(q′_{i} + q′′_{i}), q̃_{i} = \frac{1}{2}(q′_{i} − q′′_{i}), we rewrite the above expression as

: f(q + q̃, p + p̃)g(q − q̃, p − p̃) : exp (∑_{j=1}^{∞} q̃_{j}z^{j}) exp (∑_{j=1}^{∞} p̃_{j}z^{−j})

± exp (∑_{j=1}^{∞} q̃_{j}z^{j}) exp (∑_{j=1}^{∞} p̃_{j}z^{−j}) : f(q + q̃, p + p̃)g(q − q̃, p − p̃) : .

Collecting q̃’s on the left and p̃’s on the right using Lemma 3.1 (note that [p̃_{i}, q̃_{j}] = −\frac{2}{j}δ_{ij} · 1), we get

exp (∑_{j=1}^{∞} q̃_{j}z^{j}) ×

\{(exp (−2 ∑_{j=1}^{∞} \frac{z^{j}}{j} \frac{∂}{∂p̃_{j}} ) ± exp (−2 ∑_{j=1}^{∞} \frac{z^{−j}}{j} \frac{∂}{∂q̃_{j}} )) : f(q + q̃, p + p̃)g(q − q̃, p − p̃) :\}

× exp (∑_{j=1}^{∞} p̃_{j}z^{−j}) .  (3.1)

We can convert this expression into the Hirota form. Recall that the Hirota bilinear differentiation is defined as

P (D_{x}, D_{y}, …) [f(x, y, …) ∘ g(x, y, …)] :=

P (\frac{∂}{∂x̃}, \frac{∂}{∂ỹ}, …) f(x + x̃, y + ỹ, …)g(x − x̃, y − ỹ, …)|_{x̃=0,ỹ=0,...}.

Using the Taylor formula we get that (see [14], 14.11.8)

P (\frac{∂}{∂x̃}, \frac{∂}{∂ỹ}, …) f(x + x̃, y + ỹ, …)g(x − x̃, y − ỹ, …) =

P (D_{x}, D_{y}, …) exp (x̃D_{x} + ỹD_{y} + …) [f(x, y, …) ∘ g(x, y, …)] .

Applying this identity we transform (3.1) into

exp (∑_{j=1}^{∞} q̃_{j}z^{j}) \{exp (∑_{j=1}^{∞} q̃_{j}D_{q_{j}} ) (exp (−2 ∑_{j=1}^{∞} \frac{z^{j}}{j} D_{p_{j}} ) ± exp (−2 ∑_{j=1}^{∞} \frac{z^{−j}}{j} D_{q_{j}} ))

exp (∑_{j=1}^{∞} p̃_{j}D_{p_{j}} ) [: f(q, p) ∘ g(q, p) :]} exp (∑_{j=1}^{∞} p̃_{j}z^{−j}) .  (3.2)

We use the reduction to the subalgebra ŝl_{2} in a_{∞} to construct solutions of (2.5) and (2.6). We see from (2.2)-(2.4) that this reduction involves only variables p_{j}, q_{j} with j ∈ ℕ_{odd}, so the solutions we get in Corollary 2.4 do not depend on the even-indexed variables. Thus in (3.2) we may set D_{p_{j}} = 0, D_{q_{j}} = 0 for j ∈ ℕ_{ev}. In (3.2) all q’s are collected to the left of all p’s and all q̃’s are to the left of all p̃’s. Because of that we can easily evaluate the image of this expression under the map π, which gives us:

R_{±}(z)[π(f)(x) ∘ π(g)(x)],

where

R_{±}(z) = exp ( ∑_{j∈ℤ\{0}} x̃_{j}z^{j}) exp ( ∑_{j∈ℤ_{odd}} x̃_{j}D_{x_{j}} )

× \{exp (−2 ∑_{j∈ℕ_{odd}} \frac{z^{j}}{j} D_{x_{−j}} ) ± exp (−2 ∑_{j∈ℕ_{odd}} \frac{z^{−j}}{j} D_{x_{j}} )\}

and x = (…, x_{−3}, x_{−1}, x_{1}, x_{3}, …). Here R_{±}(z) is a Laurent series in z and a Taylor series in x̃_{j}. The coefficients of this series are Hirota differential operators.

Now we can rewrite (2.7) and (2.8) as the generating series for the hierarchy of Hirota bilinear equations on τ_{0} = π(τ̂_{0}) and τ_{1} = π(τ̂_{1}):

Res (z^{j}R_{−}(z)) (τ_{0} ∘ τ_{0} − τ_{1} ∘ τ_{1}) = 0, j ∈ ℤ_{ev},  (3.3)

Res (z^{j}R_{+}(z)) (τ_{0} ∘ τ_{1} + τ_{1} ∘ τ_{0}) = 0, j ∈ ℤ_{ev},  (3.4)

Res (z^{j}R_{−}(z)) (τ_{0} ∘ τ_{0} + τ_{1} ∘ τ_{1}) = 0, j ∈ ℤ_{odd},  (3.5)

Res (z^{j}R_{+}(z)) (τ_{0} ∘ τ_{1} − τ_{1} ∘ τ_{0}) = 0, j ∈ ℤ_{odd}.  (3.6)

As usual, the residue denotes the coefficient at z^{−1} of a formal Laurent series. The coefficient at each monomial in x̃’s in the formal equations above is a Hirota bilinear equation on τ_{0} and τ_{1}. We shall study the Hirota equations that occur here in the next section. Now we turn to describing solutions of this hierarchy.

From Corollary 2.4 it follows that

τ_{0} + iτ_{1} = π ((1 + α_{1}iΓ(z_{1})) … (1 + α_{N}iΓ(z_{N})))

is a solution of (3.3) - (3.6). From [14] we get that the right hand side converges to

1 + ∑_{k=1}^{N} ∑_{1≤j_{1}<...<j_{k}≤N} α_{j_{1}} … α_{j_{k}}i^{k} ∏_{1≤r<s≤k} (\frac{z_{j_{r}} − z_{j_{s}}}{z_{j_{r}} + z_{j_{s}}} )^{2} exp ( ∑_{m∈ℤ_{odd}} (∑_{s=1}^{k} z_{j_{s}}^{m}) x_{m})

(3.7)

when |z_{1}| > |z_{2}| > … > |z_{N}|. By analytic continuation we conclude that

τ_{0} = 1 + ∑_{k=1}^{[ \frac{N}{2} ]}(−1)^{k} ∑_{1≤j_{1}<...<j_{2k}≤N} α_{j_{1}} … α_{j_{2k}} ∏_{1≤r<s≤2k} (\frac{z_{j_{r}} − z_{j_{s}}}{z_{j_{r}} + z_{j_{s}}} )^{2}

× exp ( ∑_{m∈ℤ_{odd}} (∑_{s=1}^{2k} z_{j_{s}}^{m}) x_{m}) ,  (3.8)

τ_{1} = ∑_{k=0}^{[ \frac{N−1}{2} ]}(−1)^{k} ∑_{1≤j_{1}<...<j_{2k+1}≤N} α_{j_{1}} … α_{j_{2k+1}} ∏_{1≤r<s≤2k+1} (\frac{z_{j_{r}} − z_{j_{s}}}{z_{j_{r}} + z_{j_{s}}} )^{2}

× exp ( ∑_{m∈ℤ_{odd}} ( ∑_{s=1}^{2k+1} z_{j_{s}}^{m}) x_{m})  (3.9)

is a solution of the hierarchy (3.3)-(3.6) for all z_{1}, … z_{N}.

As we shall see in the next section, this hierarchy contains the sine-Gordon equation, as well as two copies of the Korteweg - de Vries hierarchy.

4. The KdV – sine-Gordon – KdV hierarchy

In this section we transform the Hirota equations (3.3)-(3.6) into non-linear PDEs. We show that among other equations, this hierarchy contains the sine-Gordon equation, the KdV and the modified KdV equations. We begin by listing some of the Hirota equations that are coefficients at monomials in (3.3)-(3.6) (we multiply them by appropriate constants to avoid fractional coefficients). For simplicity of notations, we write D_{j} for D_{x_{j}}.

From the series (3.3):

D_{−1}D_{1}(τ_{0} ∘ τ_{0} − τ_{1} ∘ τ_{1}) = 0  (at x_{1}z),  (4.1)
(D_{1}^{4} − D_{1}D_{3}) (τ_{0} ∘ τ_{0} − τ_{1} ∘ τ_{1}) = 0  (at x_{1}z^{−3}),  (4.2)
(D_{−1}D_{3} + 2D_{−1}D_{1}^{3} − 3D_{1}^{2}) (τ_{0} ∘ τ_{0} − τ_{1} ∘ τ_{1}) = 0  (at x_{−1}z^{−3}),  (4.3)
(D_{−1}D_{3} + D_{1}^{2}) (τ_{0} ∘ τ_{0} − τ_{1} ∘ τ_{1}) = 0  (at x_{3}z).  (4.4)

From the series (3.4):

(D_{−1}D_{1} − 1) (τ_{1} ∘ τ_{0}) = 0  (at x_{1}z),  (4.5)
(D_{1}^{4} − D_{1}D_{3}) (τ_{1} ∘ τ_{0}) = 0  (at x_{1}z^{−3}),  (4.6)
(D_{1}^{2} − D_{−1}D_{3}) (τ_{1} ∘ τ_{0}) = 0  (at x_{3}z),  (4.7)
(D_{1}^{2} − D_{−1}D_{1}^{3}) (τ_{1} ∘ τ_{0}) = 0  (at x_{1}^{3}z).  (4.8)

From the series (3.5):

D_{1}^{2}(τ_{0} ∘ τ_{0} + τ_{1} ∘ τ_{1}) = 0  (at x_{1}^{2}z^{0}),  (4.9)
(D_{1}^{4} + 2D_{1}D_{3}) (τ_{0} ∘ τ_{0} + τ_{1} ∘ τ_{1}) = 0  (at x_{1}x_{3}z^{0}),  (4.10)
(D_{−1}D_{1}^{3} − D_{−1}D_{3}) (τ_{0} ∘ τ_{0} + τ_{1} ∘ τ_{1}) = 0  (at x_{−1}x_{1}z^{−2}). (4.11)

And finally from the series (3.6):

(D_{1}^{3} − D_{3}) (τ_{1} ∘ τ_{0}) = 0  (at x_{1}z^{−2}),  (4.12)
(D_{−1}D_{1}^{2} − D_{1}) (τ_{1} ∘ τ_{0}) = 0  (at x_{−1}z^{−2}),  (4.13)
(D_{1}^{5} − D_{5}) (τ_{1} ∘ τ_{0}) = 0  (at x_{1}z^{−4}),  (4.14)
(2D_{1}^{5} − 5D_{1}^{2}D_{3} + 3D_{5}) (τ_{1} ∘ τ_{0}) = 0  (at x_{3}z^{−2}).  (4.15)

One of the transformations that we consider below is u = 4 arctan (\frac{τ_{1}}{τ_{0}} ) or, equivalently, \frac{τ_{1}}{τ_{0}} = tan (\frac{u}{4}). Let more generally \frac{τ_{1}}{τ_{0}} = φ(u), hence τ_{1} = φτ_{0}.

In order to rewrite systems of Hirota equations as non-linear PDEs in function u, we need the Leibnitz rule Lemma for the Hirota differential operators. For a multi-index β = (β_{1}, …, β_{n}) let D^{β} = D_{x_{1}}^{β_{1}} … D_{x_{n}}^{β_{n}} be the Hirota differential operator and let ∂^{β} = ∂_{x_{1}}^{β_{1}} … ∂_{x_{n}}^{β_{n}} be the usual differential operator. For a pair of multi-indices β and γ with

0 ≤ γ_{j} ≤ β_{j} let ( \begin{matrix}β \\ γ \end{matrix} ) = ( \begin{matrix}β_{1} \\ γ_{1} \end{matrix} ) … ( \begin{matrix}β_{n} \\ γ_{n} \end{matrix} ) .

Lemma 4.1.

(a) D^{β}(φf ∘ g) = ∑_{γ} ( \begin{matrix}β \\ γ \end{matrix} ) ∂^{γ}(φ)D^{β−γ}(f ∘ g).

(b) D^{β}(φ_{1}f ∘ φ_{2}g) = ∑_{γ} ( \begin{matrix}β \\ γ \end{matrix} ) D^{γ}(φ_{1} ∘ φ_{2})D^{β−γ}(f ∘ g).

The proof of this Lemma is straightforward. Sine-Gordon equation. Consider (4.1) and (4.5):

\{ \begin{matrix}D_{−1}D_{1}(τ_{0} ∘ τ_{0} − τ_{1} ∘ τ_{1}) & = 0 \\ (D_{−1}D_{1} − 1) (τ_{1} ∘ τ_{0}) & = 0 \end{matrix}  (4.16)

and set τ_{1} = φτ_{0}. Applying Lemma 4.1, we transform this system into

\{ \begin{matrix}(1 − φ^{2})D_{−1}D_{1}(τ_{0} ∘ τ_{0}) − D_{−1}D_{1}(φ ∘ φ)τ_{0}^{2} & = 0 \\ φD_{−1}D_{1}(τ_{0} ∘ τ_{0}) + (∂_{−1}∂_{1}(φ) − φ)τ_{0}^{2} & = 0. \end{matrix}

Interpreting this as a system of linear equations on D_{−1}D_{1}(τ_{0} ∘ τ_{0}) and τ_{0}^{2} we get that

det ( \begin{matrix}1 − φ^{2} & −D_{−1}D_{1}(φ ∘ φ) \\ φ & ∂_{−1}∂_{1}(φ) − φ \end{matrix} ) = 0  (4.17)

since this system has a non-trivial solution.

If we now substitute φ = tan (\frac{u}{4}) and multiply (4.17) by cos^{4} (\frac{u}{4}) then we obtain the sine-Gordon equation:

\frac{∂^{2}u}{∂x_{−1}∂x_{1}} = sin(u).  (4.18)

If we let parameters α_{j} in (3.8), (3.9) be purely imaginary then τ_{1} will become purely imaginary, while τ_{0} will remain real. In this case u will be purely imaginary, u = iU, \frac{τ_{1}}{iτ_{0}} = tanh (\frac{U}{4} ), and the function U satisfies the sinh-Gordon equation:

\frac{∂^{2}U}{∂x_{−1}∂x_{1}} = sinh(U).  (4.19)

Modified KdV equation. The same technique applied to (4.9) and (4.12)

\{ \begin{matrix}D_{1}^{2}(τ_{0} ∘ τ_{0} + τ_{1} ∘ τ_{1}) & = 0 \\ (D_{1}^{3} − D_{3}) (τ_{1} ∘ τ_{0}) & = 0 \end{matrix}  (4.20)

yields

\{ \begin{matrix}(1 + φ^{2})D_{1}^{2}(τ_{0} ∘ τ_{0}) + D_{1}^{2}(φ ∘ φ)τ_{0}^{2} & = 0 \\ 3∂_{1}(φ)D_{1}^{2}(τ_{0} ∘ τ_{0}) + (∂_{1}^{3}(φ) − ∂_{3}(φ))(φ ∘ φ)τ_{0}^{2} & = 0. \end{matrix}  (4.21)

After the same substitution as above, φ = tan (\frac{u}{4}), multiplying the determinant of this system by cos^{4} (\frac{u}{4}), we get the mKdV equation:

\frac{∂u}{∂x_{3}} = \frac{1}{2} (\frac{∂u}{∂x_{1}} )^{3} + \frac{∂^{3}u}{∂x_{1}^{3}},  (4.22)

or for v = \frac{∂u}{∂x_{1}} :

\frac{∂v}{∂x_{3}} = \frac{3}{2}v^{2} \frac{∂v}{∂x_{1}} + \frac{∂^{3}v}{∂x_{1}^{3}} .  (4.23)

Mixed sine-Gordon – mKdV equation. The system of three equations (4.4), (4.7) and (4.9)

\{ \begin{matrix}(D_{1}^{2} + D_{−1}D_{3}) (τ_{0} ∘ τ_{0} − τ_{1} ∘ τ_{1}) & = 0 \\ (D_{1}^{2} − D_{−1}D_{3}) (τ_{1} ∘ τ_{0}) & = 0 \\ D_{1}^{2}(τ_{0} ∘ τ_{0} + τ_{1} ∘ τ_{1}) & = 0 \end{matrix}  (4.24)

becomes a system of three linear equations on D_{1}^{2}(τ_{0} ∘ τ_{0}) , D_{−1}D_{3}(τ_{0} ∘ τ_{0}), τ_{0}^{2} and from its determinant

det ( \begin{matrix}1 − φ^{2} & 1 − φ^{2} & (D_{1}^{2} + D_{−1}D_{3})(φ ∘ φ) \\ φ & −φ & ∂_{1}^{2}(φ) − ∂_{−1}∂_{3}(φ) \\ 1 + φ^{2} & 0 & D_{1}^{2}(φ ∘ φ) \end{matrix} ) = 0  (4.25)

after substitution φ = tan (\frac{u}{4}) we obtain the equation

\frac{∂^{2}u}{∂x_{−1}∂x_{3}} = \frac{1}{2} sin(u) (\frac{∂u}{∂x_{1}} )^{2} + cos(u)\frac{∂^{2}u}{∂x_{1}^{2}}.  (4.26)

One can show that this equation follows from the sine-Gordon and mKdV equations (4.18), (4.22) taken together, however (4.26) may have an independent physical meaning.

The second mKdV equation. The system of Hirota equations (4.9), (4.10), (4.14) and (4.15)

\{ \begin{matrix}(D_{1}^{5} − D_{5}) (τ_{1} ∘ τ_{0}) & = 0 \\ (2D_{1}^{5} − 5D_{1}^{2}D_{3} + 3D_{5}) (τ_{1} ∘ τ_{0}) & = 0 \\ D_{1}^{2}(τ_{0} ∘ τ_{0} + τ_{1} ∘ τ_{1}) & = 0 \\ (D_{1}^{4} + 2D_{1}D_{3}) (τ_{0} ∘ τ_{0} + τ_{1} ∘ τ_{1}) & = 0 \end{matrix}  (4.27)

applying the same method as above and eliminating x_{3} using (4.23), is transformed into the following equation for v = 4\frac{∂}{∂x_{1}} arctan (\frac{τ_{1}}{τ_{0}} ):

\frac{∂v}{∂x_{5}} = \frac{∂}{∂x_{1}} (\frac{3}{8}v^{5} + \frac{5}{2}v^{2} \frac{∂^{2}v}{∂x_{1}^{2}} + \frac{5}{2}v (\frac{∂v}{∂x_{1}} )^{2} + \frac{∂^{4}v}{∂x_{1}^{4}} ) ,  (4.28)

which is essentially the second equation in the mKdV hierarchy constructed by the AKNS method [1].

The double KdV hierarchy. It can be easily seen that the hierarchy (3.3)-(3.6) is invariant under the transformation z ↦ z^{−1}, x_{j} ↦ x_{−j}, i.e., it is symmetric in the positive and negative directions.

If in (3.3) and (3.4) we set x̃_{k} = 0 for k < 0 and consider non-negative values of j, then

Res (z^{j}R_{±}(z)) = ±Res (z^{j}P (z)) ,

where

P (z) = exp (∑_{j=1}^{∞} x̃_{j}z^{j}) exp ( ∑_{j∈ℕ_{odd}} x̃_{j}D_{x_{j}} ) exp (−2 ∑_{j∈ℕ_{odd}} \frac{z^{−j}}{j} D_{x_{j}} ) .  (4.29)

In this case the Hirota polynomials appearing in (3.3) and (3.4) differ only by sign. Combining (3.3) and (3.4) together we get that τ = τ_{0} + iτ_{1} satisfies

Res (z^{j}P (z)) (τ ∘ τ) = 0  for j ∈ ℕ_{ev} ∪ {0}.  (4.30)

The Hirota equation

(D_{1}^{4} − D_{1}D_{3})(τ ∘ τ) = 0  (4.31)

(cf. (4.2), (4.6)) after the substitution f = \frac{∂^{2}}{∂x_{1}^{2}} ln(τ) becomes the Korteweg - de Vries equation

\frac{∂f}{∂x_{3}} = 12f \frac{∂f}{∂x_{1}} + \frac{∂^{3}f}{∂x_{1}^{3}} .  (4.32)

The series (4.30) is the KdV hierarchy [4], [14] in variables x_{1}, x_{3}, x_{5} . . .. By symmetry, the second copy of the KdV hierarchy extends in the negative direction x_{−1}, x_{−3}, …. These two KdV hierarchies are linked together by the sine-Gordon equation (4.18) which involves variables x_{1} and x_{−1}. The soliton solutions for the double KdV hierarchy are given by (3.7) with purely imaginary values of α_{1}, … α_{N}.

Remark 4.2. Another way to construct this subhierarchy is to consider the Casimir operator Ω_{aff} for affine Kac-Moody algebra ŝl_{2} (see [14]) acting on \overline{W ⊗ W}. The double KdV hierarchy then arises from the equation

(τ̂ ⊗ τ̂)Ω_{aff} = Ω_{aff}(τ̂ ⊗ τ̂).

The double KdV hierarchy that we obtain here contains more than just two copies of the KdV hierarchy. It also contains equations that involve both positive and negative variables. The simplest such an equation on the function τ = τ_{0} + iτ_{1} is

(D_{−1}D_{3} − 4D_{−1}D_{1}^{3} + 3D_{1}^{2}) (τ ∘ τ) = 0,  (4.33)

which follows from (4.3), (4.4), (4.7) and (4.8). A similar equation (without the last term) appears in the D_{4}^{(1)}-hierarchy [16], but the solutions constructed here and in [16] are inequivalent.

After the substitution g = \frac{∂}{∂x_{1}} ln(τ) we get

\frac{∂^{2}g}{∂x_{−1}∂x_{3}} = \frac{∂}{∂x_{1}} (4\frac{∂^{3}g}{∂x_{−1}∂x_{1}^{2}} + 24\frac{∂g}{∂x_{−1}} \frac{∂g}{∂x_{1}} − 3\frac{∂g}{∂x_{1}} ) ,  (4.34)

which is a generalization of the KdV with two spatial variables, but different from the Kadomtsev-Petviashvili equation.

It is interesting to note that the same equation appears in our hierarchy in a different way. If we combine (4.3), (4.4) with (4.9) and (4.11) then we get

(D_{−1}D_{3} − D_{−1}D_{1}^{3} + 3D_{1}^{2}) (τ_{0} ∘ τ_{0}) = 0  (4.35)

and

(D_{−1}D_{3} − D_{−1}D_{1}^{3} + 3D_{1}^{2}) (τ_{1} ∘ τ_{1}) = 0.  (4.36)

These can be transformed into (4.33) by rescaling the variables x_{j} = \frac{1}{2}x′_{j}. Thus in addition to the solutions g = \frac{∂}{∂x_{1}} ln(τ) of (4.34) with τ given by (3.7), we also get solutions

τ = 1 + ∑_{k=1}^{[ \frac{N}{2} ]}(−1)^{k} ∑_{1≤j_{1}<...<j_{2k}≤N} α_{j_{1}} … α_{j_{2k}} ∏_{1≤r<s≤2k} (\frac{z_{j_{r}} − z_{j_{s}}}{z_{j_{r}} + z_{j_{s}}} )^{2}

× exp (\frac{1}{2} (∑_{s=1}^{2k} z_{j_{s}}^{−1}) x_{−1} + \frac{1}{2} (∑_{s=1}^{2k} z_{j_{s}}) x_{1} + \frac{1}{2} (∑_{s=1}^{2k} z_{j_{s}}^{3}) x_{3})  (4.37)

and

τ = ∑_{k=0}^{[ \frac{N−1}{2} ]}(−1)^{k} ∑_{1≤j_{1}<...<j_{2k+1}≤N} α_{j_{1}} … α_{j_{2k+1}} ∏_{1≤r<s≤2k+1} (\frac{z_{j_{r}} − z_{j_{s}}}{z_{j_{r}} + z_{j_{s}}} )^{2}

× exp (\frac{1}{2} ( ∑_{s=1}^{2k+1} z_{j_{s}}^{−1}) x_{−1} + \frac{1}{2} ( ∑_{s=1}^{2k+1} z_{j_{s}}) x_{1} + \frac{1}{2} ( ∑_{s=1}^{2k+1} z_{j_{s}}^{3}) x_{3}) .  (4.38)

Miura transformation. It is well-known that solutions of the mKdV equation

\frac{∂V}{∂x_{3}} = −\frac{3}{2}V^{2} \frac{∂V}{∂x_{1}} + \frac{∂^{3}V}{∂x_{1}^{3}}  (4.39)

with V = \frac{1}{i}v (cf. (4.23) ) can be transformed into solutions of the KdV equation

\frac{∂f}{∂x_{3}} = 12f \frac{∂f}{∂x_{1}} + \frac{∂^{3}f}{∂x_{1}^{3}}  (4.40)

by the Miura substitution

f = \frac{1}{4} \frac{∂V}{∂x_{1}} − \frac{1}{8}V^{2}.  (4.41)

The soliton solutions of (4.39) and (4.40) are given by V = 4\frac{∂}{∂x_{1}} arctanh (\frac{1}{i} \frac{τ_{1}}{τ_{0}} ) = 2\frac{∂}{∂x_{1}} ln (\frac{τ_{0}+iτ_{1}}{τ_{0}−iτ_{1}} ) and f = \frac{∂^{2}}{∂x_{1}^{2}} ln(τ_{0} + iτ_{1}), where τ_{0} and τ_{1} are given by (3.8) and (3.9) with purely imaginary parameters α_{j}. Note that τ_{0} ± iτ_{1} is real in this case.

A natural question to ask is whether for a given pair (τ_{0}, τ_{1}), the functions V and f are linked by the Miura transform (4.41). The answer to this question is positive, and as we see from the following Lemma, the Miura transform is a feature of the whole hierarchy and not specific just to the pair of the KdV and mKdV equations.

Lemma 4.3. The functions V = 2\frac{∂}{∂x_{1}} ln (\frac{τ_{0}+iτ_{1}}{τ_{0}−iτ_{1}} ) and f = \frac{∂^{2}}{∂x_{1}^{2}} ln(τ_{0} +iτ_{1}) are related by the Miura transform (4.41) if and only if the Hirota equation (4.10) holds:

D_{1}^{2}(τ_{0} ∘ τ_{0} + τ_{1} ∘ τ_{1}) = 0.

The proof of this Lemma is straightforward.

References

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