Bidifferential graded algebras and integrable systems

In the framework of bidifferential graded algebras, we present universal solution generating techniques for a wide class of integrable systems.


Introduction
Let A be a unital associative algebra (over R or C) with identity element I, and Ω(A) = r≥0 Ω r (A) with Ω 0 (A) = A and A-bimodules Ω r (A), r = 1, 2, . . .. We call (Ω(A), d,d) a bidifferential graded algebra (BDGA), or bidifferential calculus, if (Ω(A), d) and (Ω(A),d) are both differential graded algebras, which means that Ω(A) is a graded algebra and the linear maps d,d : Ω r (A) → Ω r+1 (A) satisfy the graded Leibniz rule (antiderivation property) and d 2 =d 2 = 0 , dd +d d = 0 . (1.1) These conditions can be combined into d 2 z = 0, where d z :=d − z d with an indeterminate z. 1 In section 2 we connect this structure with 'integrable' partial differential (or difference) equations . Although this framework may not be able to cover all possible (in some sense) integrable equations, it has the advantage of admitting universal techniques for constructing exact solutions. Whereas previous work concentrated on conservation laws and Bäcklund transformations, the present work addresses Darboux transformations and presents a very effective 'linearization approach', generalizing results in [22] (see also [23,24] for related ideas). After collecting some basics in section 2, section 3 addresses universal solution generating techniques. Section 4 then presents some examples. Section 5 contains final remarks.

Dressing bidifferential graded algebras
In the following, (Ω(A), d,d) denotes a BDGA. Introducinḡ D :=d − A (2.1) 1 A generalization to an N -differential graded algebra is then obtained if d 2 z = 0 with dz = P N n=0 z n dn. But this will not be considered in this work. with a 1-form A, d andD satisfy again the BDGA relations iff dA − A A ≡ −D 2 = 0 and dA ≡ −(dD +D d) = 0 . (2.2) We are interested in cases where these equations are equivalent to a partial differential or difference equation (or a family of such equations), which requires that A depends on a set of independent variables and the differential maps d,d involve differential or difference operators. As depicted in the following diagram, we can solve either the first or the second equation.
This results in two different equations that are related by a 'Miura transformation' and this relationship is sometimes referred to as 'pseudoduality'. The conditions (2.2) can be combined into Such a zero curvature condition is at the roots of the theory of integrable systems. It is the integrability condition of the linear equation To get some more information about this equation, let us derive it from In case of (Ω(A), d,D), choosing G as multiplication by g −1 , we obtain the equivalent BDGA (Ω(A), d ′ ,D ′ ) with d ′ = d + g −1 dg andD ′ =d, by use of A = (dg) g −1 .
Remark. We can consider a simultaneous dressing of d andd by introducing Solving the first two conditions by setting A = (dg) g −1 , B = (dh) h −1 , the third (multiplied from the left by h −1 , from the right by h) becomes d[(dJ) This generalizes Yang's gauge in the case of the (anti-) self-dual Yang-Mills equation (see also [25]).

Bäcklund transformation (BT)
An elementary BT is given by where G(z) = I + F z −1 [6]. This is equivalent to Using A = dφ, the first equation can be integrated, and from the second equation we obtain the elementary BT Alternatively, using A = (dg) g −1 , the second of equations (3.2) is solved by 5) and the first of equations (3.2) becomes This equation connects the two elementary BTs.

Darboux transformation (DT)
The linear system 2d ψ = (dφ) ψ + (dψ) ∆ , (3.8) has the following integrability condition, which reduces to (2.6) ifd Let θ be an invertible solution of (3.8) with a solution ∆ ′ of (3.10), hencē As a consequence,d where This is in accordance with (3.4), i.e. φ ′ is related to φ by an elementary BT. Hence, any solution φ of (2.6) and any invertible solution θ of the linear equation (3.11) determine a new solution φ ′ of (2.6) via (3.13). This is an abstraction of what is known as a Darboux transformation (see e.g. [26]). Introducing where M satisfiesd Now we can iterate this procedure. Let θ k , k = 1, . . . , n, be invertible solutions ) ∆ with the following solution of (2.6), (3.20) below (see the next subsection). 2 Instead of (3.8), we may considerdψ = (dφ) ψ + d(ψ ∆), which results from (2.10) by setting χ = ψ ∆. In this case we have to imposed∆ = ∆ d∆ in order to obtain (2.6) as integrability condition. Some of the following formulae, also in section 3.4, then have to be modified accordingly. One can prove that the two possibilities are in fact equivalent.

Modified Miura transformation
writing g instead of ψ. The integrability condition is a modified pseudodual of (2.6), related by the modified Miura transformation (3.19). 3 (3.20) corresponds to which reduces the two equations (2.2) to a single one sincedA − A A = (dA) g ∆ g −1 . We note that (3.19) is equivalent tō

A linearization approach
Let us consider (2.8) in the formd where dQ = 0. The reason for the introduction of Q will be given below.
Next we express Φ as and impose the constraint with some P . Multiplying (3.36) by X from the right, leads tō which is a consequence of the two linear equations dY = (dY ) P ,dX = (dX) P . (3.40) The following theorem is now easily verified. 5 A somewhat weaker version of the theorem is obtained by extending (3.38) to The two equations (3.40) combine tod Z = dZ P .
and R X ′ = X ′ P respectively L X ′ + Y ′ = X ′ P .

Examples
In some examples presented below, the graded algebra will be taken of the form Ω(A) = A ⊗ C (C n ) where (C n ) is the exterior algebra of C n . It is then sufficient to define the maps d andd on A. They extend to Ω(A) in an obvious way, treating elements of (C n ) as constants. ξ 1 , . . . , ξ n denotes a basis of 1 (C n ).

Self-dual Yang-Mills (sdYM) equation
Let A be the algebra of smooth complex functions of complex variables y, z and their complex conjugates y,z. Let df = ∓f y ξ 1 + f z ξ 2 ,df = fz ξ 1 + fȳ ξ 2 (4.1) for f ∈ A. This determines a BDGA. Then (2.6), for an m × m matrix φ with entries in A, is equivalent to which is a potential form of the (Euclidean or split signature) sdYM equation (see e.g. [27]). Writing J instead of g, the Miura transformation (2.3) becomes JȳJ −1 = φ z and JzJ −1 = ∓φ y , and the pseudodual of (4.2) takes the form which is another well-known potential form of the sdYM equation.

Pseudodual chiral model hierarchy
Let M be a space with coordinates x 1 , x 2 , . . .. On smooth functions on M we define The first (non-trivial) equation is a well-known reduction of the sdYM equation. φ can be restricted to any Lie subalgebra of gl(m, C), but corresponding conditions then have to be imposed on the solution generating methods. The method of section 3.5 has been applied in [22]. In the su(m) case, a variant of corollary 3.1 has been used in particular to construct multiple lump solutions.

The potential KP (pKP) equation
On smooth functions of x, y, t we define a familiar Lax pair for the pKP equation. The same equations, with ψ replaced by g, are obtained from (3.19), which means that the DT ψ → ψ ′ acts on mKP solutions. Turning to bDTs, (3.23) reads (4.14) These are well-known formulae, see e.g. [26,29]. The ξ 1 -part of (3.45) is Z y − Z xx = 2Z x (P + ∂ x ). Choosing P = −I N ∂ x , this is the heat equation Z y = Z xx . The ξ 2 -part of (3.45) then becomes the second heat hierarchy equation, Z t = Z xxx . Setting R =R − I N ∂ x in (3.38), turns it into (4.15) Now theorem 3.1 expresses a result for the pKP equation [30,31] that extends to the whole pKP hierarchy, see the next subsection.

Kadomtsev-Petviashvili hierarchy
On smooth functions of variables x and t = (t 1 , t 2 , . . .) we define where E λ is the Miwa shift operator with an indeterminate λ, i.e.
) with [λ] = (λ, λ 2 /2, λ 3 /3, . . .). Furthermore, ∂ x is the partial derivative operator with respect to x. Let A contain the algebra of m × m matrices of smooth functions. The above expressions for df,df require that A also contains the Miwa shift operators and powers of ∂ x . (2.6) is equivalent to the following functional representation [32,33] of the (matrix) potential KP hierarchy, In particular, φ t 1 = φ x . The linear system (3.40), which is (3.45), takes the form Choosing P = −I N ∂ x and applying a Miwa shift, this reduces to which is the linear heat hierarchy Z tn = ∂ n x (Z), n = 2, 3, . . .. Choosing moreover R =R − I N ∂ x , (3.38) takes the form (4.15). Now theorem 3.1 reproduces theorem 4.1 in [30]. See also [31,34,35] for exact solutions obtained in this way.

Lotka-Volterra (LV) lattice equation
with the shift operator Λ andḟ = f t . Introducing a = ϕ + − ϕ − I where ϕ = φΛ 2 , (2.6) becomes the (noncommutative) LV lattice equationȧ = a + a − a a − . (4.29) In terms of b = g + g −1 and c =ġ g −1 , the modified Miura transformation (3.19) with ∆ = λΛ −2 reads a noncommutative version of the modified LV lattice [39]. For a DT, appropriate choices are ∆ = λΛ −2 and M = −∆ −1 . In the approach of section 3.5, we set If Q = V U t , then via ϕ n = U t Y nX −1 n V and a n = ϕ n+1 − ϕ n − 1 we obtain solutions of the scalar LV lattice equation. An extension of (4.33) in the sense of (3.43) turns out to be too restrictive.

Final remarks
Any BDGA formulation of an integrable system provides us with a Lax pair which is linear in the spectral parameter, a situation well known from the (anti-) self-dual Yang-Mills system. The restriction to a special form of Lax pairs is what enabled us to work out calculations, that appeared in the literature for specific integrable systems, in a universal way. Although many integrable models indeed fit into this framework, it is not clear to what extent this covers the existing variety of integrable systems. If some integrable system possesses a Lax pair that is non-linear in the spectral parameter, this does not exclude the existence of a Lax pair linear in such a parameter (see [40], for example).