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Non-Abelian Toda lattice and analogs of Painlev\'e III equation

In integrable models, stationary equations for higher symmetries serve as one of the main sources of reductions consistent with dynamics. We apply this method to the non-Abelian two-dimensional Toda lattice. It is shown that already the stationary equation of the simplest higher flow gives a non-trivial non-autonomous constraint that reduces the Toda lattice to a non-Abelian analog of the pumped Maxwell--Bloch equations. The Toda lattice itself is interpreted as an auto-B\"acklund transformation acting on the solutions of this system. Further self-similar reduction leads to non-Abelian analogs of the Painlev\'e III equation.


Introduction
The non-Abelian Toda lattice [1] (g n,x g −1 n ) y = g n+1 g −1 n − g n g −1 n−1 is one of the fundamental three-dimensional models (two continuous independent variables x, y and one discrete variable n). If the field variables are scalar then the substitution g n = e un is possible which leads to the equation u n,xy = e un+1−un − e un−un−1 [2]. In the non-Abelian setting, we assume that g n are elements of an arbitrary non-commutative algebra A with identity element 1 and the operation of taking the inverse (for example, a matrix algebra). The polynomial form of the Toda lattice [3] f n,y = p n − p n+1 , p n,x = f n p n − p n f n−1 (1) is obtained by the substitution p n = −g n g −1 n−1 and f n = g n,x g −1 n . The aim of our work is to construct a reduction of this lattice equation to non-Abelian analogs of the Painlevé equation The reduction is carried out in two stages. First, we reduce (1) to a two-dimensional system. The easiest way to do this is related with the 2-periodicity condition, which leads to the non-Abelian sinh-Gordon equation [3,4]. In the scalar case, other boundary conditions are also known, leading to exponential systems associated with Cartan matrices for simple Lie algebras [5]. The reduction we use is of a different type and, as far as we know, has not been considered before (although it leads to some well-known equations). It has the form of some non-autonomous constraint and is related to the stationary equation of the simplest higher symmetry from the Toda lattice hierarchy. This reduction is defined in Section 2. It leads to a two-dimensional system, which can be considered as a non-Abelian analog of the pumped Maxwell-Bloch equations [6]. In this case, the auxiliary linear equations for the Toda lattice are transformed into the zero curvature representation with variable spectral parameter for this system, and the shift in n is interpreted as a Bäcklund transformation, see Section 3. At the next stage (sections 4 and 5), we apply self-similar reductions, which lead to third order ODE systems of Painlevé type. In the non-Abelian case, these systems have no global first integrals; however, the reduction of the order of equations is possible on some special invariant submanifolds. This brings to non-Abelian analogs of P 3 with full and degenerate sets of parameters. The corresponding isomonodromic Lax pairs and Bäcklund transformations are also derived. These results generalize the scalar reductions described in [7,8,9,10].
It should be noted that analogs of the Painlevé equations under study are written not as a single second order equation, but as a coupled system of two first order equations. If the variables are scalar, then the exclusion of one of them and some additional transformations lead to P 3 in the standard form, but in the non-Abelian case this is not always possible (at least, if we restrict ourselves to the operations of addition, multiplication, and taking the inverse element). Non-Abelian analogs of the Painlevé equations are being actively studied, see for example [11,12]. It is known that a scalar equation can admit several non-equivalent non-Abelian analogs. Some classification results based on the Kovalevskaya-Painlevé test and other approaches can be found in recent papers [13,14,15]. Here we do not try to reproduce all known analogs of P 3 and restrict ourselves to only those that are obtained as a result of our reduction.

Reduction of the Toda lattice
We start by defining a constraint that is compatible with the Toda lattice. The following assertion is central to our construction, so we present it with a detailed proof. We then explain the origin of this reduction and show how it extends to linear equations for ψ-functions.

Theorem 1. The non-Abelian Toda lattice (1) admits the constraint
Proof. 1) The case ε = µ 0 = 0 amounts to the 2-periodicity condition (since the differentiation of the equality f n−1 + f n = 0 with respect to y implies p n−1 = p n+1 ). Suppose now that µ n ≡ 0, then the equality µ n = 0 is possible for at most one value of n.
2) Consider first the case µ n = 0 for all n. Then the constraint (3) leads to a separation of variables and the Toda lattice reduces to the pair of two-dimensional equations Indeed, Conversely, if the variables f n satisfy the pair (4) and (5) then we obtain a solution of the Toda lattice by setting p n = µ n (f n−1 − f n ) −1 . To prove the theorem, it suffices to verify that equations (4) and (5) are consistent, which is expressed by the equality We write this as a n − a n+1 = 0, where First, we transform one term from the right side. From (6) it follows Then (here and below we identify the number µ ∈ C with the element µ1 ∈ A, where 1 is the unit element of the noncommutative algebra), therefore a n − a n+1 = µ n − 2µ n+1 + µ n+2 , which is 0 for µ n = εn + µ 0 .
3) If µ k = 0 for some k then ε = 0 and µ 0 = −εk. We can set µ n = n without loss of generality, by scaling and changing n → n − k. Then it is easy to see that the solution satisfies the reflection symmetry with the lattice equations (5) and (4) restricted to the half-line n = 1, 2, . . . : The compatibility of these equations is proved in the same way as before, except that when checking the equation It is easy to verify that it holds identically. Remark 1. It is possible to write the constraint (3) in the form p n = µ n (f n−1 − f n ) −1 , but this is not quite equivalent, since in the case 3) for µ n = n it leads to the boundary condition p 0 = 0 instead of f −1 + f 0 = 0. Then the equation p 0,x = f 0 p 0 − p 0 f −1 holds automatically, while the reflection conditions (7) are not necessary. As a result, equations (5) and (4) turn into two independent sybsystems, one for the variables f 0 , f 1 , . . . and another for f −1 , f −2 , . . . . Each of these subsystems is equivalent to (8) with p 0 = 0. Therefore, the constraint in the form (3) is a bit more general. The equation (f −1 + f 0 )p 0 = 0 may also have solutions with zero divisors, but we will not analyze the resulting boundary conditions. Note that (4) and (5) are themselves well-known integrable lattice equations. Equation (4) defines Bäcklund transformations with zero parameters for the equation For µ n = 1, this is the modified Volterra lattice (again, one of two non-Abelian versions), and for µ n = n this is its master-symmetry. Master-symmetries for scalar equations of Volterra lattice type were studied in [16,17], some non-Abelian generalizations and reductions to the Painlevé equations were found in [18]. In the scalar case, the consistency of equations (4) and (5) was noted in [19] (for µ n = 1) and in [20] (for µ n = n), but the fact that they are both embedded in the two-dimensional Toda lattice went unnoticed. Similar results are also known regarding the compatibility of differential-difference equations with discrete equations on a square lattice, see e.g. [21].
Let us demonstrate that the reduction (3) is related to the stationary equation for the simplest higher symmetry of the Toda lattice. Recall that this lattice itself serves as a compatibility condition for the linear equations and its symmetries are defined as compatibility conditions of (9) with equations of the form The gauge h (k,1) n = f n + · · · + f n+k−1 can be chosen without loss of generality. In particular, for k = 2 we have the equation and the conditions of its compatibility with (9) are of the form h n,y = p n (f n−1 + f n ) − (f n + f n+1 )p n+1 , p n,t = h n p n − p n h n−1 .
It is easy to see that if all field variables are commutative, then the requirement that f n and p n be independent of t implies that h n is independent of x and n, and then equations are reduced to (4) and (3), up to a transformation of y. Thus, in the scalar case the constraint (3) is equivalent to the stationary equation for the higher symmetry. In the non-Abelian setting, the stationary equation is more general, but it still holds true by virtue of the constraint (3), if we choose h n = −εy − κ ∈ C. Equations (9) and the stationary equation (10) ψ n+2 + (f n+1 + f n )ψ n+1 = (εy + κ)ψ n can be transformed into zero curvature representations for the reduction under study, with κ playing the role of a spectral parameter. Indeed, let us write these equations in the matrix form Ψ n,x = U n Ψ n , Ψ n,y = V n Ψ n , Ψ n+1 = W n Ψ n , where Ψ n = (ψ n , ψ n+1 ) t and Then the compatibility conditions are and it is easy to check that these equations are equivalent to (1) and (3). The above matrices can be brought to a more symmetric form by introducing the variable spectral parameter λ 2 = εy+κ (this leads to appearance of a term with ∂ λ in the linear equations for ψ-functions) [6]. Simple calculations bring to the following representation.
with the matrices 3 Partial differential systems Due to the constraint (3), the Toda lattice (1) turns into a closed system for the variables f = f n , p = p n , q = p n+1 and parameters µ = µ n , ν = µ n+1 , for any n: (recall that µ and ν are understood as scalars multiplied by 1 ∈ A). Denoting the shift action n → n + 1 in the lattice with a tilde, we reformulate Theorem 1 and Proposition 2 as the following statements, which can also be verified by direct calculations.

Proposition 3. The Bäcklund transformatioñ
maps the solution f, p, q of the system (15) with parameters µ, ν into the solutionf ,p,q of the same system with parametersμ,ν.

Proposition 4. The system (15) admits the representation
and its Bäcklund transformation (16) admits the representation where In the commutative case, the elimination of p brings the system (15) to the equation For ν = µ, this is the so-called "negative" symmetry of the mKdV equation On the other hand, we can compare (15) with the pumped Maxwell-Bloch system introduced in [6]: where c is the pumping parameter. For ρ, E ∈ R, these equations are simplified to the system studied in [7,8,9,10] where it was shown that it admits a selfsimilar reduction to the P 3 equation. The system (20) and the scalar system (15) are related by the simple change Thus, the system (15) can be viewed as a non-Abelian generalization both for the negative mKdV flow with an additional parameter ν − µ and for the real pumped Maxwell-Bloch system with an additional parameter ν + µ.
The system (15) with µ = ν = 0 is a degenerate case corresponding to the constraint (3) with µ n = 0, that is, to the 2-periodic boundary condition Remark 2. The general 2-periodicity condition f n+2 = f n , p n+2 = p n is also equivalent to (21). Indeed, we have (f n +f n+1 ) y = p n −p n+2 = 0, that is f n +f n+1 = a(x), and it is possible to set a = 0 without loss of generality by the gauge transformation f n = Af For the system (21), an additional constraint pq = β(y) ∈ C is possible due to the relation (pq) x = [f, pq]; moreover, one can set β = 1 by the change (p, q, ∂ y ) → β 1/2 (p, q, ∂ y ). This brings to a more special reduction f 2n = f, f 2n+1 = −f, p 2n = p, p 2n+1 = p −1 and the non-Abelian sinh-Gordon equation [3] f y = p − p −1 , p x = f p + pf.
The mapping (16) in this case turns into the trivial changef = −f ,p = p −1 . The zero curvature representation for (22) is of the form Notice, that the systems (21) and (22) with scalar variables are equivalent since the relation pq = β(y) is the first integral. In the non-Abelian case, this is only a partial first integral, so the system (21) is more general than (22). Another difference is that the scalar system (22) easily reduces to the sinh-Gordon equation in rational form while in the non-Abelian case the elimination of f only by use of non-commutative algebra operations is impossible.

Sinh-Gordon equation and non-Abelian analog of P (8) 3
A self-similar reduction of the general system (15) brings to a system of three first-order ODEs. In the following sections, we show that this system has a partial first integral (which did not exist before the reduction), which allows us to reduce the order and to obtain an analog of the P 3 equation with a full set of parameters. In this section, we start with a simpler case of the sinh-Gordon equation (22), which is already second order. Obviously, the scalar equation (24) is invariant under the Lorentz group (x, y) → (εx, y/ε). Therefore, the self-similar substitution is possible p(x, y) = p(z), z = −2xy, which brings to equation (2) with parameters α = 1, β = −1, γ = δ = 0 for the variable p(z): This is the simplest and most well-studied case of P 3 known as P 3 equation (the classification of different cases of P 3 is given, for example, in [22], see also [23,24,25]).
In the non-Abelian case, the reduction is essentially just as simple, but it can be generalized slightly by adding the conjugation by elements of the form where a is an arbitrary non-Abelian constant. An analog of equation (25) is the system (26) from the following Proposition.

This system admits the isomonodromic Lax pair
Proof. Equations (26) are obtained straightforwardly. In order to obtain the matrices A and B, we apply an additional change of the spectral parameter λ = yζ, then the matrices (23) take the form , and the derivatives are replaced according to the rule Then the dependence on y in the equation U y = V x + [V, U ] is canceled out and it takes the form The above Lax pair appears as a result of the changes −2B = U and A = ζ −2 V + ζ −1 (zB + a).

Proposition 6. The reduction (27) in equations (15) brings to the system
which is invariant under the Bäcklund transformatioñ Proof. Equations (28) are obtained straightforwardly. The transformation (16) preserves the homogeneity with respect to the above weights. Therefore, if a solution f (x, y), p(x, y), q(x, y) of the system (15) possesses the self-similar structure (27), then this is true also for the new solutionf (x, y),p(x, y),q(x, y). Hence, it is also described by a system of the form (28) and we only have to rewrite equations (16) under the reduction (27).
Applying the reduction to equations (17) and (18), we obtain the following proposition. Note that the representations (30), (31) can be brought to the standard form with unit coefficient at ∂ ζ by dividing A by ζ 2 − ν + µ.

Proposition 7. The system (28) admits the isomonodromic Lax pair
and the transformation (29) is equivalent to the equations with the matrices where κ in A is an additional scalar parameter such thatκ = κ + 1.
Proof. We extend the substitution (27) with the relation λ = y 1/2 ζ, then y in the matrices (19) is separated out: U = y 1/2−a By a , V = y −1−a Cy a , W = y 1/2−a Ky a , C = 1 ζ µ+ν 2ζ The derivatives are replaced according to the rule As a result, equation (17) transforms to In order to bring this relation to the form (30) we only have to denote A = ζ(zB−2C −2a+κ).
Here κ is an arbitrary parameter, since it cancels out in (30). However, this additional term is neccessary in order to obtain the consistent representations (31) for the Bäcklund transform. The equation for K ′ is obtained from the first equation (18) automatically, while the second equation yields After replacing K ′ = BK − KB, this equation is reduced to the form which coincides with the second equation (31) under the choiceκ = κ + 1.
In the scalar case, the order of the system (28) can be reduced by use of a first integral. We will demonstrate that in the non-Abelian case this is possible due to a partial first integral, that is, equations (28) can be restricted to some invariant submanifold J = 0 which is also preserved under the Bäcklund transformation (29). This leads to a second order system, which is a non-Abelian analog of P 3 . We study the cases of ν = µ and ν = µ separately (this corresponds to the constraint (3) with ε = 0 and ε = 0).

5.2
The case ν = µ: a non-Abelian analog of P which implies the invariance of the equation J = 0. In a similar way, it is easy to check that J(κ) = qJ(κ)q −1 , which proves the invariance with respect to the Bäcklund transformation.
On the level set J = 0, we have 2q = µp −1 (zf − 2a − κ) and (28) turns into a second order system. A direct calculations bring to the following formulas.
The elimination of q by use of (35) brings to the system (36) below. As before, its Lax representation is obtained from the general formulas (30) and (31) by replacing q; the matrix A becomes rather unwieldy and we do not write it explicitly.

Data Availability Statement
Data sharing is not applicable to this article as no new data were created or analyzed in this study.