Derivation of Painlev´e type system with D (1)4 aﬃne Weyl group symmetry in a self-similarity limit

We show how the zero-curvature equations based on a loop algebra of D 4 with a principal gradation reduce via self-similarity limit to a polynomial Hamiltonian system of coupled Painlev´e III models with four canonical variables and D (1)4 aﬃne Weyl group symmetry.

]ocnmp[ Aratyn, Gomes, Lobo, Zimerman best of our knowledge, an explicit derivation of this particular model as a reduction of two-dimensional hierarchy of zero-curvature equations based on D (1)   4 affine Weyl algebra, has not been done previously.
To make the presentation self-contained we provide all the necessary algebraic background information in Appendix A and main expressions of the zero-curvature calculation in Appendix B.
For other derivations of integrable hierarchy based on D (1) 4 affine algebra the reader can consult [2] and the references therein.There also exist in the literature other approaches to applying similarity reduction to integrable hierarchy of type D (1) 4 [3] but the focus there was on deriving the sixth Painlevé equation.
The grade one component of (1) reads as ∂ x D (1) + [E (1) , D (0) ] + [A 0 , D (1) ] = 0 , with D (0) = i v i H i .The advantage of using the basis (61) is that [E (1) , where E i , i = 1, . ..E 5 are the basis elements given in expressions (61).Solving the grade one equation ( 7) in direction of E 1 yields plugging expressions (63) and taking out the total derivative gives Solving the grade one equation (7) in directions of E i , i = 2, 3, 4, 5 yields with C i , i = 2, 3, 3, 5 given in (64) and with d 1 in given in expression (10) while expressions d i , i = 2, .., 5 are given in (63).Inserting these values of d i and C i into (11) we obtain v i given in expression (65).
The grade zero component of (1) is Since [A 0 , D (0) ] = 0 the equation (12) reduces to that in components gives the t 3 flows written as When on the right hand side we insert values of v i from equation (65) we find the t 3 -flow explicitly written in equation (66) with their symmetries listed below in equations ( 16).
With definitions ]ocnmp[ Aratyn, Gomes, Lobo, Zimerman Equations (66) can be conveniently rewritten in this notation as These equations are invariant under: In addition for ε 2 = 0 these equations are invariant under: while for ε 1 = 0 they are invariant under: Obviously, for one of the parameters ε 1 or ε 2 being zero the remaining parameter can be absorbed by redefining t 3 .The above operations satisfy F 2 2 = I = F 2 3 and where T and G are automorphisms : All the above automorphisms of equations (15) are "mirror automorphisms", meaning that they square to one.
2 Self-similarity reduction for the t 3 flow We will look at the self-similar reduction of equation (15) with and correspondingly such that the KdV type of expression : is transformed to an equation fully expressible in terms of functions of z: Following these rules we are now able to take self-similarity limit of equations (15) to obtain: where C i , K i , i = 1, 2 are integration constants.
Here we comment that it is enough to chose any direction in ε 1 − ε 2 plane because of a presence of previously noticed automorphisms that establish an equivalence (by substitution) between any of the one-parameter ε models in a self-similarity limit.
We further point out that symmetry extends to any direction in the ε 1 − ε 2 plane.For example we can transform the system of equations ( 19) with ε 1 = 0 into the system of equations ( 19) with ε 1 + ε 2 = 0 with only the parameter ε such that ε = ε 1 − ε 2 as follows Thus for simplicity we will from now on only consider the self-similarity limit for the case of ε 2 = 0 rewritten as: where First, we note that equations (22) can be made independent of ε through the substitution S: It is instructive to leave the equations (22) in the current form as the change of variables we will perform to arrive at the Hamiltonian formalism will lead anyway to canonical coordinates that are invariant under the above transformation S.
The equations ( 22) are explicitly invariant under F 2 : From the last two equations of (22) we derive The first order derivatives for u, g are: Introducing for convenience we can rewrite equations ( 24),(25) as There is one further change of variables needed to end up with equations that are manifestly Hamilton equations, namely: Equations for Ḡ, Ū variables are: To end up with the polynomial Hamilton equations we further introduce : Using this notation the first two of equations ( 26) can be rewritten as: From equations ( 27) and (28) we obtain Define now the Hamiltonian : which is polynomial in all variables such that it reproduces equations ( 28)-( 29) through ]ocnmp[ Aratyn, Gomes, Lobo, Zimerman Note that the "plus" and "minus" parts of H in (30) are connected by only one term The transformation leaves only the first part of Hamiltonian (30) invariant (up to a constant).
We will now attempt to cast equations ( 28)-( 29) in a form of equations that are manifestly D (1) 4 invariant [1].First, we apply the redefinition and We now further substitute to obtain for F p , v p equations : ) . ( Introducing we can rewrite the above equations as For the "−" sector we obtain ) . (37) we can compactly rewrite the above equations as Equations ( 36) and ( 38) can be obtained from the Hamiltonian: through The author of [1] has proposed such system as two coupled Painlevé III equations involving four variables and derived by symmetry consideration as a system that admits affine Weyl group symmetry of type D 4 .Comparing equations (36) and (38) we notice presence of π 0 automorphism : that transforms equation (36) into (38) and vice-versa.
In addition we introduce a variable α 0 defined by the condition 2α 0 +α 1 +α 2 +α 3 +α 4 = const [1].The constant used to define α 0 will be fixed below by a symmetry transformation s 0 , that mixes the "+/−" sectors to be defined below.In [1] that constant is set to 1 consistently with Sasano's normalization (different from ours).
Furthermore we also find the following Bäcklund transformation s 2 : that keeps equations (36) invariant.The consequence of Similarly the following Bäcklund transformation : ]ocnmp[ Aratyn, Gomes, Lobo, Zimerman will keep equations (38) invariant.Note that s 2 2 = s 2 4 = 1, s 2 s 4 = s 4 s 2 and π 0 s 2 π 0 = s 4 .Furthermore, inspired by the automorphism (17), we define the two automorphisms: that both keep equations ( 36) -( 38) invariant and satisfy Coincidently, all the canonical coordinates v p , v m , F p , F m have been defined in such a way that they are invariant under transformation S defined in relation ( 23), while the substitution z → (ε) 1/3 z allows to eliminate ε completely from equations ( 36) -( 38).With ε being replaced by 1, one can alternatively define the automorphisms π 1 , π 3 involving a change of the sign of z → −z instead of ε → −ε, as it was done in [1].
The other two Bäcklund transformations are ( s 0 , s 3 in notation of [1]) but here relabeled as : They both square to one : s 2 1 = s 2 3 = 1.Also the Bäcklund transformations satisfy : Finally we need to prove invariance under s 0 that mixes the +/− sectors.When this Bäcklund transformation is defined as the equations ( 36) and (38) are invariant if the condition, holds.As remarked before our normalization is different from the one used by Sasano [1] and the differences also include different powers of z in equations ( 45) and in the Hamiltonian (39).
The associated step operators are ,which is the dimension of so(2n), rank of Cartan sub-algebra is n.
It holds that as long as All roots have equal length and satisfy (α, α) = 2.The basis of simple roots is given by: The inner product of simple roots defines the corresponding Cartan matrix.For so(2n) the roots and co-roots are identical, the highest root is the Coxeter number h and the dual Coxeter number h ∨ coincide and For case of so(8) these become The fundamental weights Λ i such that 2(α i , Λ j )/(α i , α i ) = δ ij are: Especially for so(8) with n = 4 we find for weights and simple roots we obtain for a sum of weights: The product of Λ and a general root α = ǫe i + ηe j (Λ, α) = 0 for all α = ǫe i + ηe j .We will use (51) to define the principal gradation operator for so(8): Note that (Λ, ψ) = (3e 1 + 2e 2 + e 3 , e 1 + e 2 ) = 5 .
For the (sub-algebras) G (3) and G (1) that have non-trivial two-and one-dimensional kernels, K (3) and K (1) , respectively, it is useful to describe their bases.
For G (3) from the relation (53) we will use the basis: with V 1 , V 2 being the two matrices from (56) that span a basis for the kernel K (3) of E (1) in G (3) , while V 3 , V 4 , V 5 , V 6 span a basis for the image of E (1) in G (3) .

B Main expressions of of the zero-curvature calculation
The coefficients M i , i = 1, . .., 4 of the matrix D (2) defined in expressions (3), are explicitly given by solving the grade 3 equation ( 4): The coefficients d i , i = 2, . .., 5 of the grade one element D (1) along the basis elements E i , i = 2, . ..E 5 given in expressions (61) are obtained from the grade 2 component of the zero curvature equations (1) to be The components of [A 0 , D (1) ] = 5 i=2 C i E i can be calculated as ( ]ocnmp[ Painlevé D (1) 4 system in a self-similarity limit 107 Inserting these values of d i and C i into equation (11) we obtain We can now insert the above values v i into the t 3 flow expression (13) to obtain

e 2 +e 3
and the same basic relation for the V -basis.