Conte. Kumamoto, 10 May 2006

14 April 2006

On the Lax pairs of the sixth Painlevé equation
Robert Conte.
Service de physique de l’état condensé (CNRS URA 2464)
CEA Saclay, F–91191 Gif-sur-Yvette Cedex, France.
Robert.Conte@cea.fr
Abstract.
The sixth Painlevé equation P6 for u(x) depends holomorphically on its four
2parameters (2α, −2β, 2γ, 1 − 2δ) = (θ∞, θ02 , θ12 , θx2 ), therefore one expects all Lax
pairs of P6 to present this dependence. This is indeed the case for the second orderscalar “Lax” pair of Richard Fuchs, whose singularities are four Fuchsian points ofcrossratio x, plus one apparent singularity located at t = u. The proof by Poincaré(1883) of the impossibility to remove this apparent singularity in the second orderscalar isomonodromic deformation certainly motivated Jimbo and Miwa to consider
matrix isomonodromy,∂x ψ = Lψ,∂t ψ = Mψ, [∂x − L, ∂t − M] = 0.

(1)

With second order matrices, there exists a choice [3] whose only singularities of M
are four Fuchsian points of crossratio x, but this choice presents the drawback tohave a meromorphic dependence on one of the four monodromy exponents θj , see [6,Eq. (3.24)] and [9, Eqs. (4.18)–(4.22)]. The discrete Lax pair for q − P6 [5] displays
the same unpleasant feature. The present work explores several directions in orderto remove this drawback in matrix Lax pairs.
A ﬁrst direction is to take a parametric representation of the four residues inthe second order monodromy matrix M which is diﬀerent from that of Jimbo andMiwa. This leads to a closed three-dimensional ﬁrst order system
Pj (u1 , u2 , u3 , x)
duj=, j, k, l = 1, 2, 3,dx
uk − ul(2)in which Pj is a polynomial and j, k, l are all diﬀerent. This system is currently
under investigation for its explicit integration with, evidently, P6.
A second direction is to start from a third order monodromy matrix presenting
one Fuchsian singularity and one nonFuchsian, then to convert it to a second order
Lax pair either by a factorization method [4] or by a Laplace transform [8]. Onegood candidate to obtain the desired result is the three-wave resonant interaction
uj,t + cj uj,x − iūk ūl = 0,ūj,t + cj ūj,x + iuk ul = 0, cj ∈ R, i2 = −1,(3)in which (j, k, l) denotes any permutation of (1, 2, 3), cj are the constant values of
the group velocities, with (c2 − c3 )(c3 − c1 )(c1 − c2 ) = 0. One of its reductions
can be integrated explicitly in terms of P6 [10], with a third order Lax pair of theabove mentioned type, but the factorization of the residue seems to be algebraic,
not rational like in [4], leading to technical diﬃculties not yet solved.
1References.
[1] R. Fuchs, C. R. Acad. Sc. Paris 141 (1905) 555–558.
[2] R. Garnier, d’ordre[3] M. Jimbo and T. Miwa, Physica D 2 (1981) 407–448.
[4] J. Harnad, Commun. Math. Phys. 166 (1994) 337–365.
[5] M. Jimbo and H. Sakai, Lett. Math. Phys. 38 (1996) 145–154.
[6] G. Mahoux, The Painlevé property, one century later, 35–76, ed. R. Conte,
CRM series in mathematical physics (Springer, New York, 1999).
[7] R. Conte and M. Musette, Chaos, solitons and fractals 11 (2000) 41–52.
http://arXiv.org/abs/solv-int/9803013 .
[8] M. Mazzocco, The Kowalevski property 219–238, CRM Proc. Lecture Notes
32 (Amer. Math. Soc., Providence, RI, 2002).
[9] Lin Run-liang, R. Conte, and M. Musette, J. Nonl. Math. Phys. 10, Supp. 2,
107–118 (2003).
http://www.sm.luth.se/~norbert/home_journal/10s2_9.pdf[10] R. Conte, A. M. Grundland, and M. Musette, J. Phys. A to appear (2006).
Special issue on Painlevé VI. http://arXiv.org/abs/nlin.SI/0604nnn

2