[Paper Review] On the Lax pairs of the sixth Painleve' equation

Robert Conte|arXiv (Cornell University)|Jan 24, 2007

Nonlinear Waves and Solitons6 references3 citations

TL;DR

This paper investigates the construction of a holomorphic matrix Lax pair for the sixth Painlevé equation (P6), addressing a key limitation in existing matrix isomonodromic formulations that exhibit meromorphic dependence on one of the monodromy exponents. By proposing a Kimura-type ansatz for the Lax matrix L, the work aims to achieve both holomorphic dependence on the parameters θj and rational dependence on u, u', and x—offering a path toward a fully holomorphic and rational matrix Lax pair, which remains an open challenge.

ABSTRACT

The dependence of the sixth equation of Painleve' on its four parameters $(2 α,-2 β,2 γ,1-2 δ) =(θ_{\infty}^2,θ_{0}^2,θ_{1}^2,θ_{x}^2)$ is holomorphic, therefore one expects all its Lax pairs to display such a dependence. This is indeed the case of the second order scalar ``Lax'' pair of Fuchs, but the second order matrix Lax pair of Jimbo and Miwa presents a meromorphic dependence on $θ_\infty$ (and a holomorphic dependence on the three other $θ_j$). We analyze the reason for this feature and make suggestions to suppress it.

Motivation & Objective

• To resolve the issue of meromorphic dependence on θ∞ in the Jimbo-Miwa matrix Lax pair for P6, which contradicts the holomorphic parameter dependence of P6 itself.
• To construct a second-order matrix Lax pair for P6 that is holomorphic in all four monodromy exponents θj = (θ∞, θ0, θ1, θx).
• To explore alternative ansatzes for the Lax matrix L beyond the Jimbo-Miwa construction, particularly by incorporating linear terms in t as suggested in prior works.
• To identify conditions under which the resulting matrix Lax pair maintains rational dependence on u, u', and x while ensuring holomorphicity in the θj parameters.

Proposed method

• Adopt a Kimura-type ansatz for the Lax matrix L, assuming L = -Mx/(t-x) + m M∞, where m is a function to be determined.
• Define the monodromy matrix M through a system of ODEs in t, with M∞ constant and M0, M1, Mx satisfying specific commutator relations.
• Impose closure conditions on the system by requiring consistency between the t-derivatives of the variables zj and the evolution equations, ensuring integrability.
• Identify the scalar Lax pair via the commutativity condition X = Sx + Cttt + CSt + 2CtS = 0, linking the matrix system to the scalar Fuchsian formulation.
• Use the condition that the apparent singularity at t = u must have Fuchs indices differing by 3 (not 2) to constrain the system, distinguishing it from the standard P6 Lax pair.
• Derive a polynomial relation F(z∞, u∞, θ∞, u0, θ0, ux, θx, u, x, e^{iφ}) = 0 that must be satisfied for consistency, with F of degree two in each variable.

Experimental results

Research questions

• RQ1 Why does the Jimbo-Miwa matrix Lax pair exhibit meromorphic dependence on θ∞ despite P6's holomorphic parameter dependence?
• RQ2 Can a second-order matrix Lax pair for P6 be constructed that is holomorphic in all four monodromy exponents θj?
• RQ3 Is it possible to achieve both holomorphic dependence on θj and rational dependence on u, u', and x in a matrix Lax formulation?
• RQ4 What alternative ansatz for the Lax matrix L can yield a holomorphic and rational matrix Lax pair, beyond the Jimbo-Miwa construction?
• RQ5 How does the transformation between the apparent singularity structure of the Kimura-type system and the standard P6 Lax pair affect the resulting Lax pair's properties?

Key findings

• The Jimbo-Miwa matrix Lax pair exhibits meromorphic dependence on θ∞, which contradicts the holomorphic parameter dependence of the P6 equation itself.
• A Kimura-type ansatz for L leads to a system with holomorphic dependence on all θj, but the resulting matrix elements are algebraic functions of u', u, and x, not rational.
• The condition for a single apparent singularity at t = u leads to a Fuchsian index difference of 3, which differs from the standard P6 case (difference of 2), implying a different underlying ODE structure.
• The consistency of the system is ensured by a polynomial relation F = 0 involving z∞, u∞, θ∞, u0, θ0, ux, θx, u, x, and the angle φ, which must be solved to close the system.
• The resulting matrix Lax pair cannot be birationally transformed into the standard scalar P6 Lax pair, indicating a distinct solution space.
• The paper concludes that a new ansatz—potentially including linear terms in t—must be used to achieve both holomorphicity in θj and rationality in u, u', x.