Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of types B₆⁽¹⁾, D₆⁽¹⁾ and D₇⁽²⁾

2007/04/19 by Yusuke Sasano, Sasano, Yusuke
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Differential Equations and Dynamical Systems #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.0704.2478

Abstract

We find four kinds of six-parameter family of coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of types B₆⁽¹⁾, D₆⁽¹⁾ and D₇⁽²⁾. Each system is the first example which gave higher-order Painlevé equations of types B_l⁽¹⁾,D_l⁽¹⁾ and D_l⁽²⁾, respectively. Each system can be expressed as a polynomial Hamiltonian system. We show that these systems are equivalent by an explicit birational and symplectic transformation, respectively. By giving each holomorphy condition, we can recover each system. These symmetries, holomorphy conditions and invariant divisors are new. We also give an explicit description of a confluence process from the system of type D₆⁽¹⁾ to the system of type A₅⁽¹⁾ by taking the coupling confluence process from the Painlevé VI system to the Painlevé V system.

Citations

• Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes

Related

• Coupled Painlevé systems with affine Weyl group symmetry of types A₇⁽²⁾,A₅⁽²⁾ and D₄⁽³⁾
• Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type E₆⁽²⁾
• Four-dimensional Painlevé systems of types D₅⁽¹⁾ and B₄⁽¹⁾
• Coupled Painlevé III systems with affine Weyl group symmetry of types B₅⁽¹⁾,D₅⁽¹⁾ and D₆⁽²⁾
• Coupled Painlevé systems in dimension four with affine Weyl group symmetry of types A₄⁽²⁾ and A₁⁽¹⁾