On the geometry of a 4-dimensional extension of a q-Painlevé I equation with symmetry type A₁⁽¹⁾

2025/06/26 by Alexander Stokes, Stokes, Alexander, Tomoyuki Takenawa +3 · 1 citation
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2506.21092

Abstract

We present a geometric study of a four-dimensional integrable discrete dynamical system which extends the autonomous form of a q-Painlevé I equation with symmetry of type A₁⁽¹⁾. By resolution of singularities it is lifted to a pseudo-automorphism of a rational variety obtained from (\mathbb P¹)^{× 4} by blowing up along 28 subvarieties and we use this to establish its integrability in terms of conserved quantities and degree growth. We embed this rational variety into a family which admits an action of the extended affine Weyl group \widetildeW(A₁⁽¹⁾)× \widetildeW(A₁⁽¹⁾) by pseudo-isomorphisms. We use this to construct two 4-dimensional analogues of q-Painlevé equations, one of which is a deautonomisation of the original autonomous integrable map.

Citations

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Cited by

• On a non-commutative sixth q-Painlevé system: from discrete system to surface theory

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• Coupled Painlevé systems in dimension four with affine Weyl group symmetry of types A₄⁽²⁾ and A₁⁽¹⁾
• On a non-commutative sixth q-Painlevé system: from discrete system to surface theory