Geometry of slow–fast Hamiltonian systems and Painlevé equations

• Published in: Indagationes mathematicae Vol. 27; no. 5; pp. 1219 - 1244
• Main Authors: Lerman, L.M., Yakovlev, E.I.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.12.2016; Elsevier Science Ltd
• Subjects: Blow-up; Bundle; Bundling; Disruption point; Geometry; Hamiltonian; Hamiltonian functions; Manifolds (mathematics); Mathematical analysis; Painlevé equations; Presymplectic manifold; Singular symplectic; Singularity (mathematics); Slow–fast; Slow–fast; Bundle; Painlevé equations; Presymplectic manifold; Disruption point; Singular symplectic; Hamiltonian; Blow-up
• ISSN: 0019-3577; 1872-6100
• DOI: 10.1016/j.indag.2016.09.003

In the first part of the paper we introduce some geometric tools needed to describe slow–fast Hamiltonian systems on smooth manifolds. We start with a smooth bundle p:M→B where (M,ω) is a C∞-smooth presymplectic manifold with a closed constant rank 2-form ω and (B,λ) is a smooth symplectic manifold. The 2-form ω is supposed to be compatible with the structure of the bundle, that is the bundle fibers are symplectic manifolds with respect to the 2-form ω and the distribution on M generated by kernels of ω is transverse to the tangent spaces of the leaves and the dimensions of the kernels and of the leaves are supplementary. This allows one to define a symplectic structure Ωε=ω+ε−1p∗λ on M for any positive small ε, where p∗λ is the lift of the 2-form λ to M. Given a smooth Hamiltonian H on M one gets a slow–fast Hamiltonian system with respect to Ωε. We define a slow manifold SM for this system. Assuming SM is a smooth submanifold, we define a slow Hamiltonian flow on SM. The second part of the paper deals with singularities of the restriction of p to SM. We show that if dimM=4,dimB=2 and Hamilton function H is generic, then the behavior of the system near a singularity of fold type is described, to the main order, by the equation Painlevé-I, and if this singularity is a cusp, then the related equation is Painlevé-II. •Geometric tools to describe slow–fast Hamiltonian systems on smooth manifolds.•Direct derivation of Painlevé-I equation near a fold point of a slow manifold.•Direct derivation of Painlevé-II equation near a cusp point of a slow manifold.