Non-autonomous Hénon-Heiles Systems

Abstract

Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered. It is shown that the resulting ODEs may be written as non-autonomous Hamiltonian equations, which are time-dependent generalizations of the well-known integrable Hénon-Heiles systems. The (time-dependent) Hamiltonians are given by logarithmic derivatives of the tau-functions (inherited from the original PDEs). The ODEs for the similarity solutions also have inherited Bäcklund transformations, which may be used to generate sequences of rational solutions as well as other special solutions related to the first Painlevé transcendent.

25 pages, Latex. Some minor corrections

References in corpus (2)

• Sixth Painlevé Equation, Universal Elliptic Curve, and Mirror of ${\mathbf{P}}^2$
• Determinant Structure of the Rational Solutions for the Painlevé II Equation

Cited by in corpus (17)

• Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach
• The Hamiltonian Structure of the Second Painleve Hierarchy
• Backlund transformations for many-body systems related to KdV
• The Yablonskii - Vorob'ev polynomials for the second Painlev'e hierarchy
• An integrable hierarchy with a perturbed Henon-Heiles system
• First degree birational transformations of the Painlevé equations and their contiguity relations
• Painleve property and the first integrals of nonlinear differential equations
• Power expansions for solution of the fourth-order analog to the first Painlevé equation
• Relations for zeros of special polynomials associated to the Painleve equations
• Special polynomials associated with the fourth order analogue to the Painlev'e equations
• Special polynomials associated with the K2 hierarchy
• New contiguity relation of the sixth Painlevé equation from a truncation
• Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville
• Non-autonomous Henon-Heiles system from Painleve class
• The Sigma Form for the PII Hierarchy
• Vector systems of Painlevé type
• A Lagrangian Description of the Higher-Order Painlevé Equations