Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · solv-int/9703005

Non-autonomous Hénon-Heiles Systems

Andrew N. W. Hone
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

solv-intnlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Andrew N. W. Hone. 1997-03-12. Non-autonomous Hénon-Heiles Systems. https://doi.org/10.1016/s0167-2789(98)00010-4

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2017

The averaging of non-local Hamiltonian structures in Whitham's method

We consider the $m$-phase Whitham's averaging method and propose a procedure of "averaging" of non-local Hamiltonian structures. The procedure is based on the existence of a sufficient number of local commuting integrals of a system and gives a Poisson bracket of Ferapontov type for the Whitham's system. The method can be considered as a generalization of the Dubrovin-Novikov procedure for the local field-theoretical brackets.

solv-int↗
arXiv2013

On two aspects of the Painleve analysis

We use the Calogero equation to illustrate the following two aspects of the Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painleve test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Backlund transformation related to the Lax pair.

solv-int↗
arXiv2009

The tetrahedral analog of Veneziano amplitude

In solv-int/9812016 it was shown that the Veneziano amplitude in string theory comes naturally from one of the simplest solutions of the functional pentagon equation (FPE). More generally, FPE is intimately connected with the duality condition for scattering processes. Here I find the amplitude that comes the same way from a solution of the functional tetrahedron equation, with the duality replaced by the local Yang - Baxter equation.

solv-int↗