Stationary problems for equation of the KdV type and dynamical $r$-matrices.

Abstract

We study a quite general family of dynamical $r$-matrices for an auxiliary loop algebra $\mathcal{L}(su(2))$ related to restricted flows for equations of the KdV type. This underlying $r$-matrix structure allows to reconstruct Lax representations and to find variables of separation for a wide set of the integrable natural Hamiltonian systems. As an example, we discuss the Henon-Heiles system and a quartic system of two degrees of freedom in detail.

25pp, LaTex

References in corpus (3)

• Separation of Variables. New Trends.
• Linear $r$-matrix algebra for classical separable systems
• Quadrics on Complex Riemannian Spaces of Constant Curvature, Separation of Variables and the Gaudin Magnet

Cited by in corpus (6)

• Separation of variables for soliton equations via their binary constrained flows
• The Kowalewski top: a new Lax representation
• Constructing N-soliton solution for the mKdV equation through constrained flows
• Deriving N-soliton solutions via constrained flows
• A new method to introduce additional separated variables for high-order binary constrained flows
• Quantum Integrability and Quantum Groups: a special issue in memory of Petr P. Kulish