1 8 Ja n 20 06 Orthogonal curvilinear coordinate systems corresponding to singular spectral curves ∗

an 2 00 6 Orthogonal curvilinear coordinate systems corresponding to singular spectral curves ∗

ar X iv : m at h / 99 02 06 8 v 3 [ m at h . A G ] 1 8 N ov 1 99 9 SPECTRAL CURVES , OPERS AND INTEGRABLE SYSTEMS

We establish a general link between integrable systems in algebraic geometry (expressed as Jacobian flows on spectral curves) and soliton equations (expressed as evolution equations on flat connections). Our main result is a natural isomorphism between a moduli space of spectral data and a moduli space of differential data, each equipped with an infinite collection of commuting flows. The spect...

Description of the n-Orthogonal Curvilinear Coordinate Systems and Hamiltonian Integrable Systems of Hydrodynamic Type. Part 1. Integration of the Lamé Equations

The classical problem of description of n-Orthogonal Curvilinear Coordinate System in flat Euclidean space is solved by the Inverse Scattering Method. The developed method allows to describe Hamiltonian and semi-Hamiltonian Integrable systems of Hydrodynamic type.

Mathematical Methods I

2 Vector Analysis 6 2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Scalar Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 The Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 The Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

ar X iv : m at h / 99 02 06 8 v 1 [ m at h . A G ] 1 1 Fe b 19 99 SPECTRAL CURVES , OPERS AND INTEGRABLE SYSTEMS

We establish a general link between integrable systems in algebraic geometry (expressed as Jacobian flows on spectral curves) and soliton equations (expressed as evolution equations on flat connections). Our main result is a natural isomorphism between a moduli space of spectral data and a moduli space of differential data, each equipped with an infinite collection of commuting flows. The spect...