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

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

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

Solving singular integral equations by using orthogonal polynomials

In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...

N ov 2 00 2 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...

2 00 1 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...