Historical remarks to finite-gap integration theory: elementary treatment of the theory

Details Historical remarks to finite-gap integration theory: elementary treatment of the theory Historical remarks to finite-gap integration theory: elementary treatment of the theory

2005-04-24

arxiv.org/abs/nlin/0504051v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

23 pages; in Russian

Scientific paper

In the example of the Schr\"odinger/KdV equation we give elementary treatment of the theory of finite-gap integration. The concept is equivalent to two kinds of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finite-dimensional Hamiltonian systems (stationary KdV--equations). Three key objects in this field: the new explicit formula for the $\Psi$-function, trace formula and the Jacobi problem provide a complete solution. Appendix contains Russian translation of two papers of J.Drach

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