What does integrability of finite-gap or soliton potentials mean?
Abstract
In the example of the Schrödinger/KdV equation we treat the theory as equivalence of two concepts 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: new explicit $\Psi$-function, trace formula and the Jacobi problem provide a complete solution. The $\Theta$-function language is derivable from these objects and used for ultimate representation of a solution to the inversion problem. Relations with non-integrable equations are discussed also.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2005
- DOI:
- arXiv:
- arXiv:nlin/0505003
- Bibcode:
- 2005nlin......5003B
- Keywords:
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- Nonlinear Sciences - Exactly Solvable and Integrable Systems
- E-Print:
- Major changes. 23 pages