Integrable systems and Riemann surfaces of infinite genus

Details Integrable systems and Riemann surfaces of infinite genus Integrable systems and Riemann surfaces of infinite genus

1994-12-21

arxiv.org/abs/solv-int/9412006v1

Physics

High Energy Physics

High Energy Physics - Theory

91 page, LaTeX, no pictures

Scientific paper

To the spectral curves of smooth periodic solutions of the $n$-wave equation the points with infinite energy are added. The resulting spaces are considered as generalized Riemann surfcae. In general the genus is equal to infinity, nethertheless these Riemann surfaces are similar to compact Riemann surfaces. After proving a Riemann Roch Theorem we can carry over most of the constructions of the finite gap potentials to all smooth periodic potentials. The symplectic form turns out to be closely related to Serre duality. Finally we prove that all non-linear PDE's, which belong to the focussing case of the non-linear Schr\"odinger equation, have global solutions for arbitrary smooth periodic inital potantials.

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