dorsal/arxiv
View SchemaIntegrable systems and Riemann surfaces of infinite genus
| Authors | Martin U. Schmidt |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9412006 |
| URL | https://arxiv.org/abs/solv-int/9412006 |
| Journal | Memoirs of the AMS Nr 581 (1996) |
Abstract
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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"abstract": "To the spectral curves of smooth periodic solutions of the $n$-wave equation\nthe points with infinite energy are added. The resulting spaces are considered\nas generalized Riemann surfcae. In general the genus is equal to infinity,\nnethertheless these Riemann surfaces are similar to compact Riemann surfaces.\nAfter proving a Riemann Roch Theorem we can carry over most of the\nconstructions of the finite gap potentials to all smooth periodic potentials.\nThe symplectic form turns out to be closely related to Serre duality. Finally\nwe prove that all non-linear PDE\u0027s, which belong to the focussing case of the\nnon-linear Schr\\\"odinger equation, have global solutions for arbitrary smooth\nperiodic inital potantials.",
"arxiv_id": "solv-int/9412006",
"authors": [
"Martin U. Schmidt"
],
"categories": [
"solv-int",
"hep-th",
"nlin.SI"
],
"journal_ref": "Memoirs of the AMS Nr 581 (1996)",
"title": "Integrable systems and Riemann surfaces of infinite genus",
"url": "https://arxiv.org/abs/solv-int/9412006"
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