Integrable systems and Riemann surfaces of infinite genus

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 Schrödinger equation, have global solutions for arbitrary smooth periodic inital potantials. 5The nonlinear Schrödinger equation is the standard example of [F-T]. 6Compare with [B-S]. 7On the line this might be different.Then all Riemann surfaces are singular.For higher groups all isospectral sets may decompose into uncountable many components with respect to the action of the Picard group.Hence there may be additional integrals of motion (see Example 10.7).Due to Proposition 10.1 this is impossible for GL(2, C) and SL(2, C).8 This matrix is the same as the monodromy matrix.

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