Baker functions for compact Riemann surfaces
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- by R. J. Schilling PDF
- Proc. Amer. Math. Soc. 98 (1986), 671-675 Request permission
Abstract:
This article contains a proof of an important theorem in soliton mathematics. The theorem, stated roughly in [4], contains necessary conditions for the existence of a vector function \[ \psi (t,p) = ({\psi _1}(t,p), \ldots ,{\psi _l}(t,p)),\quad t \in {{\mathbf {C}}^g},\quad p \in R,\] with prescribed poles and $l$ essential singularities an a compact Riemann surface $R$ of genus $g$. $\psi$ is called a Baker function in honor of the 1928 article [1] of H. F. Baker. This report clarifies Krichever’s description of $\psi$ for $l > 1$ essential singularities. The divisors ${\delta _\alpha }$ in (1) below are the key to the $l > 1$ construction. Krichever’s $(l > 1)$ construction is not easy to deal with in practical problems. E. Previato [5] noted this and applied our characterization of the ${\delta _\alpha }$ to construct the finite gap solutions to the nonlinear Schroedinger equation.References
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H. F. Baker, Commutative ordinary differential operators, Proc. Roy. Soc. A 118 (1928), 584-593.
- B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11–80 (Russian). With an appendix by I. M. Krichever. MR 616797
- Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745
- Igor Moiseevich Krichever, Integration of nonlinear equations by the methods of algebraic geometry, Funkcional. Anal. i Priložen. 11 (1977), no. 1, 15–31, 96 (Russian). MR 0494262 E. Previato, Hyperelliptic curves and solitons, Thesis, Harvard University, 1983.
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 98 (1986), 671-675
- MSC: Primary 58F07; Secondary 14H40, 14K25, 58F19
- DOI: https://doi.org/10.1090/S0002-9939-1986-0861773-X
- MathSciNet review: 861773