Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Baker functions for compact Riemann surfaces


Author: R. J. Schilling
Journal: Proc. Amer. Math. Soc. 98 (1986), 671-675
MSC: Primary 58F07; Secondary 14H40, 14K25, 58F19
MathSciNet review: 861773
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

$\displaystyle \psi (t,p) = ({\psi _1}(t,p), \ldots ,{\psi _l}(t,p)),\quad t \in {{\mathbf{C}}^g},\quad p \in R,$

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 [Enhancements On Off] (What's this?)

  • [1] H. F. Baker, Commutative ordinary differential operators, Proc. Roy. Soc. A 118 (1928), 584-593.
  • [2] 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 (83i:35149)
  • [3] Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745 (82c:30067)
  • [4] I. M. Kričever, Integration of nonlinear equations by the methods of algebraic geometry, Funkcional. Anal. i Priložen. 11 (1977), no. 1, 15–31, 96 (Russian). MR 0494262 (58 #13168)
  • [5] E. Previato, Hyperelliptic curves and solitons, Thesis, Harvard University, 1983.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F07, 14H40, 14K25, 58F19

Retrieve articles in all journals with MSC: 58F07, 14H40, 14K25, 58F19


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0861773-X
PII: S 0002-9939(1986)0861773-X
Article copyright: © Copyright 1986 American Mathematical Society