publication . Other literature type . Preprint . Article . 2019

KdV hierarchy via Abelian coverings and operator identities

B. Eichinger; T. VandenBoom; P. Yuditskii;
  • Published: 02 Jan 2019
  • Publisher: American Mathematical Society (AMS)
Abstract
We establish precise spectral criteria for potential functions $V$ of reflectionless Schr\"odinger operators $L_V = -\partial_x^2 + V$ to admit solutions to the Korteweg de-Vries (KdV) hierarchy with $V$ as an initial value. More generally, our methods extend the classical study of algebro-geometric solutions for the KdV hierarchy to noncompact Riemann surfaces by defining generalized Abelian integrals and analogues of the Baker-Akhiezer function on infinitely connected domains with a uniformly thick boundary satisfying a fractional moment condition.
Subjects
arXiv: Nonlinear Sciences::Exactly Solvable and Integrable SystemsMathematics::Analysis of PDEsNonlinear Sciences::Pattern Formation and SolitonsMathematics::Spectral Theory
free text keywords: Mathematics - Spectral Theory, Mathematical Physics, 37K10, 37K15, 35Q53, 34L40, Pure mathematics, KdV hierarchy, Bitwise operation, Abelian group, Mathematics
Funded by
FWF| Extremal polynomials on subsets of the unit circle
Project
  • Funder: Austrian Science Fund (FWF) (FWF)
  • Project Code: J 4138
  • Funding stream: Schrödinger-Programm
,
FWF| Spectral theory, abelian coverings and Iterations
Project
  • Funder: Austrian Science Fund (FWF) (FWF)
  • Project Code: P 29363
  • Funding stream: Einzelprojekte
,
NSF| RTG: Analysis, Geometry, and Topology at Rice University
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1148609
  • Funding stream: Directorate for Mathematical & Physical Sciences | Division of Mathematical Sciences

[8] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, A method of solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095{1097.

[9] J. B. Garnett, Bounded analytic functions, rst ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007.

[10] J. B. Garnett and D. E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2005.

[11] F. Gesztesy and H. Holden, Soliton equations and their algebro-geometric solutions. Vol. I, Cambridge Studies in Advanced Mathematics, vol. 79, Cambridge University Press, Cambridge, 2003, (1 + 1)-dimensional continuous models.

[12] M. Hasumi, Hardy classes on in nitely connected Riemann surfaces, Lecture Notes in Mathematics, vol. 1027, Springer-Verlag, Berlin, 1983.

[13] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979, Structure of topological groups, integration theory, group representations.

Powered by OpenAIRE Research Graph
Any information missing or wrong?Report an Issue