Differential and Integral Equations

Integrable systems in the infinite genus limit

Fritz Gesztesy

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We provide an elementary approach to integrable systems associated with hyperelliptic curves of infinite genus. In particular, we explore the extent to which the classical Burchnall-Chaundy theory generalizes in the infinite genus limit, and systematically study the effect of Darboux transformations for the KdV hierarchy on such infinite-genus curves. Our approach applies to complex-valued periodic solutions of the KdV hierarchy and naturally identifies the Riemann surface familiar from standard Floquet theoretic considerations with a limit of Burchnall-Chaundy curves.

Article information

Differential Integral Equations, Volume 14, Number 6 (2001), 671-700.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37K20: Relations with algebraic geometry, complex analysis, special functions [See also 14H70]
Secondary: 14H70: Relationships with integrable systems 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35Q58 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)


Gesztesy, Fritz. Integrable systems in the infinite genus limit. Differential Integral Equations 14 (2001), no. 6, 671--700. https://projecteuclid.org/euclid.die/1356123242

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