# ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY

@article{Dickson1999ALGEBROGEOMETRICSO,
title={ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY},
author={Ronnie Dickson and Fritz Gesztesy and Karl Unterkofler},
journal={Reviews in Mathematical Physics},
year={1999},
volume={11},
pages={823-879}
}
• Published 31 August 1998
• Mathematics
• Reviews in Mathematical Physics
We continue a recently developed systematic approach to the Bousinesq (Bsq) hierarchy and its algebro-geometric solutions. Our formalism includes a recursive construction of Lax pairs and establishes associated Burchnall–Chaundy curves, Baker–Akhiezer functions and Dubrovin-type equations for analogs of Dirichlet and Neumann divisors. The principal aim of this paper is a detailed theta function representation of all algebro-geometric quasi-periodic solutions and related quantities of the Bsq…
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## References

SHOWING 1-10 OF 68 REFERENCES
An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy
• Mathematics
• 1998
We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by
Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies
• Mathematics
• 1998
Introduction The Toda hierarchy, recursion relations, and hyperelliptic curves The stationary Baker-Akhiezer function Spectral theory for finite-gap Jacobi operators Quasi-periodic finite-gap
The KDV Hierarchy and Associated Trace Formulas
• Mathematics
• 1996
A natural algebraic approach to the KdV hierarchy and its algebro-geometric finite-gap solutions is developed. In addition, a new derivation of associated higher-order trace formulas in connection
On the Riemann Theta Function of a Trigonal Curve and Solutions of the Boussinesq and KP Equations
Recently, considerable progress has been made in understanding the nature of the algebrogeometrical superposition principles for the solutions of nonlinear completely integrable evolution equations,
A combined sine-Gordon and modified Korteweg-de Vries hierarchy and its algebro-geometric solutions
• Mathematics
• 1997
We derive a zero-curvature formalism for a combined sine-Gordon (sG) and modified Korteweg-de Vries (mKdV) equation which yields a local sGmKdV hierarchy. In complete analogy to other completely
Algebro-geometric approach to nonlinear integrable equations
A brief but self-contained exposition of the basics of Riemann surfaces and theta functions prepares the reader for the main subject of this text, namely the application of these theories to solving
Darboux Transformations for Higher-Rank Kadomtsev—Petviashvili and Krichever—Novikov Equations
• Mathematics
• 1995
It is shown that the action of a special ‘rank 2’ or ‘rank 3’ Darboux transformation, called transference, upon a pair of commuting ordinary differential operators of orders 4 and 6 implements the
Loop groups and equations of KdV type
• Mathematics
• 1985
On decrit une construction qui attribue une solution de l'equation de Korteweg-de Vries a chaque point d'un certain grassmannien de dimension infinie. On determine quelle classe on obtient par cette
Lectures on Riemann Surfaces: Jacobi Varieties
A sequel to "Lectures on Riemann Surfaces" (Mathematical Notes, 1966), this volume continues the discussion of the dimensions of spaces of holomorphic cross-sections of complex line bundles over