author={Ronnie Dickson and Fritz Gesztesy and Karl Unterkofler},
  journal={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… 
Algebro-geometric solutions for the Gerdjikov-Ivanov hierarchy
This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the Gerdjikov-Ivanov (GI) hierarchy. Our main tools include the
Algebro-Geometric Solutions for the Kadomtsev-Petviashvili Hierarchy
Based on the idea of symmetric constraint, we apply the Gesztesy-Holden's method to derive explicit representations of the Baker-Ahkiezer function $\psi_1$ of the KP hierarchy, from which we provide
Algebro-geometric constructions to the Dym-type hierarchy
Resorting to the characteristic polynomial of Lax matrix for the Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker-Akhiezer function and meromorphic function
Algebro-Geometric Solutions of the Harry Dym Hierarchy
Abstract The Harry Dym hierarchy is derived with the help of Lenard recursion equations and zero curvature equation. Based on the Lax matrix, an algebraic curve Kn $\mathcal{K}_{n}$ of arithmetic
Algebro-Geometric Solutions to a New Hierarchy of Soliton Equations
With the help of the Lenard recursion equations, we derive a new hierarchy of soliton equations associated with a 3×3 matrix spectral problem and establish Dubrovin type equations in terms of the
Real-valued algebro-geometric solutions of the Camassa–Holm hierarchy
  • F. GesztesyH. Holden
  • Mathematics
    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2007
A detailed treatment of real-valued, smooth and bounded algebro-geometric solutions of the Camassa-Holm (CH) hierarchy is provided and the associated isospectral torus is described.


An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy
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
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
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
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
Solutions of the Boussinesq equation
Darboux Transformations for Higher-Rank Kadomtsev—Petviashvili and Krichever—Novikov Equations
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
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