scholarly journals Dubrovin equations and integrable systems on hyperelliptic curves

2002 ◽  
Vol 91 (1) ◽  
pp. 91
Author(s):  
Fritz Gesztesy ◽  
Helge Holden
Keyword(s):  
Integrable Systems ◽  
Hyperelliptic Curve ◽  
Integrable Hierarchies ◽  
Hyperelliptic Curves ◽  
Completely Integrable Systems ◽  
Trace Formulas ◽  
Soliton Equations ◽  
Completely Integrable ◽  
Arbitrary Genus ◽  
Dubrovin Equations

We introduce the most general version of Dubrovin-type equations for divisors on a hyperelliptic curve $\mathcal K_g$ of arbitrary genus $g\in\boldsymbol N$, and provide a new argument for linearizing the corresponding completely integrable flows. Detailed applications to completely integrable systems, including the KdV, AKNS, Toda, and the combined sine-Gordon and mKdV hierarchies, are made. These investigations uncover a new principle for $1+1$-dimensional integrable soliton equations in the sense that the Dubrovin equations, combined with appropriate trace formulas, encode all hierarchies of soliton equations associated with hyperelliptic curves. In other words, completely integrable hierarchies of soliton equations determine Dubrovin equations and associated trace formulas and, vice versa, Dubrovin-type equations combined with trace formulas permit the construction of hierarchies of soliton equations.

BIDIFFERENTIAL CALCULI, BICOMPLEX STRUCTURE AND ITS APPLICATION TO BIHAMILTONIAN SYSTEMS

2006 ◽  
Vol 03 (02) ◽  
pp. 209-232
Author(s):  
PARTHA GUHA
Keyword(s):  
Finite Number ◽  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Integrable Hierarchies ◽  
Finsler Geometry ◽  
Lie Algebroids ◽  
Completely Integrable Systems ◽  
Completely Integrable ◽  
The Relationship

In this exposition, we study the relationship between the bihamiltonian formalism of completely integrable systems using the bidifferential calculi introduced by Dimakis and Müller-Hoissen in [1] and the bihamiltonian formulation of integrable systems with a finite number of degrees of freedom via the Frölicher–Nijenhuis geometry. This pair of bidifferetial operators are used to construct alternative Lie algebroids as shown by Camacaro and Carinena. We find its connection to Finsler geometry. We also find the dispersionless integrable hierarchies using the bidifferential ideals. Finally, we lay out its connection to Gelfand–Zakharevich bihamiltonian geometry.

A survey of the origins and physical importance of soliton equations

1985 ◽  
Vol 315 (1533) ◽  
pp. 335-365 ◽  
Keyword(s):  
Significant Role ◽  
Integrable Systems ◽  
Applied Mathematics ◽  
Theoretical Physics ◽  
Completely Integrable Systems ◽  
Soliton Equations ◽  
Completely Integrable ◽  
Physical Importance ◽  
The Subject

An introduction to the subject is given in an elementary way for the non-specialist, outlining why many completely integrable systems, although special, play a significant role in wave motions in applied mathematics and theoretical physics.

scholarly journals Transverse Hilbert schemes and completely integrable systems

2017 ◽  
Vol 4 (1) ◽  
pp. 263-272 ◽  
Author(s):  
Niccolò Lora Lamia Donin
Keyword(s):  
Integrable Systems ◽  
Hilbert Scheme ◽  
Symplectic Form ◽  
Initial Surface ◽  
Hilbert Schemes ◽  
Completely Integrable Systems ◽  
Completely Integrable ◽  
Symplectic Surface ◽  
Complex Symplectic ◽  
Minimum Polynomial

Abstract In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].

Sign in / Sign up

Export Citation Format

Share Document