423 research outputs found
Degenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equations
The bi-Hamiltonian structure of certain multi-component integrable systems,
generalizations of the dispersionless Toda hierarchy, is studies for systems
derived from a rational Lax function. One consequence of having a rational
rather than a polynomial Lax function is that the corresponding bi-Hamiltonian
structures are degenerate, i.e. the metric which defines the Hamiltonian
structure has vanishing determinant. Frobenius manifolds provide a natural
setting in which to study the bi-Hamiltonian structure of certain classes of
hydrodynamic systems. Some ideas on how this structure may be extanded to
include degenerate bi-Hamiltonian structures, such as those given in the first
part of the paper, are given.Comment: 28 pages, LaTe
Data-driven model reduction and transfer operator approximation
In this review paper, we will present different data-driven dimension
reduction techniques for dynamical systems that are based on transfer operator
theory as well as methods to approximate transfer operators and their
eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out
similarities and differences between methods developed independently by the
dynamical systems, fluid dynamics, and molecular dynamics communities such as
time-lagged independent component analysis (TICA), dynamic mode decomposition
(DMD), and their respective generalizations. As a result, extensions and best
practices developed for one particular method can be carried over to other
related methods
The Camassa-Holm Equation: A Loop Group Approach
A map is presented that associates with each element of a loop group a
solution of an equation related by a simple change of coordinates to the
Camassa-Holm (CH) Equation. Certain simple automorphisms of the loop group give
rise to Backlund transformations of the equation. These are used to find
2-soliton solutions of the CH equation, as well as some novel singular
solutions.Comment: 19 pages, 7 figures; LaTeX with psfi
Classical R-matrix theory for bi-Hamiltonian field systems
The R-matrix formalism for the construction of integrable systems with
infinitely many degrees of freedom is reviewed. Its application to Poisson,
noncommutative and loop algebras as well as central extension procedure are
presented. The theory is developed for (1+1)-dimensional case where the space
variable belongs either to R or to various discrete sets. Then, the extension
onto (2+1)-dimensional case is made, when the second space variable belongs to
R. The formalism presented contains many proofs and important details to make
it self-contained and complete. The general theory is applied to several
infinite dimensional Lie algebras in order to construct both dispersionless and
dispersive (soliton) integrable field systems.Comment: review article, 39 page
Higgs Bundles, Gauge Theories and Quantum Groups
The appearance of the Bethe Ansatz equation for the Nonlinear Schr\"{o}dinger
equation in the equivariant integration over the moduli space of Higgs bundles
is revisited. We argue that the wave functions of the corresponding
two-dimensional topological U(N) gauge theory reproduce quantum wave functions
of the Nonlinear Schr\"{o}dinger equation in the -particle sector. This
implies the full equivalence between the above gauge theory and the
-particle sub-sector of the quantum theory of Nonlinear Schr\"{o}dinger
equation. This also implies the explicit correspondence between the gauge
theory and the representation theory of degenerate double affine Hecke algebra.
We propose similar construction based on the gauged WZW model leading to
the representation theory of the double affine Hecke algebra. The relation with
the Nahm transform and the geometric Langlands correspondence is briefly
discussed.Comment: 48 pages, typos corrected, one reference adde
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