21 research outputs found
Laurent Polynomials and Superintegrable Maps
This article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure of Somos-4 recurrences, and on the Laurent property. Subsequently a family of fourth-order recurrences that share the Laurent property are considered, which are equivalent to Poisson maps in four dimensions. Two of these maps turn out to be superintegrable, and their iteration furnishes infinitely many solutions of some associated quartic Diophantine equations
Stability of stationary solutions for nonintegrable peakon equations
The Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, which include complete integrability, perhaps the most striking is the fact that in the case where linear dispersion is absent it admits weak multi-soliton solutions - "peakons" - with a peaked shape corresponding to a discontinuous first derivative. There is a one-parameter family of generalized Camassa-Holm equations, most of which are not integrable, but which all admit peakon solutions. Numerical studies reported by Holm and Staley indicate changes in the stability of these and other solutions as the parameter varies through the family.
In this article, we describe analytical results on one of these bifurcation phenomena, showing that in a suitable parameter range there are stationary solutions - "leftons" - which are orbitally stable
Discrete integrable systems and Poisson algebras from cluster maps
We consider nonlinear recurrences generated from cluster mutations applied to
quivers that have the property of being cluster mutation-periodic with period
1. Such quivers were completely classified by Fordy and Marsh, who
characterised them in terms of the skew-symmetric matrix that defines the
quiver. The associated nonlinear recurrences are equivalent to birational maps,
and we explain how these maps can be endowed with an invariant Poisson bracket
and/or presymplectic structure.
Upon applying the algebraic entropy test, we are led to a series of
conjectures which imply that the entropy of the cluster maps can be determined
from their tropical analogues, which leads to a sharp classification result.
Only four special families of these maps should have zero entropy. These
families are examined in detail, with many explicit examples given, and we show
how they lead to discrete dynamics that is integrable in the Liouville-Arnold
sense.Comment: 49 pages, 3 figures. Reduced to satisfy journal page restrictions.
Sections 2.4, 4.5, 6.3, 7 and 8 removed. All other results remain, with minor
editin
New Formula for the Eigenvectors of the Gaudin Model in the sl(3) Case
We propose new formulas for eigenvectors of the Gaudin model in the \sl(3)
case. The central point of the construction is the explicit form of some
operator P, which is used for derivation of eigenvalues given by the formula , where , fulfil
the standard well-know Bethe Ansatz equations
Properties of the series solution for Painlevé I
We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented
The Degasperis-Procesi equation as a non-metric Euler equation
In this paper we present a geometric interpretation of the periodic
Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric
linear connection on the diffeomorphism group of the circle. We also show that
for any evolution in the family of -equations there is neither gain nor loss
of the spatial regularity of solutions. This in turn allows us to view the
Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fr\'echet
space of all smooth functions on the circle.Comment: 17 page
A 2-Component Generalization of the Camassa-Holm Equation and Its Solutions
An explicit reciprocal transformation between a 2-component generalization of
the Camassa-Holm equation, called the 2-CH system, and the first negative flow
of the AKNS hierarchy is established, this transformation enables one to obtain
solutions of the 2-CH system from those of the first negative flow of the AKNS
hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH
system are presented.Comment: 15 pages, 16 figures, some typos correcte
Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation
We look for singlevalued solutions of the squared modulus M of the traveling
wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using
Clunie's lemma, we first prove that any meromorphic solution M is necessarily
elliptic or degenerate elliptic. We then give the two canonical decompositions
of the new elliptic solution recently obtained by the subequation method.Comment: 14 pages, no figure, to appear, Acta Applicandae Mathematica
An integrable discretization of the rational su(2) Gaudin model and related systems
The first part of the present paper is devoted to a systematic construction
of continuous-time finite-dimensional integrable systems arising from the
rational su(2) Gaudin model through certain contraction procedures. In the
second part, we derive an explicit integrable Poisson map discretizing a
particular Hamiltonian flow of the rational su(2) Gaudin model. Then, the
contraction procedures enable us to construct explicit integrable
discretizations of the continuous systems derived in the first part of the
paper.Comment: 26 pages, 5 figure
Cluster mutation-periodic quivers and associated Laurent sequences
We consider quivers/skew-symmetric matrices under the action of mutation (in
the cluster algebra sense). We classify those which are isomorphic to their own
mutation via a cycle permuting all the vertices, and give families of quivers
which have higher periodicity. The periodicity means that sequences given by
recurrence relations arise in a natural way from the associated cluster
algebras. We present a number of interesting new families of non-linear
recurrences, necessarily with the Laurent property, of both the real line and
the plane, containing integrable maps as special cases. In particular, we show
that some of these recurrences can be linearised and, with certain initial
conditions, give integer sequences which contain all solutions of some
particular Pell equations. We extend our construction to include recurrences
with parameters, giving an explanation of some observations made by Gale.
Finally, we point out a connection between quivers which arise in our
classification and those arising in the context of quiver gauge theories.Comment: The final publication is available at www.springerlink.com. 42 pages,
35 figure