87 research outputs found
Growth of degrees of integrable mappings
We study mappings obtained as s-periodic reductions of the lattice
Korteweg-De Vries equation. For small s=(s1,s2) we establish upper bounds on
the growth of the degree of the numerator of their iterates. These upper bounds
appear to be exact. Moreover, we conjecture that for any s1,s2 that are
co-prime the growth is ~n^2/(2s1s2), except when s1+s2=4 where the growth is
linear ~n. Also, we conjecture the degree of the n-th iterate in projective
space to be ~n^2(s1+s2)/(2s1s2).Comment: 14 pages, submitted to Journal of difference equations and
application
Initial value problems for quad equations
We describe a method to construct well-posed initial value problems for not
necessarily integrable equations on not necessarily simply connected
quad-graphs. Although the method does not always provide a well-posed initial
value problem (not all quad-graphs admit well-posed initial value problems) it
is effective in the class of rhombic embeddable quad-graphs.Comment: 22 pages, 34 figures, submitted to Discrete & Computational Geometr
On the Fourier transform of the greatest common divisor
The discrete Fourier transform of the greatest common divisor is a
multiplicative function that generalises both the gcd-sum function and Euler's
totient function. On the one hand it is the Dirichlet convolution of the
identity with Ramanujan's sum, and on the other hand it can be written as a
generalised convolution product of the identity with the totient function. We
show that this arithmetic function of two integers (a,m) counts the number of
elements in the set of ordered pairs (i,j) such that i*j is equivalent to a
modulo m. Furthermore we generalise a dozen known identities for the totient
function, to identities which involve the discrete Fourier transform of the
greatest common divisor, including its partial sums, and its Lambert series.Comment: 3 figures, submitted to Proceedings of the American Mathematical
Societ
Symmetry condition in terms of Lie brackets
A passive orthonomic system of PDEs defines a submanifold in the
corresponding jet manifold, coordinated by so called parametric derivatives. We
restrict the total differential operators and the prolongation of an
evolutionary vector field v to this submanifold. We show that the vanishing of
their commutators is equivalent to v being a generalized symmetry of the
system.Comment: 10 pages, no figures, unpublishe
Somos-4 and Somos-5 are arithmetic divisibility sequences
We provide an elementary proof to a conjecture by Robinson that multiples of
(powers of) primes in the Somos-4 sequence are equally spaced. We also show,
almost as a corollary, for the generalised Somos-4 sequence defined by
and initial
values , that the polynomial
is a divisor of for all
and establish a similar result for the generalized Somos-5
sequence.Comment: 9 page
From integrable equations to Laurent recurrences
Based on a recursive factorisation technique we show how integrable
difference equations give rise to recurrences which possess the Laurent
property. We derive non-autonomous Somos- sequences, with , whose
coefficients are periodic functions with period 8 for , and period 7 for
, and which possess the Laurent property. We also apply our method to the
DTKQ- equation, with , and derive Laurent recurrences with
terms, of order . In the case the recurrence has periodic
coefficients with period 8. We demonstrate that recursive factorisation also
provides a proof of the Laurent property
New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves
For cubic pencils we define the notion of an involution curve. This is a
curve which intersects each curve of the pencil in exactly one non-base point
of the pencil. Involution curves can be used to construct integrable maps of
the plane which leave invariant a cubic pencil
Discrete Painlev\'e equations and their Lax pairs as reductions of integrable lattice equations
We present a method of determining a Lax representation for similarity
reductions of autonomous and non-autonomous partial difference equations. This
method may be used to obtain Lax representations that are general enough to
provide the Lax integrability for entire hierarchies of reductions. A main
result is, as an example of this framework, how we may obtain the q-Painlev\'e
equation whose group of B\"acklund transformations is an affine Weyl group of
type E_6^{(1)} as a similarity reduction of the discrete Schwarzian Korteweg-de
Vries equation.Comment: 23 pages, 5 figure
Duality for discrete integrable systems II
We generalise the concept of duality to lattice equations. We derive a novel
3 dimensional lattice equation, which is dual to the lattice AKP equation.
Reductions of this equation include Rutishauser's quotient-difference (QD)
algorithm, the higher analogue of the discrete time Toda (HADT) equation and
its corresponding quotient-quotient-difference (QQD) system, the discrete
hungry Lotka-Volterra system, discrete hungry QD, as well as the hungry forms
of HADT and QQD. We provide three conservation laws, we conjecture the equation
admits N-soliton solutions and that reductions have the Laurent property and
vanishing algebraic entropy.Comment: 11 pages, 2 figure
Symbolic Computation of Lax Pairs of Partial Difference Equations Using Consistency Around the Cube
A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair
for scalar partial difference equations (P\Delta Es) is reviewed. The method
assumes that the P\Delta Es are defined on a quadrilateral, and consistent
around the cube. Next, the method is extended to systems of P\Delta Es where
one has to carefully account for equations defined on edges of the
quadrilateral. Lax pairs are presented for scalar integrable P\Delta Es
classified by Adler, Bobenko, and Suris and systems of P\Delta Es including the
integrable 2-component potential Korteweg-de Vries lattice system, as well as
nonlinear Schroedinger and Boussinesq-type lattice systems. Previously unknown
Lax pairs are presented for P\Delta Es recently derived by Hietarinta (J. Phys.
A: Math. Theor., 44, 2011, Art. No. 165204). The method is algorithmic and is
being implemented in Mathematica.Comment: Paper dedicated to Peter Olver as part of a special issue of FoCM in
honor of his 60th birthda
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