30 research outputs found
Algebraic entropy, symmetries and linearization of quad equations consistent on the cube
We discuss the non autonomous nonlinear partial difference equations
belonging to Boll classification of quad graph equations consistent around the
cube. We show how starting from the compatible equations on a cell we can
construct the lattice equations, its B\"acklund transformations and Lax pairs.
By carrying out the algebraic entropy calculations we show that the
trapezoidal and the families are linearizable and in a few examples we
show how we can effectively linearize them
Lagrangians and integrability for additive fourth-order difference equations
We use a recently found method to characterise all the invertible
fourth-order difference equations linear in the extremal values based on the
existence of a discrete Lagrangian. We also give some result on the
integrability properties of the obtained family and we put it in relation with
known classifications. Finally, we discuss the continuum limits of the
integrable cases.Comment: 40 pages, 1 figur
On Partial Differential and Difference Equations with Symmetries Depending on Arbitrary Functions
In this note we present some ideas on when Lie symmetries, both point and
generalized, can depend on arbitrary functions. We show on a few examples, both
in partial differential and partial difference equations when this happens.
Moreover we show that the infinitesimal generators of generalized symmetries
depending on arbitrary functions, both for continuous and discrete equations,
effectively play the role of master symmetries
Algebraic entropy for systems of quad equations
In this work I discuss briefly the calculation of the algebraic entropy for
systems of quad equations. In particular, I observe that since systems of
multilinear equations can have algebraic solution, in some cases one might need
to restrict the direction of evolution only to the pair of vertices yielding a
birational evolution. Some examples from the exiting literature are presented
and discussed within this framework.Comment: 24 pages (amsart style), 5 figures. This paper is dedicated to the
memory of Prof. Decio Lev
Darboux Integrability of Trapezoidal and Families of Lattice Equations II: General Solutions
In this paper we construct the general solutions of two families of
quad-equations, namely the trapezoidal equations and the
equations. These solutions are obtained exploiting the properties of the first
integrals in the Darboux sense, which were derived in [Gubbiotti G., Yamilov
R.I., J. Phys. A: Math. Theor. 50 (2017), 345205, 26 pages, arXiv:1608.03506].
These first integrals are used to reduce the problem to the solution of some
linear or linearizable non-autonomous ordinary difference equations which can
be formally solved
Growth and integrability of some birational maps in dimension three
Motivated by the study of the Kahan--Hirota--Kimura discretisation of the
Euler top, we characterise the growth and integrability properties of a
collection of elements in the Cremona group of a complex projective 3-space
using techniques from algebraic geometry. This collection consists of maps
obtained by composing the standard Cremona transformation
with projectivities that permute
the fixed points of and the points over which
performs a divisorial contraction. More specifically, we show that three
behaviour are possible: (A) integrable with quadratic degree growth and two
invariants, (B) periodic with two-periodic degree sequences and more than two
invariants, and (C) non-integrable with submaximal degree growth and one
invariant.Comment: 46 pages, 6 figures, 7 tables, comments are welcom
An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps
We show how to construct in an elementary way the invariant of the KHK
discretisation of a cubic Hamiltonian system in two dimensions. That is, we
show that this invariant is expressible as the product of the ratios of affine
polynomials defining the prolongation of the three parallel sides of a hexagon.
On the vertices of such a hexagon lie the indeterminacy points of the KHK map.
This result is obtained analysing the structure of the singular fibres of the
known invariant. We apply this construction to several examples, and we prove
that a similar result holds true for a case outside the hypotheses of the main
theorem, leading us to conjecture that further extensions are possible.Comment: 30 pages, 7 figure
Discrete integrable systems and random Lax matrices
We study properties of Hamiltonian integrable systems with random initial
data by considering their Lax representation. Specifically, we investigate the
spectral behaviour of the corresponding Lax matrices when the number of
degrees of freedom of the system goes to infinity and the initial data is
sampled according to a properly chosen Gibbs measure. We give an exact
description of the limit density of states for the exponential Toda lattice and
the Volterra lattice in terms of the Laguerre and antisymmetric Gaussian
-ensemble in the high temperature regime. For generalizations of the
Volterra lattice to short range interactions, called INB additive and
multiplicative lattices, the focusing Ablowitz--Ladik lattice and the focusing
Schur flow, we derive numerically the density of states. For all these systems,
we obtain explicitly the density of states in the ground states.Comment: 35 pages, 8 figures, 1 tabl