264 research outputs found
Boussinesq-like multi-component lattice equations and multi-dimensional consistency
We consider quasilinear, multi-variable, constant coefficient, lattice
equations defined on the edges of the elementary square of the lattice, modeled
after the lattice modified Boussinesq (lmBSQ) equation, e.g., . These equations are classified into three canonical forms and
the consequences of their multidimensional consistency
(Consistency-Around-the-Cube, CAC) are derived. One of the consequences is a
restriction on form of the equation for the variable, which in turn implies
further consistency conditions, that are solved. As result we obtain a number
of integrable multi-component lattice equations, some generalizing lmBSQ.Comment: 24 page
On a two-parameter extension of the lattice KdV system associated with an elliptic curve
A general structure is developed from which a system of integrable partial
difference equations is derived generalising the lattice KdV equation. The
construction is based on an infinite matrix scheme with as key ingredient a
(formal) elliptic Cauchy kernel. The consistency and integrability of the
lattice system is discussed as well as special solutions and associated
continuum equations.Comment: Submitted to the proceedings of the Oeresund PDE-symposium, 23-25 May
2002; 17 pages LaTeX, style-file include
Quantum discrete Dubrovin equations
The discrete equations of motion for the quantum mappings of KdV type are
given in terms of the Sklyanin variables (which are also known as quantum
separated variables). Both temporal (discrete-time) evolutions and spatial
(along the lattice at a constant time-level) evolutions are considered. In the
classical limit, the temporal equations reduce to the (classical) discrete
Dubrovin equations as given in a previous publication. The reconstruction of
the original dynamical variables in terms of the Sklyanin variables is also
achieved.Comment: 25 page
A new two-dimensional lattice model that is "consistent around a cube"
For two-dimensional lattice equations one definition of integrability is that
the model can be naturally and consistently extended to three dimensions, i.e.,
that it is "consistent around a cube" (CAC). As a consequence of CAC one can
construct a Lax pair for the model. Recently Adler, Bobenko and Suris conducted
a search based on this principle and certain additional assumptions. One of
those assumptions was the "tetrahedron property", which is satisfied by most
known equations. We present here one lattice equation that satisfies the
consistency condition but does not have the tetrahedron property. Its Lax pair
is also presented and some basic properties discussed.Comment: 8 pages in LaTe
An algebraic method of classification of S-integrable discrete models
A method of classification of integrable equations on quad-graphs is
discussed based on algebraic ideas. We assign a Lie ring to the equation and
study the function describing the dimensions of linear spaces spanned by
multiple commutators of the ring generators. For the generic case this function
grows exponentially. Examples show that for integrable equations it grows
slower. We propose a classification scheme based on this observation.Comment: 11 pages, workshop "Nonlinear Physics. Theory and Experiment VI",
submitted to TM
Solutions of Adler's lattice equation associated with 2-cycles of the Backlund transformation
The BT of Adler's lattice equation is inherent in the equation itself by
virtue of its multidimensional consistency. We refer to a solution of the
equation that is related to itself by the composition of two BTs (with
different Backlund parameters) as a 2-cycle of the BT. In this article we will
show that such solutions are associated with a commuting one-parameter family
of rank-2 (i.e., 2-variable), 2-valued mappings. We will construct the explicit
solution of the mappings within this family and hence give the solutions of
Adler's equation that are 2-cycles of the BT.Comment: 10 pages, contribution to the NEEDS 2007 proceeding
Integrability and Fusion Algebra for Quantum Mappings
We apply the fusion procedure to a quantum Yang-Baxter algebra associated
with time-discrete integrable systems, notably integrable quantum mappings. We
present a general construction of higher-order quantum invariants for these
systems. As an important class of examples, we present the Yang-Baxter
structure of the Gel'fand-Dikii mapping hierarchy, that we have introduced in
previous papers, together with the corresponding explicit commuting family of
quantum invariants.Comment: 26 page
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