11 research outputs found
Complete list of Darboux Integrable Chains of the form
We study differential-difference equation of the form with unknown
depending on continuous and discrete variables and . Equation
of such kind is called Darboux integrable, if there exist two functions and
of a finite number of arguments , ,
, such that and , where
is the operator of total differentiation with respect to , and is
the shift operator: . Reformulation of Darboux integrability in
terms of finiteness of two characteristic Lie algebras gives an effective tool
for classification of integrable equations. The complete list of Darboux
integrable equations is given in the case when the function is of the
special form
Integrable nonlinear equations on a circle
The concept of integrable boundary value problems for soliton equations on
and is extended to bounded regions enclosed by
smooth curves. Classes of integrable boundary conditions on a circle for the
Toda lattice and its reductions are found.Comment: 23 page
Characteristic Lie rings, finitely-generated modules and integrability conditions for 2+1 dimensional lattices
Characteristic Lie rings for Toda type 2+1 dimensional lattices are defined.
Some properties of these rings are studied. Infinite sequence of special kind
modules are introduced. It is proved that for known integrable lattices these
modules are finitely generated. Classification algorithm based on this
observation is briefly discussed.Comment: 11 page
Classification of integrable discrete Klein-Gordon models
The Lie algebraic integrability test is applied to the problem of
classification of integrable Klein-Gordon type equations on quad-graphs. The
list of equations passing the test is presented containing several well-known
integrable models. A new integrable example is found, its higher symmetry is
presented.Comment: 12 pages, submitted to Physica Script
On the classification of Darboux integrable chains
Cataloged from PDF version of article.We study differential-difference equation (d/dx) t (n+1,x) =f (t (n,x),t (n+1,x), (d/dx) t (n,x)) with unknown t (n,x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, { t (n+k,x) } k=-∞ ∞, {(dk /d xk) t (n,x) } k=1 ∞, such that Dx F=0 and DI=I, where D x is the operator of total differentiation with respect to x and D is the shift operator: Dp (n) =p (n+1). Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function f is of the special form f (u,v,w) =w+g (u,v). © 2009 American Institute of Physics
Miura-Type Transformations for Integrable Lattices in 3D
This article studies a class of integrable semi-discrete equations with one continuous and two discrete independent variables. At present, in the literature there are nine integrable equations of the form un+1,xj=f(un,xj,unj+1,unj,un+1j,un+1j−1) up to point transformations. An efficient method based on some relation that generalizes the notion of the local conservation law is proposed for searching for Miura-type transformations relating to semi-discrete equations in 3D. The efficiency of the method is illustrated with the equations from the list. For one of the equations, which is little studied, the continuum limit is calculated. For this equation, the problem of finite-field reductions in the form of Darboux integrable systems of equations of a hyperbolic type is discussed. For reductions of small orders, N=1 and N=2, complete sets of characteristic integrals are presented. Note that the existence of characteristic integrals makes it possible to construct particular solutions to the original lattice. For the case N=1, an explicit solution was found in this paper. A new semi-discrete equation is found that lies beyond the considered class. For this equation, the Lax pair is presented