38 research outputs found
Towards the theory of integrable hyperbolic equations of third order
The examples are considered of integrable hyperbolic equations of third order
with two independent variables. In particular, an equation is found which
admits as evolutionary symmetries the Krichever--Novikov equation and the
modified Landau--Lifshitz system. The problem of choice of dynamical variables
for the hyperbolic equations is discussed.Comment: 22
Discrete analogues of the Liouville equation
The notion of Laplace invariants is transferred to the lattices and discrete
equations which are difference analogs of hyperbolic PDE's with two independent
variables. The sequence of Laplace invariants satisfy the discrete analog of
twodimensional Toda lattice. The terminating of this sequence by zeroes is
proved to be the necessary condition for existence of the integrals of the
equation under consideration. The formulae are presented for the higher
symmetries of the equations possessing integrals. The general theory is
illustrated by examples of difference analogs of Liouville equation.Comment: LaTeX, 15 pages, submitted to Teor. i Mat. Fi
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=-â â, {(dk /d xk) t (n,x) } k=1 â, 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
On Darboux Integrable Semi-Discrete Chains
Differential-difference equation
with unknown
depending on continuous and discrete variables and is studied.
We call an equation of such kind Darboux integrable, if there exist two
functions (called integrals) and of a finite number of dynamical
variables such that and , where is the operator of total
differentiation with respect to , and is the shift operator:
. It is proved that the integrals can be brought to some
canonical form. A method of construction of an explicit formula for general
solution to Darboux integrable chains is discussed and for a class of chains
such solutions are found.Comment: 19 page
Prolongation Approach to B\"{a}cklund Transformation of Zhiber-Mikhailov-Shabat Equation
The prolongation structure of Zhiber-Mikhailov-Shabat (ZMS) equation is
studied by using Wahlquist-Estabrook's method. The Lax-pair for ZMS equation
and Riccati equations for pseudopotentials are formulated respectively from
linear and nonlinear realizations of the prolongation structure. Based on
nonlinear realization of the prolongation structure, an auto-Bcklund
transformation of ZMS equation is obtained.Comment: Revtex, no figures, to appear in J. Math. Phys. (1996
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
Tzitzeica solitons versus relativistic CalogeroâMoser three-body clusters
We establish a connection between the hyperbolic relativistic CalogeroâMoser systems and a class of soliton solutions to the Tzitzeica equation (also called the DoddâBulloughâZhiberâShabatâMikhailov equation). In the 6N-dimensional phase space Omega of the relativistic systems with 2N particles and N antiparticles, there exists a 2N-dimensional PoincarĂ©-invariant submanifold OmegaP corresponding to N free particles and N bound particle-antiparticle pairs in their ground state. The Tzitzeica N-soliton tau functions under consideration are real valued and obtained via the dual Lax matrix evaluated in points of OmegaP. This correspondence leads to a picture of the soliton as a cluster of two particles and one antiparticle in their lowest internal energy state