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

Infinite series solutions of the symmetry equation for the 1+21 +2 dimensional continuous Toda chain

A sequence of solutions to the equation of symmetry for the continuous Toda chain in $1+2$ dimensions is represented in explicit form. This fact leads to the supposition that this equation is completely integrable.Comment: 9 pages, latex, no figure

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

The Klein-Gordon Equation and Differential Substitutions of the Form v=φ(u,ux,uy)v=\varphi(u,u_x,u_y)

We present the complete classification of equations of the form $u_{xy}=f(u,u_x,u_y)$ and the Klein-Gordon equations $v_{xy}=F(v)$ connected with one another by differential substitutions $v=\varphi(u,u_x,u_y)$ such that $\varphi_{u_x}\varphi_{u_y}\neq 0$ over the ring of complex-valued variables

On Darboux Integrable Semi-Discrete Chains

Differential-difference equation $\frac{d}{dx}t(n+1,x)=f(x,t(n,x),t(n+1,x),\frac{d}{dx}t(n,x))$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$ is studied. We call an equation of such kind Darboux integrable, if there exist two functions (called integrals) $F$ and $I$ of a finite number of dynamical variables such that $D_xF=0$ and $DI=I$, where $D_x$ is the operator of total differentiation with respect to $x$, and $D$ is the shift operator: $Dp(n)=p(n+1)$. 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

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

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-B$\ddot{a}$cklund transformation of ZMS equation is obtained.Comment: Revtex, no figures, to appear in J. Math. Phys. (1996

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