Discretization of hyperbolic type Darboux integrable equations preserving integrability

Abstract

A method of integrable discretization of the Liouville type nonlinear partial differential equations is suggested based on integrals. New examples of discrete Liouville type models are presented.

Generalized symmetry classification of discrete equations of a class depending on twelve parameters

“…F (u n,m , u n+1,m , u n,m+1 , u n+1,m+1 ) = F (u n+1,m , u n,m , u n+1,m+1 , u n,m+1 ) = F (u n,m+1 , u n+1,m+1 , u n,m , u n+1,m ), and it depends on 7 arbitrary constant parameters. As it has been shown in [31], the Q V equation has the generalized symmetries (11) for all values of these parameters. We are interested in the intersection of our class (3) and the Q V equation, which has the form (u n,m u n+1,m + u n,m+1 u n+1,m+1 )k 1 + (u n,m u n+1,m+1 + u n+1,m u n,m+1 )k 2 + (u n,m + u n+1,m + u n,m+1 + u n+1,m+1 )k 3 + k 4 = 0.…”

“…The existence of symmetries of the form (11) implies the existence of a symmetry given by (10). The generalized symmetries (11) as themselves are integrable differential-difference equations. In autonomous case they are integrable equations of the Volterra type of a complete list obtained in [33], see [34] for details.…”

Discretization of Liouville type nonautonomous equations preserving integrals

“…Evidently, the coefficient before t xx in (9) vanishes, that is f tx = 0. Now collection of the coefficients before t x in (9) gives f t − f = 0, or f = A(x, y, t y )e t .…”

“…Evidently, the coefficient before t xx in (9) vanishes, that is f tx = 0. Now collection of the coefficients before t x in (9) gives f t − f = 0, or f = A(x, y, t y )e t . We substitute the expression f = A(x, y, t y )e t into (9) and get A x e t + A ty e 2t = 0 which immediately implies A x = A ty = 0.…”

“…Moreover, in [9] we did not evaluate the continuum limit equations of the chains obtained by the discretization.…”