We study differential-difference equation of the formwith 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, such that D x 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).
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.
We study a differential-difference equation of the form t x ͑n +1͒ = f͑t͑n͒ , t͑n +1͒ , t x ͑n͒͒ with unknown t = t͑n , x͒ depending on x and n. The equation is called a Darboux integrable if there exist functions F ͑called an x-integral͒ and I ͑called an n-integral͒, both of a finite number of variables x , t͑n͒ , t͑n Ϯ 1͒ , t͑n Ϯ 2͒ , ... , t x ͑n͒ , t xx ͑n͒ ,..., such that D x F = 0 and DI = I, where D x is the operator of total differentiation with respect to x and D is the shift operator: Dp͑n͒ = p͑n +1͒. The Darboux integrability property is reformulated in terms of characteristic Lie algebras that give an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial x-integrals is given in the case when the function f is of the special form f͑x , y , z͒ = z + d͑x , y͒.
Differential-difference equation d dx t(n + 1, x) = f (x, 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 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 x 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).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.1 two decades the discrete phenomena have become very popular due to various important applications (for more details see [1]-[3] and references therein).Below we use a subindex to indicate the shift of the discrete argument: t k = t(n+k, x), k ∈ Z, and derivatives with respect to x: t [1] We denote through D and D x the shift operator and the operator of the total derivative with respect to x correspondingly. For instance, Dh(n, x) = h(n + 1, x) and D x h(n, x) = d dx h(n, x). Functions I and F , both depending on x, n, and a finite number of dynamical variables, are called respectively n-and x-integrals of (1), if DI = I and D x F = 0 (see also [4]). Clearly, any function depending on n only, is an x-integral, and any function, depending on x only, is an n-integral. Such integrals are called trivial integrals. One can see that any n-integral I does not depend on variables t m , m ∈ Z\{0}, and any x-integral F does not depend on variables t [m] , m ∈ N.Chain (1) is called Darboux integrable if it admits a nontrivial n-integral and a nontrivial xintegral.The basic ideas on integration of partial differential equations of the hyperbolic type go back to classical works by Laplace, Darboux, Goursat, Vessiot, Monge, Ampere, Legendre, Egorov, etc.Notice that understanding of integration as finding an explicit formula for a general solution was later replaced by other, in a sense less obligatory, definitions. For instance, the Darboux method for integration of hyperbolic type equations consists of searching for integrals in both directions followed by the reduction of the equation to two ordinary differential equations. In order to find integrals, provided that they exist, Darboux used the Laplace cascade method. An alternative, more algebraic approach based on the characteristic vector fields was used by Goursat and Vessiot. Namely this method allowed Goursat to get a list of integrable equations [5]. An important contribution to the development of the algebraic method investigating Darboux integrable equations was made by A.B. Shabat who introduced the notion of the characteristic algebra of the hyperbolic equation u x,y = f (x, y, u, u x , u y ) .(2)It turned out that the operator D y of total differentiation, with respect to the variable y, defines a derivative in the characteristic algebra in the direction of x. Mo...
The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral. For the Liouville type integrable equations from the well-known Goursat list for which the integrals of minimal order are of the order less than or equal to two we presented a list of corresponding semi-discrete versions.The list contains new examples of non-autonomous Darboux integrable chains.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.