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.
Deep Dive into 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.
In this paper we study Darboux integrable semi-discrete chains of the form
Here unknown function t = t(n, x) depends on discrete and continuous variables n and x respectively; function f = f (x, t, t 1 , t x ) is assumed to be locally analytic, and ∂f ∂tx is not identically zero. The last 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] = t x = d dx t(n, x), t [2] = t xx = d 2 dx 2 t(n, x), t [m] = d m dx m t(n, x), m ∈ N. Introduce the set of dynamical variables containing {t k } ∞ k=-∞ ; {t [m] } ∞ m=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
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. Moreover, the operator ad Dy defined according to the rule ad Dy X = [D y , X] acts on the generators of the algebra in a very simple way.
This makes it possible to obtain effective integrability conditions for the equation ( 2) (see [6]).
A.V. Zhiber and F.Kh. Mukminov investigated the structure of the characteristic algebras for the so-called quadratic systems containing the Liouville equation and the sine-Gordon equation (see [7]). In [7] and [8] the very nontrivial connection between characteristic algebras and Lax pairs of the hyperbolic S-integrable equations and systems of equations is studied, and perspectives on the application of the characteristic algebras to classify such kinds of equations are discussed.
Recently the concept of the characteristic algebras has been defined for discrete models. In our articles [9]- [11] an effective algorithm was worked out to classify Darboux integrable models. By using this algorithm some new classification results were obtained. In [12] a method of classification of S-integrable discrete models is suggested based on the concept of characteristic algebra.
The article is organized as follows. In Section 2 characteristic algebras are defined for the chain (1). In Section 3 we describe the structure of n-integrals and x-integrals of the Darboux integrable chains of general form (1) (see Theorems 3.1 and 3.2). Then we show that one can choose the minimal order n-integral and the minimal order x-integral of a special canonical form, important for the purpose of classification (see Theorems 3.3 and 3.4).
The complete classification of a particular case t 1x = t x + d(t 1 , t) in [11] was done due to the finiteness of the characteristic algebras in both directions. However, algebras themselves were not described. In Subsections 4.1 and 4.2 we fill up this gap and represent the tables of multiplications for all of these algebras.
The problem of finding explicit solutions for Darboux integrable models is rather complicated.
Even in the mostly studied case of PDE u xy = f (x, y, u, u x , u y ) this problem is not completely solved.
In Subsection 4.3 we give the explicit formulas for general solutions of the integrable chains in
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