Characteristic algebras and classification of discrete equations Ismagil Habibullin Ufa, Institute of Mathematics, Russian Academy of Science Russia

Content What is the characteristic algebra of the completely continuous equation? (explain with the example of Liouville equation) What is the characteristic algebra of the discrete equation? Definition, examples. Characteristic algebra and classification of integrable equations Examples of new equations

Characteristic Lie algebra for the Liouville equation In order to explain the notion of the characteristic algebra we start with the purely continuous model – the Liuoville equation

It is known that any solution of the Liouville equation satisfies the following conditions and

This allows one to reduce the Liuoville equation to a system of two ordinary differential equations The l.h.s. of these equations are called y- and x- integrals.

How to find such kind relations? Any x-integral should satisfy the following equation Evaluating the total derivative by chain rule one gets Introduce two vector fields and.

Then will satisfy the equations and so on. The Lie algebra generated by and with the usual commutator is called characteristic Lie algebra of the equation. Evidently any hyperbolic type equation admits a characteristic algebra, but in some cases the algebra is of finite dimension. Only in these cases the integrals exist. For the Liouville equation one gets

Discrete equations Consider a discrete nonlinear equation of the form where depending on the integers Introduce the shift operators and acting as follows and. For the iterated shifts we use the notations and, What is the integral in the discrete case? is an unknown function (1)

Integrals and vector fields A function, depending on and a finite number of the dynamical variables is called -integral, if it is a stationary "point" of the shift with respect to really functionsolves the functional equation

Lemma 1 The-integral doesn’t depend on the variables in the set If F is is a solution of the following ordinary discrete equation whereis a function on - integral, then each solution of the equation (1) Due to Lemma 1 the equation can be rewritten as

The left hand side of the equation contains right hand side does not. while the Hence the total derivative of with respect tovanishes.In other words the annulates the-integral operator In a similar way one can check that any operator of the form where, satisfies the equation

Up to now we shifted the variables forward, shift them backward now and use the equation It can be represented in the form Due to the original equation written as By introducing the notation One gets They satisfy Define the operators

Operators annulating the invariant Summarizing one gets that all the operators in the infinite set below should annulate the invariant F Remind that the operators are defined as follows and Linear envelope of the operators and all of the multiple commutators constitute a Lie algebra. We call it characteristic algebra of the equation (1)

Equations of Liouville type Algebraic criterion of existence of the integrals Equation is of the Liouville type if it admits integrals in both directions. Theorem 1. Equation (1) admits a nontrivial -invariant if and only if algebra is of finite dimension. Example. Consider discrete analogue of the Liouville equation (found by Zabrodin, Protogenov, 1997)

Characteristic algebra of the discrete Liouville equation Explicit form of the operators

Basis of the char. algebra For the discrete Liouville equation the algebra L is of dimension 4. The basis contains the operators Two of them satisfy the condition below and all the other commutators vanish

Semi discrete equations In the same way one can define the characteristic algebra for the semi discrete equations, with one discrete and one continuous variables here, and. Defining the operators below introduce the characteristic Lie algebra, generated by multiple commutators and (2)

Classification problem The main classification problem is to find all equations of the form (1) and form (2) of the Liouville type i.e. equations with finite dimensional characteristic Lie algebras. It is a hard problem. The algebra usually generated by an infinite set of the operators One can use the necessary condition of the Liouville integrability: any subalgebra of the characteristic algebra is of finite dimension.

Example of classification Suppose that subalgebra generated by the following two operators is two-dimensional. Then the r.h.s. of the equation (2) should satisfy the differential equation where

New equations put an additional constraint The following two equations admit n-integrals

n-integrals The corresponding n-integrals are