Characteristic algebras and classification of discrete equations

1. Characteristic algebras and classification of discrete equations Ismagil Habibullin Ufa, Institute of Mathematics, Russian Academy of Science e-mail: ihabib@imat.rb.ru Russia

2. 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

3. 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

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

5. 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.

6. 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

7. 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

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

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

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

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

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

13. 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)

14. 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)

15. Characteristic algebra of the discrete Liouville equation • Explicit form of the operators

16. 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

17. 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 (2) here and , . Defining the operators below introduce the characteristic Lie algebra, generated by multiple commutators and

18. 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 Classification problem One can use the necessary condition of the Liouville integrability: any subalgebra of the characteristic algebra is of finite dimension.

19. 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

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

21. n-integrals • The corresponding n-integrals are