Characteristic algebras and classification of discrete equations

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Characteristic algebras and classification of
discrete equations

• Ismagil Habibullin
• Ufa, Institute of Mathematics, Russian Academy of
  Science
• e-mail ihabib_at_imat.rb.ru
• Russia

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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

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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

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

and
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• 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.
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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
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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
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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?
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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
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Lemma 1 The
-integral doesnt depend on the
variables in the set
If F is
- integral, then each solution of the equation
(1)
is a solution of the following ordinary discrete
equation
where
is a function on
Due to Lemma 1 the equation
can be rewritten as
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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
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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
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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)
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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)
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Characteristic algebra of the discrete Liouville
equation

• Explicit form of the operators

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

• The corresponding n-integrals are