Beals-Kartashova Factorization
Operator of order 2
Consider an operator

with smooth coefficients and look for a factorization

Let us write down the equations on
explicitly, keeping in mind the rule of left composition, i.e. that

Then in all cases






where the notation
is used.
Without loss of generality,
i.e.
and it can be taken as 1,
Now solution of the system of 6 equations on the variables



can be found in three steps.
At the first step, the roots of a quadratic polynomial have to be found.
At the second step, a linear system of two algebraic equations has to be solved.
At the third step, one algebraic condition has to be checked.
Step 1. Variables



can be found from the first three equations,



The (possible) solutions are then the functions of the roots of a quadratic polynomial:

Let
be a root of the polynomial
then




Step 2. Substitution of the results obtained at the first step, into the next two equations


yields linear system of two algebraic equations:


In particularly, if the root
is simple, i.e.
then these
equations have the unique solution:


At this step, for each root of the polynomial
a corresponding set of coefficients
is computed.
Step 3. Check factorization condition (which is the last of the initial 6 equations)

written in the known variables
and
):

If

the operator
is factorizable and explicit form for the factorization coefficients
is given above.
Operator of order 3
Consider an operator

with smooth coefficients and look for a factorization

Similar to the case of the operator
the conditions of factorization are described by the following system:










with
and again
i.e.
and three-step procedure yields:
At the first step, the roots of a cubic polynomial

have to be found. Again
denotes a root and first four coefficients are





At the second step, a linear system of three algebraic equations has to be solved:



At the third step, two algebraic conditions have to be checked.
Definition The operators
,
are called equivalent if there is a gauge transformation that takes one to the other:

BK-factorization is then pure algebraic procedure which allows to construct explicitly a factorization of an arbitrary order LPDO
in the form

with first-order operator
where
is an arbitrary simple root of the characteristic polynomial

Factorization is possible then for each simple root
iff
for 
for 
for 
and so on. All functions
are known functions, for instance,



and so on.
Theorem All functions

are invariants under gauge transformations.
Definition Invariants
are called generalized invariants of a bivariate operator of arbitrary order.
In particular case of the bivariate hyperbolic operator its generalized invariants coincide with Laplace invariants (see Laplace invariant).
Corollary If an operator
is factorizable, then all operators equivalent to it, are also factorizable.
Equivalent operators are easy to compute:


and so on. Some example are given below:




Transpose
Factorization of an operator is the first step on the way of solving corresponding equation. But for solution we need right factors and BK-factorization constructs left factors which are easy to construct. On the other hand, the existence of a certain right factor of a LPDO is equivalent to the existence of a corresponding left factor of the transpose of that operator.
Definition The transpose
of an operator
is defined as
and the identity
implies that 
Now the coefficients are


with a standard convention for binomial coefficients in several variables (see Binomial coefficient), e.g. in two variables

In particular, for the operator
the coefficients are 

For instance, the operator

is factorizable as

and its transpose
is factorizable then as 