Invariant factorization of LPDOs


The factorization of a linear partial differential operator is an important issue in the theory of integrability, due to the Laplace-Darboux transformations, which allow construction of integrable LPDEs. Laplace solved the factorization problem for a bivariate hyperbolic operator of the second order, constructing two Laplace invariants. Each Laplace invariant is an explicit polynomial condition of factorization; coefficients of this polynomial are explicit functions of the coefficients of the initial LPDO. The polynomial conditions of factorization are called invariants because they have the same form for equivalent operators.
Beals-Kartashova-factorization is a constructive procedure to factorize a bivariate operator of the arbitrary order and arbitrary form. Correspondingly, the factorization conditions in this case also have polynomial form, are invariants and coincide with Laplace invariants for bivariate hyperbolic operators of the second order. The factorization procedure is purely algebraic, the number of possible factorizations depending on the number of simple roots of the Characteristic polynomial of the initial LPDO and reduced LPDOs appearing at each factorization step. Below the factorization procedure is described for a bivariate operator of arbitrary form, of order 2 and 3. Explicit factorization formulas for an operator of the order can be found in General invariants are defined in and invariant formulation of the Beals-Kartashova factorization is given in

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 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.
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
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.

Operator of order n

Invariant Formulation

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.
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, 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