Invariant factorization of LPDOs

Operator of order 2

Consider an operator

with smooth coefficients and look for a factorization

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

Then in all cases

where the notation L = p 1 ∂ x + p 2 ∂ y is used.

Without loss of generality, a 20 ≠ 0 , i.e. p 1 ≠ 0 , and it can be taken as 1, p 1 = 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 ω be a root of the polynomial P 2 , 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 ω is simple, i.e.

equations have the unique solution:

At this step, for each root of the polynomial P 2 a corresponding set of coefficients p j is computed.

Step 3. Check factorization condition (which is the last of the initial 6 equations)

written in the known variables p j and ω ):

If

the operator A 2 is factorizable and explicit form for the factorization coefficients p j 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 A 2 , the conditions of factorization are described by the following system:

with L = p 1 ∂ x + p 2 ∂ y , and again a 30 ≠ 0 , i.e. p 1 = 1 , and three-step procedure yields:

At the first step, the roots of a cubic polynomial

have to be found. Again ω 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.

Invariant Formulation

Definition The operators A , A ~ 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 A ~ in the form

with first-order operator L = ∂ x − ω ∂ y + p where ω is an arbitrary simple root of the characteristic polynomial

Factorization is possible then for each simple root ω ~ iff

for n = 2     → l 2 = 0 ,

for n = 3     → l 3 = 0 , l 31 = 0 ,

for n = 4     → l 4 = 0 , l 41 = 0 , l 42 = 0 ,

and so on. All functions l 2 , l 3 , l 31 , l 4 , l 41 ,     l 42 , . . . are known functions, for instance,

and so on.

Theorem All functions

are invariants under gauge transformations.

Definition Invariants l 2 = a 00 − L ( p 6 ) + p 3 p 6 , l 3 = a 00 − L ( p 9 ) + p 3 p 9 , l 31 , . . . . . 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 A ~ 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 A t of an operator A = ∑ a α ∂ α , ∂ α = ∂ 1 α 1 ⋯ ∂ n α n . is defined as A t u = ∑ ( − 1 ) | α | ∂ α ( a α u ) . and the identity ∂ γ ( u v ) = ∑ ( γ α ) ∂ α u , ∂ γ − α v implies that A t = ∑ ( − 1 ) | α + β | ( α + β α ) ( ∂ β a α + β ) ∂ α .

Now the coefficients are

A t = ∑ a ~ α ∂ α , a ~ α = ∑ ( − 1 ) | α + β | ( α + β α ) ∂ β ( a α + β ) .

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

In particular, for the operator A 2 the coefficients are a ~ j k = a j k , j + k = 2 ; a ~ 10 = − a 10 + 2 ∂ x a 20 + ∂ y a 11 , a ~ 01 = − a 01 + ∂ x a 11 + 2 ∂ y a 02 ,

For instance, the operator

is factorizable as

and its transpose A 1 t is factorizable then as [ . . . ] [ ∂ x − ∂ y + 1 2 ( y + x ) ] .