Invariant factorization of LPDOs

From Wikipedia, the free encyclopedia
Jump to: navigation, search


The factorization of a linear partial differential operator (LPDO) is an important issue in the theory of integrability, due to the Laplace-Darboux transformations,[1] which allow construction of integrable LPDEs. Laplace solved the factorization problem for a bivariate hyperbolic operator of the second order (see Hyperbolic partial differential equation), 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 (i.e. self-adjoint) operators.

Beals-Kartashova-factorization (also called BK-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 (also called symbol) 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[2] General invariants are defined in[3] and invariant formulation of the Beals-Kartashova factorization is given in[4]

Beals-Kartashova Factorization[edit]

Operator of order 2[edit]

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[edit]

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 [edit]

Invariant Formulation[edit]

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[edit]

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

See also[edit]

Notes[edit]

  1. ^ Weiss (1986)
  2. ^ R. Beals, E. Kartashova. Constructively factoring linear partial differential operators in two variables. Theor. Math. Phys. 145(2), pp. 1510-1523 (2005)
  3. ^ E. Kartashova. A Hierarchy of Generalized Invariants for Linear Partial Differential Operators. Theor. Math. Phys. 147(3), pp. 839-846 (2006)
  4. ^ E. Kartashova, O. Rudenko. Invariant Form of BK-factorization and its Applications. Proc. GIFT-2006, pp.225-241, Eds.: J. Calmet, R. W. Tucker, Karlsruhe University Press (2006); arXiv

References[edit]

  • J. Weiss. Bäcklund transformation and the Painlevé property. [1] J. Math. Phys. 27, 1293-1305 (1986).
  • R. Beals, E. Kartashova. Constructively factoring linear partial differential operators in two variables. Theor. Math. Phys. 145(2), pp. 1510-1523 (2005)
  • E. Kartashova. A Hierarchy of Generalized Invariants for Linear Partial Differential Operators. Theor. Math. Phys. 147(3), pp. 839-846 (2006)
  • E. Kartashova, O. Rudenko. Invariant Form of BK-factorization and its Applications. Proc. GIFT-2006, pp.225-241, Eds.: J. Calmet, R. W. Tucker, Karlsruhe University Press (2006); arXiv