The factorization of a linear partial differential operator (LPDO) 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 (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 n {\displaystyle n} 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
A 2 = a 20 ∂ x 2 + a 11 ∂ x ∂ y + a 02 ∂ y 2 + a 10 ∂ x + a 01 ∂ y + a 00 . {\displaystyle {\mathcal {A}}_{2}=a_{20}\partial _{x}^{2}+a_{11}\partial _{x}\partial _{y}+a_{02}\partial _{y}^{2}+a_{10}\partial _{x}+a_{01}\partial _{y}+a_{00}.}
with smooth coefficients and look for a factorization
A 2 = ( p 1 ∂ x + p 2 ∂ y + p 3 ) ( p 4 ∂ x + p 5 ∂ y + p 6 ) . {\displaystyle {\mathcal {A}}_{2}=(p_{1}\partial _{x}+p_{2}\partial _{y}+p_{3})(p_{4}\partial _{x}+p_{5}\partial _{y}+p_{6}).}
Let us write down the equations on p i {\displaystyle p_{i}} explicitly, keeping in mind the rule of left composition, i.e. that
∂ x ( α ∂ y ) = ∂ x ( α ) ∂ y + α ∂ x y . {\displaystyle \partial _{x}(\alpha \partial _{y})=\partial _{x}(\alpha )\partial _{y}+\alpha \partial _{xy}.}
Then in all cases
a 20 = p 1 p 4 , {\displaystyle a_{20}=p_{1}p_{4},}
a 11 = p 2 p 4 + p 1 p 5 , {\displaystyle a_{11}=p_{2}p_{4}+p_{1}p_{5},}
a 02 = p 2 p 5 , {\displaystyle a_{02}=p_{2}p_{5},}
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