About: Invariant factorization of LPDOs

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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.

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rdfs:label	Invariant factorization of LPDOs (en)
rdfs:comment	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. (en)
dcterms:subject	Differential operators Partial differential equations Multivariable calculus
Wikipage page ID	17143362 (xsd:integer)
Wikipage revision ID	940541496 (xsd:integer)
Link from a Wikipage to another Wikipage	Binomial coefficient Hyperbolic partial differential equation Characteristic polynomial Invariant (mathematics) Partial derivative Laplace invariant Differential operators Invariant theory Partial differential equations Laplace Multivariable calculus
Link from a Wikipage to an external page	https://archive.today/20020906093934/http:/www2.appmath.com:8080/site/few/few.html https://arxiv.org/abs/math-ph/0607040/ https://doi.org/10.1007%2Fs11232-005-0178-7 https://doi.org/10.1007%2Fs11232-006-0079-4
sameAs	Invariant factorization of LPDOs Invariant factorization of LPDOs Invariant factorization of LPDOs Invariant factorization of LPDOs
dbp:wikiPageUsesTemplate	dbt:Context
has abstract	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 can be found in General invariants are defined in and invariant formulation of the Beals-Kartashova factorization is given in (en)
prov:wasDerivedFrom	wikipedia-en:Invariant_factorization_of_LPDOs?oldid=940541496&ns=0
page length (characters) of wiki page	15611 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Invariant_factorization_of_LPDOs
is Link from a Wikipage to another Wikipage of	Invariant differential operator Laplace invariant Jean Gaston Darboux
is foaf:primaryTopic of	wikipedia-en:Invariant_factorization_of_LPDOs

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