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- Laplace_invariant abstract "In differential equations, the Laplace invariant of any of certain differential operators is a certain function of the coefficients and their derivatives. Consider a bivariate hyperbolic differential operator of the second orderwhose coefficientsare smooth functions of two variables. Its Laplace invariants have the formTheir importance is due to the classical theorem:Theorem: Two operators of the form are equivalent under gauge transformations if and only if their Laplace invariants coincide pairwise.Here the operatorsare called equivalent if there is a gauge transformation that takes one to the other:Laplace invariants can be regarded as factorization "remainders" for the initial operator A:If at least one of Laplace invariants is not equal to zero, i.e.then this representation is a first step of the Laplace–Darboux transformations used for solvingnon-factorizable bivariate linear partial differential equations (LPDEs).If both Laplace invariants are equal to zero, i.e.then the differential operator A is factorizable and corresponding linear partial differential equation of second order is solvable.Laplace invariants have been introduced for a bivariate linear partial differential operator (LPDO) of order 2 and of hyperbolic type. They are a particular case of generalized invariants which can be constructed for a bivariate LPDO of arbitrary order and arbitrary type; see Invariant factorization of LPDOs.".
- Laplace_invariant wikiPageExternalLink n426ttx757676531.
- Laplace_invariant wikiPageID "17140644".
- Laplace_invariant wikiPageRevisionID "569300346".
- Laplace_invariant hasPhotoCollection Laplace_invariant.
- Laplace_invariant subject Category:Differential_operators.
- Laplace_invariant subject Category:Multivariable_calculus.
- Laplace_invariant type Abstraction100002137.
- Laplace_invariant type DifferentialOperators.
- Laplace_invariant type Function113783816.
- Laplace_invariant type MathematicalRelation113783581.
- Laplace_invariant type Operator113786413.
- Laplace_invariant type Relation100031921.
- Laplace_invariant comment "In differential equations, the Laplace invariant of any of certain differential operators is a certain function of the coefficients and their derivatives. Consider a bivariate hyperbolic differential operator of the second orderwhose coefficientsare smooth functions of two variables.".
- Laplace_invariant label "Laplace invariant".
- Laplace_invariant sameAs m.043rlz7.
- Laplace_invariant sameAs Q6153242.
- Laplace_invariant sameAs Q6153242.
- Laplace_invariant sameAs Laplace_invariant.
- Laplace_invariant wasDerivedFrom Laplace_invariant?oldid=569300346.
- Laplace_invariant isPrimaryTopicOf Laplace_invariant.