Differential operators | Multivariable calculus

Laplace invariant

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 order whose coefficients are smooth functions of two variables. Its Laplace invariants have the form Their 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 operators are 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 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. (Wikipedia).

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Related pages

Differential operator | Differential equation | Invariant theory | Invariant (mathematics) | Partial derivative | Derivative | Invariant factorization of LPDOs