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 Laplace–Darboux transformations used for solving non-factorizable bivariate linear partial differential equations (LPDEs).
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.