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
\partialx\partialy+a\partialx+b\partialy+c,
whose coefficients
a=a(x,y), b=c(x,y), c=c(x,y),
are smooth functions of two variables. Its Laplace invariants have the form
\hat{a}=c-ab-ax and \hat{b}=c-ab-by.
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
A and \tildeA
are called equivalent if there is a gauge transformation that takes one to the other:
\tildeAg=e-\varphiA(e\varphig)\equivA\varphig.
Laplace invariants can be regarded as factorization "remainders" for the initial operator A:
\partialx\partialy+a\partialx+b\partialy+c=\left\{\begin{array}{c} (\partialx+b)(\partialy+a)-ab-ax+c,\\ (\partialy+a)(\partialx+b)-ab-by+c. \end{array}\right.
If at least one of Laplace invariants is not equal to zero, i.e.
c-ab-ax ≠ 0 and/or c-ab-by ≠ 0,
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.
c-ab-ax=0 and c-ab-by=0,
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.