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
∂ x ∂ y + a ∂ x + b ∂ y + c , {\displaystyle \partial _{x}\,\partial _{y}+a\,\partial _{x}+b\,\partial _{y}+c,\,}
whose coefficients
a = a ( x , y ) , b = c ( x , y ) , c = c ( x , y ) , {\displaystyle a=a(x,y),\ \ b=c(x,y),\ \ c=c(x,y),}
are smooth functions of two variables. Its Laplace invariants have the form
a ^ = c − a b − a x and b ^ = c − a b − b y . {\displaystyle {\hat {a}}=c-ab-a_{x}\quad {\text{and}}\quad {\hat {b}}=c-ab-b_{y}.}
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 A ~ {\displaystyle A\quad {\text{and}}\quad {\tilde {A}}}
are called equivalent if there is a gauge transformation that takes one to the other:
A ~ g = e − φ A ( e φ g ) ≡ A φ g . {\displaystyle {\tilde {A}}g=e^{-\varphi }A(e^{\varphi }g)\equiv A_{\varphi }g.}
Laplace invariants can be regarded as factorization "remainders" for the initial operator A:
∂ x ∂ y + a ∂ x + b ∂ y + c = { ( ∂ x + b ) ( ∂ y + a ) − a b − a x + c , ( ∂ y + a ) ( ∂ x + b ) − a b − b y + c . {\displaystyle \partial _{x}\,\partial _{y}+a\,\partial _{x}+b\,\partial _{y}+c=\left\{{\begin{array}{c}(\partial _{x}+b)(\partial _{y}+a)-ab-a_{x}+c,\\(\partial _{y}+a)(\partial _{x}+b)-ab-b_{y}+c.\end{array}}\right.}
If at least one of Laplace invariants is not equal to zero, i.e.
c − a b − a x ≠ 0 and/or c − a b − b y ≠ 0 , {\displaystyle c-ab-a_{x}\neq 0\quad {\text{and/or}}\quad c-ab-b_{y}\neq 0,}
then this representation is a first step of the Laplace–Darboux transformations used for solving non-factorizable bivariate linear partial differential equations (LPDEs). If both Laplace invariants are equal to zero, i.e.
c − a b − a x = 0 and c − a b − b y = 0 , {\displaystyle c-ab-a_{x}=0\quad {\text{and}}\quad c-ab-b_{y}=0,}
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