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

${\displaystyle \partial _{x}\,\partial _{y}+a\,\partial _{x}+b\,\partial _{y}+c,\,}$

whose coefficients

${\displaystyle a=a(x,y),\ \ b=c(x,y),\ \ c=c(x,y),}$

are smooth functions of two variables. Its Laplace invariants have the form

${\displaystyle {\hat {a}}=c-ab-a_{x}\quad {\text{and}}\quad {\hat {b}}=c-ab-b_{y}.}$

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

${\displaystyle A\quad {\text{and}}\quad {\tilde {A}}}$

are called equivalent if there is a gauge transformation that takes one to the other:

${\displaystyle {\tilde {A}}g=e^{-\varphi }A(e^{\varphi }g)\equiv A_{\varphi }g.}$

Laplace invariants can be regarded as factorization "remainders" for the initial operator A:

${\displaystyle \partial _{x}\,\partial _{y}+a\,\partial _{x}+b\,\partial _{y}+c=\left\{{\begin{array}{c}(\partial _{x}+b)(\partial _{y}+a)-ab-a_{x}+c,\\(\partial _{y}+a)(\partial _{x}+b)-ab-b_{y}+c.\end{array}}\right.}$

If at least one of Laplace invariants is not equal to zero, i.e.

${\displaystyle c-ab-a_{x}\neq 0\quad {\text{and/or}}\quad c-ab-b_{y}\neq 0,}$

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.

${\displaystyle c-ab-a_{x}=0\quad {\text{and}}\quad c-ab-b_{y}=0,}$

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.