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


 * $$\partial_x \, \partial_y + a\,\partial_x + b\,\partial_y + c, \,$$

whose coefficients


 * $$ a=a(x,y), \ \ b=c(x,y), \ \ c=c(x,y), $$

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


 * $$\hat{a}= c- ab -a_x \quad \text{and} \quad \hat{b}=c- ab -b_y.$$

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

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


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

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


 * $$\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- 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- 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.