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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

\( \hat{a}= c- ab -a_x \quad \mbox{and} \quad \hat{b}=c- ab -b_y.\)

Their importance is due to the classical theorem:

Theorem: Two operators of the form are equivalent under gauge transformations if and only if when their Laplace invariants coincide pairwise.

Here the operators

\( A \quad \mbox{and} \quad \tilde A\)

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.

\( c- ab -a_x \neq 0 \quad \mbox{and/or} \quad c- ab -b_y \neq 0,\)

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.

\( c- ab -a_x=0 \quad \mbox{and} \quad c- ab -b_y =0,\)

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.
See also

Partial derivative
Invariant (mathematics)
Invariant theory

References

G. Darboux, "Leçons sur la théorie général des surfaces" , Gauthier-Villars (1912) (Edition: Second)
G. Tzitzeica G., "Sur un theoreme de M. Darboux". Comptes Rendu de l'Academie des Aciences 150 (1910), pp.955-956; 971-974
L. Bianchi, "Lezioni di geometria differenziale", Zanichelli, Bologna, (1924)
A. B. Shabat, "On the theory of Laplace-Darboux transformations". J. Theor. Math. Phys. Vol. 103, N.1,pp. 170-175 (1995) [1]
A.N. Leznov, M.P. Saveliev. "Group-theoretical methods for integration on non-linear dynamical systems" (Russian), Moscow, Nauka (1985). English translation: Progress in Physics, 15. Birkhauser Verlag, Basel (1992)

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