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
A and \( {\displaystyle A\quad {\text{and}}\quad {\tilde {A}}} \)
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.
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 Sciences 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)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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