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

:$hat\{a\}=\; c-\; ab\; -a\_x\; quad\; mbox\{and\}\; quad\; hat\{b\}=c-\; ab\; -b\_y.$

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 :$A\; quad\; mbox\{and\}\; quad\; ilde\; A$

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

:$ilde\; Ag=\; e^\{-varphi\}A(e^\{varphi\}g)equiv\; A\_varphi\; g.$

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

:

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

:$c-\; ab\; -a\_x\; eq\; 0\; quad\; mbox\{and/or\}\; quadc-\; ab\; -b\_y\; eq\; 0,$

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.

:$c-\; ab\; -a\_x=0\; quad\; mbox\{and\}\; quadc-\; ab\; -b\_y\; =0,$

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.

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) [http://www.springerlink.com/content/n426ttx757676531/]
* 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)

ee also

* Partial derivative
* Invariant (mathematics)
* Invariant theory
* Invariant factorization of LPDOs