View source for Laplace invariant

In [[differential equation]]s, the '''Laplace invariant''' of any of certain [[differential operator]]s is a certain function of the coefficients and their [[derivative]]s.  Consider a bivariate hyperbolic differential operator of the second order

:<math>\partial_x \, \partial_y + a\,\partial_x + b\,\partial_y + c, \,</math>

whose coefficients

:<math> a=a(x,y), \ \ b=c(x,y), \ \ c=c(x,y), </math>

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

:<math>\hat{a}= c- ab -a_x \quad \text{and} \quad \hat{b}=c- ab  -b_y.</math>

Their importance is due to the classical theorem:

'''Theorem''':  ''Two operators of the form are equivalent under [[gauge transformation]]s if and only if their Laplace invariants coincide pairwise.''

Here the operators
 
:<math>A \quad \text{and} \quad \tilde A</math>

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

:<math> \tilde Ag= e^{-\varphi}A(e^{\varphi}g)\equiv A_\varphi g. </math>

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

:<math>\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.</math>

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

:<math> c- ab -a_x \neq 0 \quad \text{and/or} \quad
c- ab  -b_y  \neq 0,</math>

then this representation is a first step of the [[Laplace–Darboux transformation]]s used for solving
''non-factorizable'' bivariate linear partial differential equations (LPDEs).

If both Laplace invariants are equal to zero, i.e.

:<math> c- ab -a_x=0 \quad \text{and} \quad
c- ab  -b_y =0,</math>

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.&nbsp;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.&nbsp;170–175 (1995) [https://doi.org/10.1007%2FBF02069791]
* 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)

[[Category:Multivariable calculus]]
[[Category:Differential operators]]