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

[math]\displaystyle{ \partial_x \, \partial_y + a\,\partial_x + b\,\partial_y + c, \, }[/math]

whose coefficients

[math]\displaystyle{ 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]\displaystyle{ \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 transformations if and only if their Laplace invariants coincide pairwise.

Here the operators

[math]\displaystyle{ A \quad \text{and} \quad \tilde A }[/math]

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

[math]\displaystyle{ \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]\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. }[/math]

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

[math]\displaystyle{ 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 transformations used for solving non-factorizable bivariate linear partial differential equations (LPDEs).

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

[math]\displaystyle{ 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. 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)