differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s, the Laplace invariant of any of certain

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

s is a certain function of the coefficients and their

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

s. 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= c- ab -a_x \quad \text \quad \hat=c- ab  -b_y.

Their importance is due to the classical theorem: Theorem: ''Two operators of the form are equivalent under

gauge transformation In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...

s if and only if their Laplace invariants coincide pairwise.'' Here the operators :

A \quad \text \quad \tilde A

are called ''equivalent'' if there is a

that takes one to the other: :

\tilde Ag= e^A(e^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 \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. :

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.

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

* 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) Multivariable calculus Differential operators

See also

References