Characteristic

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One of the basic concepts in the theory of partial differential equations. The role of characteristics manifests itself in essential properties of these equations such as the local properties of solutions, the solvability of various problems, their being well posed, etc.

Suppose that

is a linear partial differential operator of order , and let

be its symbol. Here , is a multi-index, , ,

Let be the hypersurface defined in by the equation , where for , and let

 (1)

In this case is called a characteristic surface or a characteristic for the operator . Other names are: characteristic manifold, characteristic line (in case ).

The example of the Cauchy problem is discussed below. Let be the arbitrary (not necessarily characteristic) hypersurface in defined by the relations

Let be functions defined on in a neighbourhood of , and let

be the Cauchy problem for the unknown function . Here is a given function, is a given linear differential operator of order , and is a vector orthonormal to . Assume, to be definite, that , . Then, by the change of variables

one arrives at the equation

 (2)

The expression under the sign that is not written out does not contain partial derivatives of with respect to of order . Two cases arise:

1) , ;

2) , .

In the first case division of (2) by leads to an equation that can be solved for the highest partial derivative of , that is, can be written in normal form. The initial conditions can be put in the form

For this case the Cauchy problem has been well studied. For example, when the functions in the equations and when the initial data are real-analytic, there exists a unique solution of this problem in the class of real-analytic functions in a sufficiently small neighbourhood of (the Cauchy–Kovalevskaya theorem). In the second case is a characteristic point, and if (1) holds for all , then is called a characteristic. In this case (2) implies that the initial data cannot be arbitrary, and the study of the Cauchy problem becomes complicated.

For example, for the equation

 (3)

initial data can be given on one of its characteristics :

 (4)

If the function is not constant, then the Cauchy problem (3), (4) has no solution in the space . But if is constant, for example equal to , then a solution is not unique in , since it may be any function of the form

where

Thus, the Cauchy problem differs substantially, depending on whether the initial data are given on a characteristic surface or not.

A characteristic has the property of invariance under invertible transformations of the independent variables: If is a solution of (1) and if the transformation leads to , , then satisfies the equation

where

Another property of a characteristic is that is, relative to a characteristic , an interior differential operator.

Elliptic linear differential operators are defined as operators for which there are no (real) characteristics. The definitions of hyperbolic and parabolic operators are also closely connected with the concept of a characteristic. For example, a second-order differential operator in two variables (i.e. ) is of hyperbolic type if it has two families of characteristics and of parabolic type if it has one such family. The knowledge of the characteristics of a differential equation makes it possible to reduce the equation to simpler form. For example, let the equation

 (5)

be hyperbolic. That is, equation (1), which now reads

determines two distinct families of characteristics:

For any selected pair the change of variables by the formula

transforms (5) to the canonical form

For a non-linear differential equation

 (6)

where are multi-indices and , , the characteristic is defined as the hypersurface in with the equation , where and for . In this case the symbol for the operator (6) given by the function is defined as follows:

with the usual assumption . Evidently, may depend, apart from the variables and , also on , and . Suppose, for example, that a first-order equation is given . For simplicity, suppose in addition that . Then (6) takes the form

with a function . The equation of the characteristics is:

Since a solution of this equation may, in fact, depend on , it can be given in parametric form

where these functions satisfy the ordinary differential equations

Geometrically the -tuple determines the so-called characteristic strip (for ). The projection of this strip onto the space determines a curve in such that at every point of it, it touches the plane with direction coefficients . This curve is also called a characteristic of the equation (6).

References

 [1] S. Misohata, "The theory of partial differential equations" , Cambridge Univ. Press (1973) [2] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion , Akad. Verlagsgesell. (1944) [3] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) [4] I.G. Petrovskii, "Partial differential equations" , Saunders (1967) (Translated from Russian) [5] N.S. Koshlyakov, E.B. Gliner, M.M. Smirnov, "Partial differential equations" , Moscow (1970) (In Russian) [6] V.S. Vladimirov, "Die Gleichungen der mathematischen Physik" , MIR (1984) (Translated from Russian) [7] S.G. Mikhlin, "A course of mathematical physics" , Moscow (1968) (In Russian) [8] A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)