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.
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
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
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
initial data can be given on one of its characteristics :
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
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
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
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
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).
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It has to be stressed that for first-order partial differential equations that are non-linear with respect to there is a whole family of characteristics through a given point (a conoid). A classic notion in this connection is the one of Monge cones (cf. also Monge cone). Referring again to the case , the normal vectors to possible integral surfaces through a given point are defined by the equation . The envelope of the associated one-parameter family of tangent planes , i.e. the set of characteristic directions , is called the Monge cone at the point .
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Characteristic. Yu.V. Komlenko (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Characteristic&oldid=18652