Characteristic

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

$$ L ( x, D) = \ \sum _ {| \nu | \leq m } a _ \nu ( x) D ^ \nu $$

is a linear partial differential operator of order $ m $, and let

$$ \sigma ( x, \xi ) = \ \sum _ {| \nu | = m } a _ \nu ( x) \xi ^ \nu $$

be its symbol. Here $ x, \xi \in \mathbf R ^ {n} $, $ \nu \in \mathbf Z _ {+} ^ {n} $ is a multi-index, $ | \nu | = \nu _ {1} + \dots + \nu _ {n} $, $ a _ \nu : \Omega \subseteq \mathbf R ^ {n} \rightarrow \mathbf R $,

$$ D _ {j} ^ {\nu _ {j} } = \ \left ( \frac \partial {\partial x _ {j} } \right ) ^ {\nu _ {j} } ,\ \ 1 \leq j \leq n, $$

$$ D ^ \nu = D _ {1} ^ {\nu _ {1} } \cdots D _ {n} ^ {\nu _ {n} } ,\ \ \xi ^ {\nu} = \xi _ {1} ^ {\nu _ {1} } \cdots \xi _ {n} ^ {\nu _ {n} } ,\ j, m, n \in \mathbf N . $$

Let $ S $ be the hypersurface defined in $ \mathbf R ^ {n} $ by the equation $ \phi ( x) = 0 $, where $ \phi _ {x} ( x) = \mathop{\rm grad} \phi ( x) \neq 0 $ for $ x \in S $, and let

$$ \tag{1 } \sigma ( x, \phi _ {x} ( x)) = 0. $$

In this case $ S $ is called a characteristic surface or a characteristic for the operator $ L ( x, D) $. Other names are: characteristic manifold, characteristic line (in case $ \mathbf R ^ {n} = \mathbf R ^ {2} $).

The example of the Cauchy problem is discussed below. Let $ S $ be the arbitrary (not necessarily characteristic) hypersurface in $ \mathbf R ^ {n} $ defined by the relations

$$ \phi ( x) = 0,\ \ \phi _ {x} ( x) \neq 0. $$

Let $ u _ {0}, \dots, u _ {m - 1 } $ be functions defined on $ S $ in a neighbourhood $ U $ of $ x _ {0} \in S $, and let

$$ L ( x, D) u = f,\ \ x \in U, $$

$$ u = u _ {0} ,\ \frac{\partial u }{\partial \mathbf n } = u _ {1}, \dots, \frac{\partial ^ {m - 1 } u }{ \partial \mathbf n ^ {m - 1 } } = u _ {m - 1 } ,\ x \in S, $$

be the Cauchy problem for the unknown function $ u $. Here $ f $ is a given function, $ L ( x, D) $ is a given linear differential operator of order $ m $, and $ \mathbf n $ is a vector orthonormal to $ S $. Assume, to be definite, that $ ( \partial / \partial x _ {n} ) \phi ( x) \neq 0 $, $ x \in U $. Then, by the change of variables

$$ x \rightarrow x ^ \prime ,\ \ \textrm{ where } \ x _ {j} ^ \prime = x _ {j} ,\ j = 1, \dots, n - 1; \ x _ {n} ^ \prime = \phi ( x), $$

one arrives at the equation

$$ \tag{2 } \sigma ( x, \phi _ {x} ( x)) \left ( \frac \partial {\partial x _ {n} ^ \prime } \right ) ^ {m} u + \sum \dots = f. $$

The expression under the sign $ \sum $ that is not written out does not contain partial derivatives of $ u $ with respect to $ x _ {n} ^ \prime $ of order $ m $. Two cases arise:

1) $ \sigma ( x, \phi _ {x} ( x)) \neq 0 $, $ x \in U $;

2) $ \sigma ( x, \phi _ {x} ( x)) = 0 $, $ x = x _ {0} $.

In the first case division of (2) by $ \sigma $ leads to an equation that can be solved for the highest partial derivative of $ x _ {n} ^ \prime $, that is, can be written in normal form. The initial conditions can be put in the form

$$ \left ( \frac \partial {\partial x _ {n} ^ \prime } \right ) ^ {j} u ( x _ {1} ^ \prime , \dots, x _ {n - 1 } ^ \prime , 0) = \ u _ {j} ( x _ {1} ^ \prime , \dots, x _ {n - 1 } ^ \prime ), $$

$$ j = 0, \dots, m - 1. $$

For this case the Cauchy problem has been well studied. For example, when the functions $ a _ \nu , f $ in the equations and when the initial data $ u _ {0}, \dots, u _ {m- 1} $ are real-analytic, there exists a unique solution of this problem in the class of real-analytic functions in a sufficiently small neighbourhood of $ x _ {0} $ (the Cauchy–Kovalevskaya theorem). In the second case $ x _ {0} $ is a characteristic point, and if (1) holds for all $ x \in S $, then $ S $ 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

$$ \tag{3 } \frac{\partial ^ {2} u }{\partial x _ {1} \partial x _ {2} } = 0 $$

initial data can be given on one of its characteristics $ x _ {1} = 0 $:

$$ \tag{4 } u ( 0, x _ {2} ) = \ u _ {0} ( x _ {2} ),\ \ \frac{\partial u }{\partial x _ {1} } ( 0, x _ {2} ) = \ u _ {1} ( x _ {2} ). $$

If the function $ u _ {1} $ is not constant, then the Cauchy problem (3), (4) has no solution in the space $ C ^ {2} $. But if $ u _ {1} $ is constant, for example equal to $ a \in \mathbf R $, then a solution is not unique in $ C ^ {2} $, since it may be any function of the form

$$ u ( x _ {1} , x _ {2} ) = \ ax _ {1} + b ( x _ {1} ) + u _ {0} ( x _ {2} ), $$

where

$$ b, u _ {0} \in C ^ {2} ,\ \ b ( 0) = b ^ \prime ( 0) = 0. $$

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 $ \phi ( x) $ is a solution of (1) and if the transformation $ x \rightarrow x ^ \prime $ leads to $ \phi ( x) \rightarrow \psi ( x ^ \prime ) $, $ a _ \nu ( x) \rightarrow b _ \nu ( x ^ \prime ) $, then $ \psi ( x ^ \prime ) $ satisfies the equation

$$ \sigma _ {1} ( x ^ \prime , \psi _ {x ^ \prime } ( x ^ \prime )) = 0, $$

where

$$ \sigma _ {1} ( x ^ \prime , \xi ) = \ \sum _ {| \nu | = m } b _ \nu ( x ^ \prime ) \xi ^ \nu . $$

Another property of a characteristic is that $ L ( x, D) $ is, relative to a characteristic $ S $, 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. $ n = 2 $) 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

$$ \tag{5 } a _ {20} ( x _ {1} , x _ {2} ) \frac{\partial ^ {2} u }{\partial x _ {1} ^ {2} } + a _ {11} ( x _ {1} , x _ {2} ) \frac{\partial ^ {2} u }{\partial x _ {1} \partial x _ {2} } + $$

$$ + a _ {02} ( x _ {1} , x _ {2} ) \frac{\partial ^ {2} u }{\partial x _ {2} ^ {2} } = 0 $$

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

$$ a _ {20} \left [ \frac{\partial \phi }{\partial x _ {1} } \right ] ^ {2} + a _ {11} \frac{\partial \phi }{\partial x _ {1} } \frac{\partial \phi }{\partial x _ {2} } + a _ {02} \left [ \frac{\partial \phi }{\partial x _ {2} } \right ] ^ {2} = 0 $$

determines two distinct families of characteristics:

$$ \phi _ {1} ( x) = \ \psi _ {1} ( x) - c _ {1} = 0,\ \ c _ {1} \in \mathbf R , $$

$$ \phi _ {2} ( x) = \psi _ {2} ( x) - c _ {2} = 0,\ c _ {2} \in \mathbf R . $$

For any selected pair $ ( c _ {1} , c _ {2} ) $ the change of variables $ x \rightarrow x ^ \prime $ by the formula

$$ x _ {1} ^ \prime = \ \phi _ {1} ( x),\ \ x _ {2} ^ \prime = \ \phi _ {2} ( x), $$

transforms (5) to the canonical form

$$ \frac{\partial ^ {2} u }{\partial x _ {1} ^ \prime \partial x _ {2} ^ \prime } + \textrm{ first- order terms } = 0. $$

For a non-linear differential equation

$$ \tag{6 } F ( x, u , D ^ \nu u , D ^ \mu u) = 0, $$

where $ \mu , \nu \in \mathbf Z _ {+} ^ {n} $ are multi-indices and $ | \nu | \leq m - 1 $, $ | \mu | = m $, the characteristic $ S $ is defined as the hypersurface in $ \mathbf R ^ {n} $ with the equation $ \phi ( x) = 0 $, where $ \phi _ {x} ( x) \neq 0 $ and $ \sigma ( x, \phi _ {x} ( x)) = 0 $ for $ x \in S $. In this case the symbol for the operator (6) given by the function $ F ( x, u , v, w) $ is defined as follows:

$$ \sigma ( x, \xi ) = \ \sum _ {| \mu | = m } F _ {w} ( x, u , D ^ \nu u , D ^ \mu u) \xi ^ \mu , $$

with the usual assumption $ F _ {w} \neq 0 $. Evidently, $ \sigma $ may depend, apart from the variables $ x $ and $ \xi $, also on $ u , D ^ \nu u $, and $ D ^ \mu u $. Suppose, for example, that a first-order equation is given $ ( m = 1) $. For simplicity, suppose in addition that $ n = 2 $. Then (6) takes the form

$$ F \left ( x _ {1} , x _ {2} , u ,\ \frac{\partial u }{\partial x _ {1} } ,\ \frac{\partial u }{\partial x _ {2} } \right ) = 0 $$

with a function $ F ( x, y, z, p, q) $. The equation of the characteristics is:

$$ F _ {p} \left ( x _ {1} , x _ {2} , u ,\ \frac{\partial u }{\partial x _ {1} } ,\ \frac{\partial u }{\partial x _ {2} } \right ) \frac{\partial \phi }{\partial x _ {1} } + $$

$$ + F _ {q} \left ( x _ {1} , x _ {2} , u , \frac{ \partial u }{\partial x _ {1} } , \frac{\partial u }{\partial x _ {2} } \right ) \frac{\partial \phi }{\partial x _ {2} } = 0. $$

Since a solution $ \phi ( x _ {1} , x _ {2} ) $ of this equation may, in fact, depend on $ u , \partial u/ \partial x _ {1} , \partial u/ \partial x _ {2} $, it can be given in parametric form

$$ x _ {1} = x ( t),\ \ x _ {2} = y ( t),\ \ u = z ( t), $$

$$ \frac{\partial u }{\partial x _ {1} } = p ( t),\ \ \frac{\partial u }{\partial x _ {2} } = q ( t), $$

where these functions satisfy the ordinary differential equations

$$ x ^ \prime ( t) = F _ {p} ,\ \ y ^ \prime ( t) = F _ {q} ,\ \ z ^ \prime ( t) = pF _ {p} + qF _ {q} , $$

$$ p ^ \prime ( t) = - F _ {x} - pF _ {z} ,\ q ^ \prime ( t) = - F _ {y} - qF _ {z} . $$

Geometrically the $ 5 $- tuple $ ( x ( t) , y ( t) , z ( t) , p ( t) , q ( t)) $ determines the so-called characteristic strip (for $ \alpha < t < \beta $). The projection of this strip onto the space $ ( x ( t), y ( t), z ( t)) $ determines a curve in $ \mathbf R ^ {3} $ such that at every point of it, it touches the plane with direction coefficients $ p ( t), q ( t) $. 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 $ D u $ 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 $ n = 2 $, the normal vectors to possible integral surfaces $ z = u ( x , y ) $ through a given point $ ( x _ {0} , y _ {0} , z _ {0} ) $ are defined by the equation $ F ( x _ {0} , y _ {0} , z _ {0} , p , q ) = 0 $. The envelope of the associated one-parameter family of tangent planes $ p ( x - x _ {0} ) + q ( y - y _ {0} ) = z - z _ {0} $, i.e. the set of characteristic directions $ ( x - x _ {0} ) / F _ {p} = ( y - y _ {0} ) / F _ {q} + ( z - z _ {0} ) / ( p F _ {p} + q F _ {q} ) $, is called the Monge cone at the point $ ( x _ {0} , y _ {0} , z _ {0} ) $.

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