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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.
 
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
 
Suppose that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c0216101.png" /></td> </tr></table>
+
$$
 +
L ( x, D)  = \
 +
\sum _ {| \nu | \leq  m }
 +
a _  \nu  ( x) D  ^  \nu
 +
$$
  
is a linear partial differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c0216102.png" />, and let
+
is a linear partial differential operator of order $  m $,  
 +
and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c0216103.png" /></td> </tr></table>
+
$$
 +
\sigma ( x, \xi )  = \
 +
\sum _ {| \nu | = m }
 +
a _  \nu  ( x) \xi  ^  \nu
 +
$$
  
be its symbol. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c0216104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c0216105.png" /> is a multi-index, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c0216106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c0216107.png" />,
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c0216108.png" /></td> </tr></table>
+
$$
 +
D _ {j} ^ {\nu _ {j} }  = \
 +
\left (
 +
\frac \partial {\partial  x _ {j} }
 +
\right ) ^ {\nu _ {j} } ,\ \
 +
1 \leq  j \leq  n,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c0216109.png" /></td> </tr></table>
+
$$
 +
D  ^  \nu  = D _ {1} ^ {\nu _ {1} } \cdots D _ {n} ^ {\nu _ {n} } ,\ \
 +
\xi  ^ {\nu}  = \xi _ {1} ^ {\nu _ {1} } \cdots \xi _ {n} ^ {\nu _ {n} } ,\  j, m, n \in \mathbf N .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161010.png" /> be the hypersurface defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161011.png" /> by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161014.png" />, and let
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\sigma ( x, \phi _ {x} ( x))  = 0.
 +
$$
  
In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161016.png" /> is called a characteristic surface or a characteristic for the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161017.png" />. Other names are: characteristic manifold, characteristic line (in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161018.png" />).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161019.png" /> be the arbitrary (not necessarily characteristic) hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161020.png" /> defined by the relations
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161021.png" /></td> </tr></table>
+
$$
 +
\phi ( x)  = 0,\ \
 +
\phi _ {x} ( x)  \neq  0.
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161022.png" /> be functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161023.png" /> in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161025.png" />, and let
+
Let $  u _ {0}, \dots, u _ {m - 1 }  $
 +
be functions defined on $  S $
 +
in a neighbourhood $  U $
 +
of $  x _ {0} \in S $,  
 +
and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161026.png" /></td> </tr></table>
+
$$
 +
L ( x, D) u  = f,\ \
 +
x \in U,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161027.png" /></td> </tr></table>
+
$$
 +
= u _ {0} ,\ 
 +
\frac{\partial  u }{\partial  \mathbf n
 +
}
 +
  = u _ {1}, \dots,
 +
\frac{\partial  ^ {m - 1 } u }{
 +
\partial  \mathbf n ^ {m - 1 } }
 +
  = u _ {m - 1 }  ,\  x \in S,
 +
$$
  
be the Cauchy problem for the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161028.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161029.png" /> is a given function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161030.png" /> is a given linear differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161031.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161032.png" /> is a vector orthonormal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161033.png" />. Assume, to be definite, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161035.png" />. Then, by the change of variables
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161036.png" /></td> </tr></table>
+
$$
 +
x  \rightarrow  x  ^  \prime  ,\ \
 +
\textrm{ where } \
 +
x _ {j}  ^  \prime  = x _ {j} ,\
 +
j = 1, \dots, n - 1; \
 +
x _ {n}  ^  \prime  = \phi ( x),
 +
$$
  
 
one arrives at the equation
 
one arrives at the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161038.png" /> that is not written out does not contain partial derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161039.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161040.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161041.png" />. Two cases arise:
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161043.png" />;
+
1) $  \sigma ( x, \phi _ {x} ( x)) \neq 0 $,  
 +
$  x \in U $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161045.png" />.
+
2) $  \sigma ( x, \phi _ {x} ( x)) = 0 $,  
 +
$  x = x _ {0} $.
  
In the first case division of (2) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161046.png" /> leads to an equation that can be solved for the highest partial derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161047.png" />, that is, can be written in normal form. The initial conditions can be put in the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161048.png" /></td> </tr></table>
+
$$
 +
\left (
 +
\frac \partial {\partial  x _ {n}  ^  \prime  }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161049.png" /></td> </tr></table>
+
\right )  ^ {j}
 +
u ( x _ {1}  ^  \prime , \dots, x _ {n - 1 }  ^  \prime  , 0= \
 +
u _ {j} ( x _ {1}  ^  \prime , \dots, x _ {n - 1 }  ^  \prime  ),
 +
$$
  
For this case the Cauchy problem has been well studied. For example, when the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161050.png" /> in the equations and when the initial data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161051.png" /> are real-analytic, there exists a unique solution of this problem in the class of real-analytic functions in a sufficiently small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161052.png" /> (the [[Cauchy–Kovalevskaya theorem|Cauchy–Kovalevskaya theorem]]). In the second case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161053.png" /> is a characteristic point, and if (1) holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161055.png" /> 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.
+
$$
 +
j  =  0, \dots, m - 1.
 +
$$
 +
 
 +
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|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
 
For example, for the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
  
initial data can be given on one of its characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161057.png" />:
+
\frac{\partial  ^ {2} u }{\partial  x _ {1} \partial  x _ {2} }
 +
  = 0
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
initial data can be given on one of its characteristics  $  x _ {1} = 0 $:
  
If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161059.png" /> is not constant, then the Cauchy problem (3), (4) has no solution in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161060.png" />. But if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161061.png" /> is constant, for example equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161062.png" />, then a solution is not unique in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161063.png" />, since it may be any function of the form
+
$$ \tag{4 }
 +
u ( 0, x _ {2} ) = \
 +
u _ {0} ( x _ {2} ),\ \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161064.png" /></td> </tr></table>
+
\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
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161065.png" /></td> </tr></table>
+
$$
 +
b, u _ {0}  \in  C  ^ {2} ,\ \
 +
b ( 0)  = b  ^  \prime  ( 0)  = 0.
 +
$$
  
 
Thus, the Cauchy problem differs substantially, depending on whether the initial data are given on a characteristic surface or not.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161066.png" /> is a solution of (1) and if the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161067.png" /> leads to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161069.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161070.png" /> satisfies the equation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161071.png" /></td> </tr></table>
+
$$
 +
\sigma _ {1} ( x  ^  \prime  , \psi _ {x  ^  \prime  } ( x  ^  \prime  ))  = 0,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161072.png" /></td> </tr></table>
+
$$
 +
\sigma _ {1} ( x  ^  \prime  , \xi )  = \
 +
\sum _ {| \nu | = m }
 +
b _  \nu  ( x  ^  \prime  ) \xi  ^  \nu  .
 +
$$
  
Another property of a characteristic is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161073.png" /> is, relative to a characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161074.png" />, an [[Interior differential operator|interior differential operator]].
+
Another property of a characteristic is that $  L ( x, D) $
 +
is, relative to a characteristic $  S $,  
 +
an [[Interior differential operator|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. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161075.png" />) 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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161076.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
a _ {20} ( x _ {1} , x _ {2} )
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161077.png" /></td> </tr></table>
+
\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
 
be hyperbolic. That is, equation (1), which now reads
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161078.png" /></td> </tr></table>
+
$$
 +
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:
 
determines two distinct families of characteristics:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161079.png" /></td> </tr></table>
+
$$
 +
\phi _ {1} ( x)  = \
 +
\psi _ {1} ( x) -
 +
c _ {1}  = 0,\ \
 +
c _ {1} \in \mathbf R ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161080.png" /></td> </tr></table>
+
$$
 +
\phi _ {2} ( x)  = \psi _ {2} ( x) - c _ {2}  = 0,\  c _ {2} \in \mathbf R .
 +
$$
  
For any selected pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161081.png" /> the change of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161082.png" /> by the formula
+
For any selected pair $  ( c _ {1} , c _ {2} ) $
 +
the change of variables $  x \rightarrow x  ^  \prime  $
 +
by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161083.png" /></td> </tr></table>
+
$$
 +
x _ {1}  ^  \prime  = \
 +
\phi _ {1} ( x),\ \
 +
x _ {2}  ^  \prime  = \
 +
\phi _ {2} ( x),
 +
$$
  
 
transforms (5) to the canonical form
 
transforms (5) to the canonical form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161084.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  ^ {2} u }{\partial  x _ {1}  ^  \prime  \partial  x _ {2}  ^  \prime  }
 +
+  \textrm{ first- order
 +
terms  }  = 0.
 +
$$
  
 
For a non-linear differential equation
 
For a non-linear differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161085.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
F ( x, u , D  ^  \nu  u , D  ^  \mu  u)  = 0,
 +
$$
 +
 
 +
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:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161086.png" /> are multi-indices and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161088.png" />, the characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161089.png" /> is defined as the hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161090.png" /> with the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161091.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161093.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161094.png" />. In this case the symbol for the operator (6) given by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161095.png" /> is defined as follows:
+
$$
 +
\sigma ( x, \xi )  = \
 +
\sum _ {| \mu | = m }
 +
F _ {w} ( x, u , D  ^  \nu  u , D  ^  \mu  u)
 +
\xi  ^  \mu  ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161096.png" /></td> </tr></table>
+
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
  
with the usual assumption <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161097.png" />. Evidently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161098.png" /> may depend, apart from the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c02161099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610100.png" />, also on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610101.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610102.png" />. Suppose, for example, that a first-order equation is given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610103.png" />. For simplicity, suppose in addition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610104.png" />. Then (6) takes the form
+
$$
 +
F \left (
 +
x _ {1} , x _ {2} , u ,\
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610105.png" /></td> </tr></table>
+
\frac{\partial  u }{\partial  x _ {1} }
 +
,\
  
with a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610106.png" />. The equation of the characteristics is:
+
\frac{\partial  u }{\partial  x _ {2} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610107.png" /></td> </tr></table>
+
\right )  = 0
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610108.png" /></td> </tr></table>
+
with a function  $  F ( x, y, z, p, q) $.
 +
The equation of the characteristics is:
  
Since a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610109.png" /> of this equation may, in fact, depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610110.png" />, it can be given in parametric form
+
$$
 +
F _ {p} \left (
 +
x _ {1} , x _ {2} , u ,\
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610111.png" /></td> </tr></table>
+
\frac{\partial  u }{\partial  x _ {1} }
 +
,\
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610112.png" /></td> </tr></table>
+
\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
 +
 
 +
$$
 +
x _ {1}  = x ( t),\ \
 +
x _ {2}  = y ( t),\ \
 +
= z ( t),
 +
$$
 +
 
 +
$$
 +
 
 +
\frac{\partial  u }{\partial  x _ {1} }
 +
  = p ( t),\ \
 +
 
 +
\frac{\partial  u }{\partial  x _ {2} }
 +
  = q ( t),
 +
$$
  
 
where these functions satisfy the ordinary differential equations
 
where these functions satisfy the ordinary differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610113.png" /></td> </tr></table>
+
$$
 +
x  ^  \prime  ( t)  = F _ {p} ,\ \
 +
y  ^  \prime  ( t)  = F _ {q} ,\ \
 +
z  ^  \prime  ( t)  = pF _ {p} + qF _ {q} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610114.png" /></td> </tr></table>
+
$$
 +
p  ^  \prime  ( t)  = - F _ {x} - pF _ {z} ,\  q  ^  \prime  ( t)  = - F _ {y} - qF _ {z} .
 +
$$
  
Geometrically the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610115.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610116.png" /> determines the so-called [[Characteristic strip|characteristic strip]] (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610117.png" />). The projection of this strip onto the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610118.png" /> determines a curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610119.png" /> such that at every point of it, it touches the plane with direction coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610120.png" />. This curve is also called a characteristic of the equation (6).
+
Geometrically the $  5 $-
 +
tuple $  ( x ( t) , y ( t) , z ( t) , p ( t) , q ( t)) $
 +
determines the so-called [[Characteristic strip|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).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Misohata,  "The theory of partial differential equations" , Cambridge Univ. Press  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.G. Petrovskii,  "Partial differential equations" , Saunders  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.S. Koshlyakov,  E.B. Gliner,  M.M. Smirnov,  "Partial differential equations" , Moscow  (1970)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.S. Vladimirov,  "Die Gleichungen der mathematischen Physik" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.G. Mikhlin,  "A course of mathematical physics" , Moscow  (1968)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Misohata,  "The theory of partial differential equations" , Cambridge Univ. Press  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.G. Petrovskii,  "Partial differential equations" , Saunders  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.S. Koshlyakov,  E.B. Gliner,  M.M. Smirnov,  "Partial differential equations" , Moscow  (1970)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.S. Vladimirov,  "Die Gleichungen der mathematischen Physik" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.G. Mikhlin,  "A course of mathematical physics" , Moscow  (1968)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
It has to be stressed that for first-order partial differential equations that are non-linear with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610121.png" /> there is a whole family of characteristics through a given point (a [[Conoid|conoid]]). A classic notion in this connection is the one of Monge cones (cf. also [[Monge cone|Monge cone]]). Referring again to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610122.png" />, the normal vectors to possible integral surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610123.png" /> through a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610124.png" /> are defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610125.png" />. The [[Envelope|envelope]] of the associated one-parameter family of tangent planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610126.png" />, i.e. the set of characteristic directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610127.png" />, is called the Monge cone at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021610/c021610128.png" />.
+
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|conoid]]). A classic notion in this connection is the one of Monge cones (cf. also [[Monge cone|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|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} ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''1–2''' , Interscience  (1953–1962)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Garabedian,  "Partial differential equations" , Wiley  (1964)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1–4''' , Springer  (1983–1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. John,  "Partial differential equations" , Springer  (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Jeffrey,  "Quasilinear hyperbolic systems and waves" , Pitman  (1976)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Cartan,  "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann  (1945)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I.G. Petrovskii,  "Lectures on partial differential equations" , Interscience  (1954)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''1–2''' , Interscience  (1953–1962)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Garabedian,  "Partial differential equations" , Wiley  (1964)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1–4''' , Springer  (1983–1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. John,  "Partial differential equations" , Springer  (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Jeffrey,  "Quasilinear hyperbolic systems and waves" , Pitman  (1976)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Cartan,  "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann  (1945)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I.G. Petrovskii,  "Lectures on partial differential equations" , Interscience  (1954)  (Translated from Russian)</TD></TR></table>

Latest revision as of 02:16, 15 June 2022


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).

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)

Comments

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} ) $.

References

[a1] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 1–2 , Interscience (1953–1962) (Translated from German)
[a2] P.R. Garabedian, "Partial differential equations" , Wiley (1964)
[a3] L.V. Hörmander, "The analysis of linear partial differential operators" , 1–4 , Springer (1983–1985)
[a4] F. John, "Partial differential equations" , Springer (1974)
[a5] A. Jeffrey, "Quasilinear hyperbolic systems and waves" , Pitman (1976)
[a6] E. Cartan, "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945)
[a7] I.G. Petrovskii, "Lectures on partial differential equations" , Interscience (1954) (Translated from Russian)
How to Cite This Entry:
Characteristic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characteristic&oldid=18652
This article was adapted from an original article by Yu.V. Komlenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article