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A partial differential equation (or system) of the form

$$ \tag{1 }
\sum _ {| \alpha | \leq  m }
a _  \alpha  ( x) D  ^  \alpha  u = f
$$

for which at any point  $  x = ( x _ {0}, \dots, x _ {n} ) $
of its domain of definition  $  \Omega $
one can distinguish among the real variables  $  y _ {0}, \dots, y _ {n} $(
if necessary, after a suitable affine transformation of the independent variables) one variable, say  $  y _ {0} = \lambda $, 
in a such a way that for all points  $  Y = ( y _ {1}, \dots, y _ {n} ) $
of the  $  n $-
dimensional Euclidean space  $  \mathbf R  ^ {n} $
the characteristic equation (in  $  \lambda $)

$$ \tag{2 }
 \mathop{\rm det}  \left . \sum _ {| \alpha | \leq  m }
a _  \alpha  ( x) y  ^  \alpha  \right | _ {y _ {0}  = \lambda } ,\ \ 
y  ^  \alpha   =  ( y _ {0} ^ {\alpha _ {0} }, \dots,
y _ {n} ^ {\alpha _ {n} } ) ,
$$

has exactly  $  N m $
real roots. Here  $  \alpha = ( \alpha _ {0}, \dots, \alpha _ {n} ) $
is a vector with non-negative integer coordinates,  $  | \alpha | = \sum _ {j=} 0  ^ {n} \alpha _ {j} $
is the order of the differential operator  $  D  ^  \alpha  = D ^ {\alpha _ {0} } \dots D _ {n} ^ {\alpha _ {n} } $, 
$  D _ {j} = \partial  / \partial  x _ {j} $, 
$  j = 0, \dots, n $, 
$  m $
is the order of the system (1),  $  a _  \alpha  ( x) $
is a real square matrix of order  $  N $, 
defined in  $  \Omega $, 
$  u ( x) = \| u _ {j} ( x) \| $, 
$  j = 1, \dots, N $, 
is an unknown column vector, and  $  f ( x) $
is a vector with  $  N $
components, defined in  $  \Omega $.

A typical example is the [[Wave equation|wave equation]]

$$ \tag{3 }
u _ {x _ {0}  x _ {0} }  = \ 
\sum _ { j= 1} ^ { n } 
u _ {x _ {j}  x _ {j} } .
$$

Many problems in mathematical physics reduce to linear hyperbolic partial differential equations or systems of equations.

A subset  $  S : \phi ( x) = 0 $
is said to be characteristic at a point  $  x $
if  $   \mathop{\rm grad}  \phi \neq 0 $
and  $  Q ( x ,  \mathop{\rm grad}  \phi ) = 0 $, 
where

$$ 
Q ( x , y )  \equiv  Q
( x , y _ {0} , Y )  =   \mathop{\rm det} \ 
\sum _ {| \alpha | = m }
a _  \alpha  ( x) y  ^  \alpha 
$$

is the characteristic form of the system (1). If  $  Q ( x , y ) = 0 $, 
then one says that the vector  $  y $
defines a characteristic direction or characteristic normal at the point  $  x $. 
The surface  $  S $
is called a characteristic surface (or characteristic) of the system (1) if

$$ 
Q ( x ,  \mathop{\rm grad}  \phi )  =  0 \ \ 
\textrm{ for  all  }  x \in S .
$$

A surface which does not have characteristic normals at any point is called a free surface. On a free surface the rank of the characteristic matrix

$$ 
A ( x , y )  \equiv  A ( x , y _ {0} , Y )  = \ 
\sum _ {| \alpha | = m }
a _  \alpha  ( x) y  ^  \alpha  ,\ \ 
y  =   \mathop{\rm grad}  \phi ,
$$

is equal to  $  N $, 
while on a characteristic surface  $  S $
it is less than  $  N $. 
A characteristic  $  S $
is said to be simple if for some  $  j $
and all  $  x \in S $,

$$ 
\left ( 
\frac \partial {\partial  x _ {j} }
 \right ) A
( x ,  \mathop{\rm grad}  \phi )  \neq  0 .
$$

Otherwise the characteristic is said to be multiple. A characteristic is sometimes said to be simple if the rank of the matrix  $  A ( x ,  \mathop{\rm grad}  \phi ) $
is  $  N - 1 $.

The system (1) is said to be hyperbolic at the point  $  x $
with respect to the hyperplane  $  S _ {0} $: 
$  x _ {0} = 0 $
if the matrix  $  A ( x , 1 , Y ) = a _ {m , 0 \dots 0 }  $
is non-singular (that is, the surface  $  S $
is free) and if all roots  $  \lambda = \lambda _ {k} $, 
$  k = 1, \dots, m N $, 
of the characteristic equation  $  Q ( x , \lambda , Y ) = 0 $
are real for all points  $  Y \in \mathbf R  ^ {n} $. 
The system (1) is said to be hyperbolic in the domain  $  \Omega $
with respect to  $  S _ {0} $
if it is hyperbolic with respect to  $  S _ {0} $
at every point  $  x \in \Omega $.

An important class of hyperbolic equations and systems consists of strictly hyperbolic equations and systems, which are sometimes called fully hyperbolic systems, or systems, hyperbolic in the narrow sense. A system (1) is called a strictly hyperbolic system if all roots  $  \lambda = \lambda _ {k} $
of the characteristic equation are distinct for any non-zero vector  $  Y \in \mathbf R  ^ {n} $. 
The characteristics of a strictly hyperbolic equation (or system) are simple. Strictly hyperbolic (with respect to  $  S _ {0} $) 
systems are notable for the fact that the Cauchy problem

$$ \tag{4 }
D _ {0}  ^ {k} u \mid  _ {x _ {0}  = 0 }  = \ 
\psi _ {k} ( x) ,\  k = 0, \dots, m - 1 ,
$$

for them is well-posed under the single assumption of sufficient smoothness of the coefficients  $  a _  \alpha  ( x) $, 
$  f ( x) $
of the system (1) and of the initial data (initial functions)  $  \psi _ {k} ( x) $, 
$  x = ( x _ {0}, \dots, x _ {n} ) $. 
There are examples of hyperbolic but not strictly hyperbolic equations of the form (1) (even with constant coefficients in front of the derivatives of order  $  m $) 
for which the Cauchy problem is ill-posed.

The solution  $  u ( x) $
of the Cauchy problem (4) for the wave equation (3) can be written out explicitly, and when  $  m \equiv 0 (  \mathop{\rm mod}  2) $, 
and only then, it has the property that the value of  $  u ( x) $
at the vertex  $  x  ^ {*} $
of the characteristic cone  $  | X - X  ^ {*} | = x _ {0}  ^ {*} - x _ {0} $, 
$  X = ( x _ {1}, \dots, x _ {n} ) $, 
depends only on the value of the initial data  $  \psi _ {0} ( x) $
and  $  \psi _ {1} ( x) $
and their derivatives on the base  $  | X - X  ^ {*} | = x _ {0}  ^ {*} $, 
$  x _ {0} = 0 $, 
of this cone (the so-called [[Huygens principle|Huygens principle]]).

For strictly hyperbolic (with respect to  $  S _ {0} $) 
equations and systems the question of the diffusion of waves and the related question of gaps have been investigated (cf. [[Lacuna|Lacuna]]). An exhaustive answer has been given in the case of an equation with constant coefficients of the form

$$ 
\sum _ {| \alpha | = m } a _  \alpha  D  ^  \alpha  u  =  0 ,\ \ 
a _  \alpha   =  \textrm{ const } .
$$

For the case of one (scalar) equation with constant coefficients  $  a _  \alpha  ( x) = a _  \alpha  = \textrm{ const } $
the definition of strict hyperbolicity has been generalized as follows. Equation (1) is said to be hyperbolic with respect to a non-zero vector  $  y \in \mathbf R  ^ {n+ 1} $
if

$$ 
\sum _ {| \alpha | = m } a _  \alpha  y  ^  \alpha   \neq  0
$$

and if there is a real number  $  \lambda _ {0} $
such that for all  $  \xi = ( \xi _ {0}, \dots, \xi _ {n} ) \in \mathbf R  ^ {n+} 1 $
and  $  \lambda > \lambda _ {0} $,

$$ 
\sum _ {| \alpha | \leq  m }
a _  \alpha  ( \lambda y + i \xi )  ^  \alpha   \neq  0 .
$$

Of all linear equations with constant coefficients, only for equations that are hyperbolic in this sense the Cauchy problem is well-posed for arbitrary sufficiently smooth initial functions defined on the hyperplane

$$ 
\sum _ { j= 0} ^ { n }  x _ {j} y _ {j}  =  0 .
$$

In particular, the wave equation (3) is hyperbolic in this sense with respect to any vector for which

$$ 
y _ {0}  ^ {2}  >  \sum _ { j= 1} ^ { n } 
y _ {j}  ^ {2} .
$$

There are various generalizations of the definition of strict hyperbolicity of equations and systems. These are mainly equations and systems that are completely characterized by the fact that the Cauchy problem with data on a free surface is uniquely solvable for them for any sufficiently smooth initial functions, without any restrictions on the growth at infinity.

Another important class of linear hyperbolic systems of the first order is the class of symmetric hyperbolic systems. The system

$$ \tag{5 }
\sum _ { j= } 0 ^ { n }  a _ {j} ( x) u _ {x _ {j}  }
+ b ( x) u  =  f ( x) ,
$$

where  $  a _ {j} ( x) $, 
$  b ( x) $
are square matrices of order  $  N $, 
defined in  $  \Omega $, 
and  $  u ( x) $
is an unknown vector of  $  N $
components, is called a symmetric hyperbolic system in  $  \Omega $
if the matrices  $  a _ {j} ( x) $
are symmetric (or are symmetrizable simultaneously by the same transformation) and if at every point there is a spatially-oriented hyperplane (or, space-like hyperplane), that is, a hyperplane whose normal  $  y = ( y _ {0}, \dots, y _ {n} ) $
has the property that the matrix  $  \sum _ {j=} 0  ^ {n} y _ {j} a _ {j} ( x) $
is positive definite. If for a symmetric hyperbolic system (5) with sufficiently smooth coefficients the given initial functions and the right-hand side have square-integrable generalized partial derivatives of order  $  p $, 
then there is a unique generalized solution of the Cauchy problem with the same number of square-integrable partial derivatives. Any strictly hyperbolic partial differential equation of the second order reduces to a symmetric hyperbolic system.

An equation (1) of the second order in the class of solutions regular in a domain  $  \Omega $
can be written in the form

$$ \tag{6 }
\sum _ {k , j = 0 } ^ { n } 
a _ {kj} ( x) u _ {x _ {k}  x _ {j} } +
\sum _ { j= } 0 ^ { n }  b _ {j} ( x) u _ {x _ {j}  } +
c ( x) u  =  f ( x) ,
$$

where  $  a _ {kj} ( x) = a _ {jk} ( x) $, 
$  b _ {j} ( x) $, 
$  c ( x) $, 
and  $  f ( x) $
are functions defined in  $  \Omega $. 
Equation (6) is hyperbolic in  $  \Omega $
if at every point of  $  \Omega $
all eigen values of the matrix of leading coefficients  $  a ( x) = \| a _ {jk} ( x) \| $, 
$  k , j = 0, \dots, n $, 
are non-zero, and one of these eigen values differs in sign from all others. With respect to (6), along with the characteristic surface one can distinguish two types of smooth surfaces: spatially-oriented surfaces and time-oriented surfaces (also called space-like and time-like surfaces). If the surfaces are given by an equation of the form  $  \phi ( x) = 0 $, 
then on a surface of the first type  $  Q ( x ,  \mathop{\rm grad}  \phi ) > 0 $, 
while on a surface of the second type  $  Q ( x ,  \mathop{\rm grad}  \phi ) < 0 $, 
where

$$ 
Q ( x , y )  =  \sum _ {k , j = 0 } ^ { n } 
a _ {kj} ( x) y _ {k} y _ {j} .
$$

The Cauchy problem for hyperbolic partial differential equations with initial data on a time-like surface is generally not well-posed.

A hyperbolic partial differential equation

$$ 
u _ {x _ {0}  x _ {0} } -
\sum _ {k , j = 1 } ^ { n } 
a _ {kj} ( x) u _ {x _ {k}  x _ {j} } +
\sum _ { j= } 0 ^ { n }  b _ {j} ( x) u _ {x _ {j}  } +
c ( x) u  =  f ( x)
$$

is said to be uniformly (or regularly) hyperbolic in a domain  $  \Omega $
if there is a positive number  $  \epsilon $
such that

$$ 
\sum _ {k , j = 1 } ^ { n } 
a _ {kj} ( x) \xi _ {k} \xi _ {j}  > \ 
\epsilon \sum _ { j= } 1 ^ { n } 
\xi _ {j}  ^ {2}
$$

for all  $  x $
in  $  \overline \Omega \; $
and for any non-zero vector  $  ( \xi _ {1}, \dots, \xi _ {n} ) \in \mathbf R  ^ {n} $. 
For  $  n = 1 $
the inequality

$$ 
a _ {01}  ^ {2} - a _ {00} a _ {11}  <  0 \ \ 
\textrm{ for  all  }  x \in \overline \Omega
$$

is a necessary and sufficient condition for (6) to be uniformly hyperbolic in  $  \Omega $. 
The equation for the vibration of a string,

$$ 
u _ {x _ {0}  x _ {0} } - u _ {x _ {1}  x _ {1} }  =  0,
$$

is a typical representative of a linear uniformly hyperbolic partial differential equation of the second order with two independent variables. The general solution of this equation in any convex domain  $  \Omega $
of the plane  $  \mathbf R  ^ {2} $
is given by the [[D'Alembert formula|d'Alembert formula]]:

$$ 
u ( x)  =  f ( x _ {0} + x _ {1} ) + g ( x _ {0} - x _ {1} ) ,
$$

where  $  f $
and  $  g $
are arbitrary functions.

After a non-singular real change of variables  $  x _ {0} $
and  $  x _ {1} $, 
the hyperbolic partial differential equation (6) with  $  n = 1 $
reduces to the normal (canonical) form

$$ \tag{7 }
u _ {y _ {0}  y _ {0} } -
u _ {y _ {1}  y _ {1} } +
A ( y) u _ {y _ {0}  } + B ( y) u _ {y _ {1}  } + C
( y) u  =  F ( y) ,
$$

$$ 
y  =  ( y _ {0} , y _ {1} ) .
$$

For hyperbolic systems written in the form (7), where  $  A ( y) $, 
$  B ( y) $
and  $  C ( y) $
are given real square matrices of order  $  N $, 
$  F ( y) $
is a given vector and  $  u $
an unknown vector, both with  $  N $
components, the question of the well-posedness of the Cauchy problem with initial data on a non-characteristic (free) curve and the [[Goursat problem|Goursat problem]] with data on two intersecting characteristics have been completely investigated by the [[Riemann method|Riemann method]].

The main problems about hyperbolic equations are the following: the [[Cauchy problem|Cauchy problem]], the [[Cauchy characteristic problem|Cauchy characteristic problem]] and the [[Mixed problem|mixed problem]] (see also [[Mixed and boundary value problems for hyperbolic equations and systems|Mixed and boundary value problems for hyperbolic equations and systems]]).

In the investigation of the main problems an important role is played by fundamental solutions, which make it possible to obtain explicit (integral) representations of regular and generalized solutions and to establish their structural and qualitative properties, in particular to study the question of wave fronts and the propagation of discontinuities.

Equation (6) is called an ultra-hyperbolic equation in a domain  $  \Omega $
if at every point  $  x \in \Omega $
all eigenvalues of the matrix  $  a ( x) $
are non-zero and at least two of them differ in sign from all the others, of which there are at least two. An example of an ultra-hyperbolic equation is an equation of the form

$$ \tag{8 }
\sum _ { j= } 0 ^ { n } 
( u _ {x _ {j}  x _ {j} } -
u _ {y _ {j}  y _ {j} } )  =  0 ,\ \ 
n \geq  1 ,
$$

which has the following property: If  $  u ( x , y ) $
is a regular solution of (8) in a domain  $  \Omega $
of the Euclidean space of the variables  $  x = ( x _ {0}, \dots, x _ {n} ) $, 
$  y = ( y _ {0}, \dots, y _ {n} ) $
and if  $  ( x  ^ {*} , y  ^ {*} ) $
is an arbitrary point of  $  \Omega $, 
then the mean value of the function  $  u ( x , y  ^ {*} ) $
calculated on the sphere  $  \sum _ {j=} 0  ^ {n} ( x _ {j} - x _ {j}  ^ {*} )  ^ {2} = r  ^ {2} $
with centre at the point  $  x  ^ {*} = ( x _ {0}  ^ {*}, \dots, x _ {n}  ^ {*} ) $
and radius  $  r $, 
is equal to the mean value of the function  $  u ( x  ^ {*} , y ) $
calculated on the sphere  $  \sum _ {j=} 0  ^ {n} ( y _ {j} - y _ {j}  ^ {*} )  ^ {2} = r  ^ {2} $
with centre at the point  $  y  ^ {*} = ( y _ {0}  ^ {*}, \dots, y _ {n}  ^ {*} ) $
and the same radius  $  r $. 
This theorem is extensively used in the theory of linear hyperbolic partial differential equations of the second order with constant coefficients.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Hadamard, "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint (1952)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964) {{MR|0163043}} {{ZBL|0126.00207}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.V. Bitsadse, "Equations of mixed type" , Pergamon (1964) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.V. Bitsadze, "Equations of mathematical physics" , MIR (1980) (Translated from Russian) {{MR|0587310}} {{MR|0581247}} {{ZBL|0499.35002}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) {{MR|0764399}} {{ZBL|0954.35001}} {{ZBL|0652.35002}} {{ZBL|0695.35001}} {{ZBL|0699.35005}} {{ZBL|0607.35001}} {{ZBL|0506.35001}} {{ZBL|0223.35002}} {{ZBL|0231.35002}} {{ZBL|0207.09101}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> I.M. Gel'fand, G.E. Shilov, "Generalized functions" , Acad. Press (1964) (Translated from Russian) {{MR|435831}} {{ZBL|0115.33101}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> L. Gårding, "Cauchy's problem for hyperbolic equations" , ''Third Congress Scand. Math. Helsinki, 1957'' , Mercators Tryckeri (1958) pp. 104–109 {{MR|116143}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.A. Dezin, "Boundary problems for certain symmetric linear first-order systems" ''Mat. Sb.'' , '''49''' : 4 (1959) pp. 459–484 (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience (1965) (Translated from German) {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> O.A. Ladyzhenskaya, "A mixed problem for a hyperbolic equation" , Moscow (1953) (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> J. Leray, L. Gårding, T. Kotake, "Uniformisation et developpement asymptotique de la solution du problème de Cauchy linéaire à données holomorphes; analogie avec la théorie des ondes asymptotiques et approchées" ''Bull. Soc. Math. France'' , '''92''' (1964) pp. 236–361 {{MR|0196280}} {{ZBL|0156.33101}} {{ZBL|0147.08101}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> I.G. Petrovskii, "Partial differential equations" , Saunders (1967) (Translated from Russian) {{MR|0211021}} {{ZBL|}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> V.I. Smirnov, "A course of higher mathematics" , '''2''' , Addison-Wesley (1964) (Translated from Russian) {{MR|0182690}} {{MR|0182688}} {{MR|0182687}} {{MR|0177069}} {{MR|0168707}} {{ZBL|0122.29703}} {{ZBL|0121.25904}} {{ZBL|0118.28402}} {{ZBL|0117.03404}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) {{MR|104888}} {{ZBL|}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> L. Hörmander, "Linear partial differential operators" , Springer (1976) {{MR|0404822}} {{ZBL|0321.35001}} </TD></TR></table>

====Comments====
A fully hyperbolic system, or system hyperbolic in the narrow sense, is also called a totally hyperbolic system or a system hyperbolic in the sense of Petrovskii.

For the definition of hyperbolicity in the case of scalar equations with constant coefficients, see [[#References|[a1]]], Vol. II, Chapt. 12.

For fundamental solutions and propagation of singularities, see also [[#References|[a1]]], Vol. III, Chapts. 13, 14.

The theorem mentioned in the last line of the main article above is called Asgeirsson's theorem, cf. [[#References|[9]]], Chapt. 6.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.V. Hörmander, "The analysis of linear partial differential operators" , '''1–4''' , Springer (1983–1985) {{MR|2512677}} {{MR|2304165}} {{MR|2108588}} {{MR|1996773}} {{MR|1481433}} {{MR|1313500}} {{MR|1065993}} {{MR|1065136}} {{MR|0961959}} {{MR|0925821}} {{MR|0881605}} {{MR|0862624}} {{MR|1540773}} {{MR|0781537}} {{MR|0781536}} {{MR|0717035}} {{MR|0705278}} {{ZBL|1178.35003}} {{ZBL|1115.35005}} {{ZBL|1062.35004}} {{ZBL|1028.35001}} {{ZBL|0712.35001}} {{ZBL|0687.35002}} {{ZBL|0619.35002}} {{ZBL|0619.35001}} {{ZBL|0612.35001}} {{ZBL|0601.35001}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.R. Garabedian, "Partial differential equations" , Wiley (1964) {{MR|0162045}} {{ZBL|0124.30501}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> F. John, "Partial differential equations" , Springer (1974) {{MR|2098409}} {{MR|1713869}} {{MR|1185075}} {{MR|0765229}} {{MR|0831655}} {{MR|0614918}} {{MR|0598466}} {{MR|0514404}} {{MR|0502136}} {{MR|0470380}} {{MR|0447820}} {{MR|0390518}} {{MR|0350162}} {{MR|1536882}} {{MR|0304828}} {{MR|0261133}} {{MR|1534312}} {{MR|0208121}} {{MR|0163043}} {{MR|0130456}} {{ZBL|0304.90142}} </TD></TR></table>