View source for Linear hyperbolic partial differential equation and system

A partial differential equation (or system) of 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/l/l059/l059270/l0592701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

for which at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l0592702.png" /> of its domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l0592703.png" /> one can distinguish among the real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l0592704.png" /> (if necessary, after a suitable affine transformation of the independent variables) one variable, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l0592705.png" />, in a such a way that for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l0592706.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l0592707.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l0592708.png" /> the characteristic equation (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l0592709.png" />)

<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/l/l059/l059270/l05927010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927011.png" /> real roots. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927012.png" /> is a vector with non-negative integer coordinates, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927013.png" /> is the order of the differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927017.png" /> is the order of the system (1), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927018.png" /> is a real square matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927019.png" />, defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927022.png" />, is an unknown column vector, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927023.png" /> is a vector with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927024.png" /> components, defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927025.png" />.

A typical example is the [[Wave equation|wave 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/l/l059/l059270/l05927026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

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

A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927027.png" /> is said to be characteristic at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927028.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927030.png" />, 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/l/l059/l059270/l05927031.png" /></td> </tr></table>

is the characteristic form of the system (1). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927032.png" />, then one says that the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927033.png" /> defines a characteristic direction or characteristic normal at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927034.png" />. The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927035.png" /> is called a characteristic surface (or characteristic) of the system (1) if

<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/l/l059/l059270/l05927036.png" /></td> </tr></table>

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

<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/l/l059/l059270/l05927037.png" /></td> </tr></table>

is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927038.png" />, while on a characteristic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927039.png" /> it is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927040.png" />. A characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927041.png" /> is said to be simple if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927042.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927043.png" />,

<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/l/l059/l059270/l05927044.png" /></td> </tr></table>

Otherwise the characteristic is said to be multiple. A characteristic is sometimes said to be simple if the rank of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927045.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927046.png" />.

The system (1) is said to be hyperbolic at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927047.png" /> with respect to the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927048.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927049.png" /> if the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927050.png" /> is non-singular (that is, the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927051.png" /> is free) and if all roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927053.png" />, of the characteristic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927054.png" /> are real for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927055.png" />. The system (1) is said to be hyperbolic in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927056.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927057.png" /> if it is hyperbolic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927058.png" /> at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927059.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927060.png" /> of the characteristic equation are distinct for any non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927061.png" />. The characteristics of a strictly hyperbolic equation (or system) are simple. Strictly hyperbolic (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927062.png" />) systems are notable for the fact that the Cauchy problem

<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/l/l059/l059270/l05927063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

for them is well-posed under the single assumption of sufficient smoothness of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927065.png" /> of the system (1) and of the initial data (initial functions) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927067.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927068.png" />) for which the Cauchy problem is ill-posed.

The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927069.png" /> of the Cauchy problem (4) for the wave equation (3) can be written out explicitly, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927070.png" />, and only then, it has the property that the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927071.png" /> at the vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927072.png" /> of the characteristic cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927074.png" />, depends only on the value of the initial data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927076.png" /> and their derivatives on the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927078.png" />, of this cone (the so-called [[Huygens principle|Huygens principle]]).

For strictly hyperbolic (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927079.png" />) 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

<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/l/l059/l059270/l05927080.png" /></td> </tr></table>

For the case of one (scalar) equation with constant coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927081.png" /> the definition of strict hyperbolicity has been generalized as follows. Equation (1) is said to be hyperbolic with respect to a non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927082.png" /> if

<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/l/l059/l059270/l05927083.png" /></td> </tr></table>

and if there is a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927084.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927086.png" />,

<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/l/l059/l059270/l05927087.png" /></td> </tr></table>

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

<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/l/l059/l059270/l05927088.png" /></td> </tr></table>

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

<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/l/l059/l059270/l05927089.png" /></td> </tr></table>

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

<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/l/l059/l059270/l05927090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927092.png" /> are square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927093.png" />, defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927094.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927095.png" /> is an unknown vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927096.png" /> components, is called a symmetric hyperbolic system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927097.png" /> if the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927098.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l05927099.png" /> has the property that the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270100.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270101.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270102.png" /> can be written 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/l/l059/l059270/l059270103.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270106.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270107.png" /> are functions defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270108.png" />. Equation (6) is hyperbolic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270109.png" /> if at every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270110.png" /> all eigen values of the matrix of leading coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270112.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270113.png" />, then on a surface of the first type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270114.png" />, while on a surface of the second type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270115.png" />, 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/l/l059/l059270/l059270116.png" /></td> </tr></table>

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

<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/l/l059/l059270/l059270117.png" /></td> </tr></table>

is said to be uniformly (or regularly) hyperbolic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270118.png" /> if there is a positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270119.png" /> such 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/l/l059/l059270/l059270120.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270121.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270122.png" /> and for any non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270123.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270124.png" /> the inequality

<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/l/l059/l059270/l059270125.png" /></td> </tr></table>

is a necessary and sufficient condition for (6) to be uniformly hyperbolic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270126.png" />. The equation for the vibration of a string,

<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/l/l059/l059270/l059270127.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270128.png" /> of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270129.png" /> is given by the [[D'Alembert formula|d'Alembert 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/l/l059/l059270/l059270130.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270132.png" /> are arbitrary functions.

After a non-singular real change of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270134.png" />, the hyperbolic partial differential equation (6) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270135.png" /> reduces to the normal (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/l/l059/l059270/l059270136.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

<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/l/l059/l059270/l059270137.png" /></td> </tr></table>

For hyperbolic systems written in the form (7), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270139.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270140.png" /> are given real square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270142.png" /> is a given vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270143.png" /> an unknown vector, both with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270144.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270145.png" /> if at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270146.png" /> all eigen values of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270147.png" /> 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

<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/l/l059/l059270/l059270148.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

which has the following property: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270149.png" /> is a regular solution of (8) in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270150.png" /> of the Euclidean space of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270152.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270153.png" /> is an arbitrary point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270154.png" />, then the mean value of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270155.png" /> calculated on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270156.png" /> with centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270157.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270158.png" />, is equal to the mean value of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270159.png" /> calculated on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270160.png" /> with centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270161.png" /> and the same radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059270/l059270162.png" />. 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>