# Hyperbolic partial differential equation

at a given point $M( x _ {1} \dots x _ {n} )$

A partial differential equation for which the Cauchy problem is uniquely solvable for initial data specified in a neighbourhood of $M$ on any non-characteristic surface (cf. Characteristic surface). In particular, a partial differential equation for which the normal cone has no imaginary zones is a hyperbolic partial differential equation. The differential equation

$$\tag{* } L ( u) = H ( D _ {1} \dots D _ {n} ) u + F ( D _ {1} \dots D _ {n} ) u + G ( x) = 0,$$

where $D _ {i} = \partial / \partial x _ {i}$( $i = 1 \dots n$), $H( D _ {1} \dots D _ {n} )$ is a homogeneous polynomial of degree $m$, while the polynomial $F$ is of lower degree than $m$, is a hyperbolic partial differential equation if its characteristic equation

$$Q ( \xi _ {1} \dots \xi _ {n} ) = H ( \xi _ {1} \dots \xi _ {n} ) = 0$$

has $m$ different real solutions with respect to one of the variables $\xi _ {1} \dots \xi _ {n}$, the remaining ones being fixed. Any equation (*) of the first order $( m = 1 )$ with real coefficients is a hyperbolic partial differential equation. A second-order equation

$$L ( u) = u _ {tt} - \sum _ {i, j = 1 } ^ { n } a _ {ij} D _ {i} D _ {j} u + Fu + G = 0$$

is hyperbolic if the quadratic form

$$\sum _ {i, j = 1 } ^ { n } a _ {ij} \xi _ {i} \xi _ {j}$$

is positive definite.

The special variable among the $\xi _ {1} \dots \xi _ {n}$ such that $H ( \xi _ {1} \dots \xi _ {n} )$ has $m$ different real solutions for each set of fixed values of the other $n - 1$ is often taken to be $t$( time). One speaks then of a (strictly) hyperbolic equation or an equation of (strictly) hyperbolic type with respect to the $t$- direction. More generally one considers hyperbolicity with respect to a vector $N$[a1].

A polynomial $P$ of degree $m$ with principal part $P _ {m}$ is called hyperbolic with respect to the real vector $N$ if $P _ {m} ( N) \neq 0$ and there exists a number $\tau _ {0} > 0$ such that

$$P ( \xi + i \tau N) \neq 0 \ \ \textrm{ if } \ \xi \in \mathbf R ^ {n} ,\ \tau < \tau _ {0} .$$

If $P _ {m}$ is such that $P _ {m} ( N) \neq 0$ and $P _ {m} ( \xi + \tau N)$ has only simple real roots for every real $\xi \neq 0$, then $P$ is said to be strictly hyperbolic or hyperbolic in the sense of Petrovskii.

The Cauchy problem for a constant-coefficient differential operator $P$ with data on a non-characteristic plane is well posed for arbitrary lower-order terms if and only if $P$ is strictly hyperbolic. For a discussion of similar matters for polynomials $P$ with variable coefficients cf. [a2].

For a system of higher-order linear partial differential equations

$$\sum _ {j = 1 } ^ { l } \sum _ {| \alpha | \leq N _ {j} } a _ \alpha ^ {ij} ( x) \frac{\partial ^ \alpha }{\partial x ^ \alpha } u _ {j} = 0,\ \ i = 1 \dots l ,$$

where $\alpha = ( \alpha _ {0} \dots \alpha _ {n} )$, is a hyperbolic system of partial differential equations in the sense of Petrovskii if the determinant

$$\mathop{\rm det} \ \left ( \sum _ {| \alpha | \leq N _ {j} } a _ \alpha ^ {ij} \frac{\partial ^ \alpha }{\partial x ^ \alpha } \right )$$

calculated in the ring of differential operators is a hyperbolic polynomial in the sense of Petrovskii (as a polynomial of degree $N = \sum N _ {j}$). The Cauchy problem for a system that is hyperbolic in this sense is well posed [a3], [a4].

Instead of strictly hyperbolic one also finds the term strongly hyperbolic and instead of hyperbolic also weakly hyperbolic (which is therefore the case in which the lower-order terms of $P$ do matter).

How to Cite This Entry:
Hyperbolic partial differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_partial_differential_equation&oldid=47289
This article was adapted from an original article by B.L. Rozhdestvenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article