# Characteristic strip

of a first-order partial differential equation

A family

$$x = x ( t),\ \ u = y ( t),\ \ u _ {x} = p ( t)$$

of continuously-differentiable functions in an interval $\alpha < t < \beta$, satisfying the equations

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

where the multiplication of the vectors is the scalar product, and where

$$\tag{* } F ( x, u , u _ {x} ) = 0$$

is a non-linear first-order partial differential equation in the unknown function $u: \Omega \subseteq \mathbf R ^ {n} \rightarrow \mathbf R$. Here $u _ {x} = \mathop{\rm grad} u$, $F ( x, y, p): \Omega \times \mathbf R \times \mathbf R ^ {n} \rightarrow \mathbf R$, $x, p \in \mathbf R ^ {n}$, $y \in \mathbf R$, $n \in \mathbf N$.

The importance of a characteristic strip consists in the fact that it is used in the study of, and in the search for, solutions of equation (*).

#### References

 [1] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion , Akad. Verlagsgesell. (1944) [2] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)

A characteristic strip is sometimes called a bicharacteristic.

In the modern theory, the characteristic strips of a partial differential equation carry the wave front sets of solutions of a partial differential equation.

#### References

 [a1] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1962) (Translated from German) [a2] L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983)
How to Cite This Entry:
Characteristic strip. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characteristic_strip&oldid=46323
This article was adapted from an original article by Yu.V. Komlenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article