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 (*).

See also Characteristic.

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)

Comments

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)