# Bicharacteristic

bicharacteristic strip, of a linear partial differential operator

A line of tangency of any two characteristics (cf. Characteristic)

$$\phi(x_1,\dots,x_n)=0,\quad\psi(x_1,\dots,x_n)=0$$

of this linear partial differential operator. If the parameter $s$ is introduced on the bicharacteristic strip, then its equations $x_i=x_i(s)$, $i=1,\dots,n$, are defined by solving a system of $2n$ ordinary differential equations

Here $Q(\xi_1,\dots,\xi_n,x_1,\dots,x_n)$ is the principal symbol of the linear partial differential operator, the dot indicates differentiation with respect to the parameter $s$ and, if $\xi_i=\phi_{x_i}$, the equation $Q=0$ is the characteristic equation of the differential operator. Thus, the solution $x_i=x_i(s)$, $\xi_i=\xi_i(s)$, $i=1,\dots,n$, of the system \eqref{*} for $Q=0$ defines the bicharacteristic strip $Q=0$. This bicharacteristic strip belongs to the characteristic $\phi(x_1,\dots,x_n)=0$, i.e. $\phi(x_1(s),\dots,x_n(s))\equiv0$, if the equations

$$\phi(x_1(s),\dots,x_n(s))=0$$

and

$$\xi_i(s)=\phi_{x_i}(x_1(s),\dots,x_n(s)),\quad i=1,\dots,n,$$

are valid for at least one value of $s$, then it follows that they are valid for all values of $s$.

How to Cite This Entry:
Bicharacteristic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bicharacteristic&oldid=43524
This article was adapted from an original article by B.L. Rozhdestvenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article