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''bicharacteristic strip, of a linear partial differential operator''

A line of tangency of any two characteristics (cf. [[Characteristic|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

\begin{equation}\dot x_i(s)=Q_{\xi_i},\quad\dot\xi_i=-Q_{x_i},\quad i=1,\dots,n.\label{*}\end{equation}

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 $s$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR></table>



====Comments====
The projections $x_i=x_i(s)$, $i=1,\dots,n$, into $x$-space are called the bicharacteristic curves (or rays). The bicharacteristic curves are tangent to the characteristic hypersurfaces $\phi(x_1,\dots,x_n)=0$ due to the homogeneity of the principal symbol, as a function of $(\xi_1,\dots,\xi_n)$, of degree equal to the order of the linear partial differential operator (cf. also [[Principal part of a differential operator|Principal part of a differential operator]]; [[Symbol of an operator|Symbol of an operator]]).

Nowadays, the standard reference on these matters is [[#References|[a1]]], or the older, more concise, [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  pp. 271; 302</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Hörmander,   "Linear partial differential operators" , Springer  (1963)  pp. 29; 31</TD></TR></table>