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

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

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

[1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)

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; Symbol of an operator).

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

References

[a1]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) pp. 271; 302
[a2]	L. Hörmander, "Linear partial differential operators" , Springer (1963) pp. 29; 31