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''bicharacteristic strip, of a linear partial differential operator''
 
''bicharacteristic strip, of a linear partial differential operator''
  
 
A line of tangency of any two characteristics (cf. [[Characteristic|Characteristic]])
 
A line of tangency of any two characteristics (cf. [[Characteristic|Characteristic]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b0160701.png" /></td> </tr></table>
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$$\phi(x_1,\dots,x_n)=0,\quad\psi(x_1,\dots,x_n)=0$$
  
of this linear partial differential operator. If the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b0160702.png" /> is introduced on the bicharacteristic strip, then its equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b0160703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b0160704.png" />, are defined by solving a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b0160705.png" /> ordinary differential equations
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b0160706.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b0160707.png" /> is the principal symbol of the linear partial differential operator, the dot indicates differentiation with respect to the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b0160708.png" /> and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b0160709.png" />, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607010.png" /> is the characteristic equation of the differential operator. Thus, the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607013.png" />, of the system (*) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607014.png" /> defines the bicharacteristic strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607015.png" />. This bicharacteristic strip belongs to the characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607016.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607017.png" />, if the equations
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607018.png" /></td> </tr></table>
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$$\phi(x_1(s),\dots,x_n(s))=0$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607019.png" /></td> </tr></table>
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$$\xi_i(s)=\phi_{x_i}(x_1(s),\dots,x_n(s)),\quad i=1,\dots,n,$$
  
are valid for at least one value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607020.png" />, then it follows that they are valid for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607021.png" />.
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are valid for at least one value of $s$, then it follows that they are valid for all values of $s$.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
The projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607023.png" />, into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607024.png" />-space are called the bicharacteristic curves (or rays). The bicharacteristic curves are tangent to the characteristic hypersurfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607025.png" /> due to the homogeneity of the principal symbol, as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016070/b01607026.png" />, 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]]).
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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]]].
 
Nowadays, the standard reference on these matters is [[#References|[a1]]], or the older, more concise, [[#References|[a2]]].

Latest revision as of 22:32, 10 December 2018

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
How to Cite This Entry:
Bicharacteristic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bicharacteristic&oldid=15970
This article was adapted from an original article by B.L. Rozhdestvenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article