View source for Bicharacteristic

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

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

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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

<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>

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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and

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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" />.

====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 <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]]).

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>