View source for Hyperbolic partial differential equation

''at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483001.png" />''

A partial differential equation for which the [[Cauchy problem|Cauchy problem]] is uniquely solvable for initial data specified in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483002.png" /> on any non-characteristic surface (cf. [[Characteristic surface|Characteristic surface]]). In particular, a partial differential equation for which the normal cone has no imaginary zones is a hyperbolic partial differential equation. The differential equation

<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/h/h048/h048300/h0483003.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483004.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483005.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483006.png" /> is a homogeneous polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483007.png" />, while the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483008.png" /> is of lower degree than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483009.png" />, is a hyperbolic partial differential equation if its characteristic equation

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has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830011.png" /> different real solutions with respect to one of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830012.png" />, the remaining ones being fixed. Any equation (*) of the first order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830013.png" /> with real coefficients is a hyperbolic partial differential equation. A second-order equation

<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/h/h048/h048300/h04830014.png" /></td> </tr></table>

is hyperbolic if the quadratic form

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is positive definite.



====Comments====
The special variable among the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830017.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830018.png" /> different real solutions for each set of fixed values of the other <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830019.png" /> is often taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830020.png" /> (time). One speaks then of a (strictly) hyperbolic equation or an equation of (strictly) hyperbolic type with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830021.png" />-direction. More generally one considers hyperbolicity with respect to a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830022.png" /> [[#References|[a1]]].

A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830023.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830024.png" /> with principal part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830025.png" /> is called hyperbolic with respect to the real vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830027.png" /> and there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830028.png" /> such that

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830030.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830032.png" /> has only simple real roots for every real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830033.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830034.png" /> is said to be strictly hyperbolic or hyperbolic in the sense of Petrovskii.

The Cauchy problem for a constant-coefficient differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830035.png" /> with data on a non-characteristic plane is well posed for arbitrary lower-order terms if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830036.png" /> is strictly hyperbolic. For a discussion of similar matters for polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830037.png" /> with variable coefficients cf. [[#References|[a2]]].

For a system of higher-order linear partial differential equations

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830039.png" />, is a hyperbolic system of partial differential equations in the sense of Petrovskii if the determinant

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calculated in the ring of differential operators is a hyperbolic polynomial in the sense of Petrovskii (as a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830041.png" />). The Cauchy problem for a system that is hyperbolic in this sense is well posed [[#References|[a3]]], [[#References|[a4]]].

Instead of strictly hyperbolic one also finds the term strongly hyperbolic and instead of hyperbolic also weakly hyperbolic (which is therefore the case in which the lower-order terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830042.png" /> do matter).

====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. Chapt. XII</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''III''' , Springer  (1985)  pp. Chapt. XXIII</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.G. Petrovskii,   "Ueber das Cauchysche Problem für Systeme von partiellen Differentialgleichungen"  ''Mat. Sb. (N.S.)'' , '''2(44)'''  (1937)  pp. 815–870</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Mizohata,   "The theory of partial differential equations" , Cambridge Univ. Press  (1973)  (Translated from Japanese)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Chaillou,   "Hyperbolic differential polynomials" , Reidel  (1979)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Chazarain,   "Opérateurs hyperboliques à characteristique de multiplicité constante"  ''Ann. Inst. Fourier'' , '''24'''  (1974)  pp. 173–202</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Gårding,   "Linear hyperbolic equations with constant coefficients"  ''Acta Math.'' , '''85'''  (1951)  pp. 1–62</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  O.A. Oleinik,   "On the Cauchy problem for weakly hyperbolic equations"  ''Comm. Pure Appl. Math.'' , '''23'''  (1970)  pp. 569–586</TD></TR></table>