View source for Liouville equation

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l1101502.png" />''

The Liouville 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/l/l110/l110150/l1101503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

is a non-linear partial differential equation (cf. [[Differential equation, partial|Differential equation, partial]]) that can be linearized and subsequently solved. Namely, it can be transformed into the linear [[Wave equation|wave 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/l/l110/l110150/l1101504.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

by any of the following two differential substitutions (see [[#References|[a1]]], formulas (4) and (2)):

<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/l/l110/l110150/l1101505.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

In other words, the formulas (a3) provide the general solution to the Liouville equation, in terms of the well-known general solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l1101506.png" /> of the wave equation (a2).

The Liouville equation appears also in Lie's classification [[#References|[a2]]] of second-order differential equations of the form

<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/l/l110/l110150/l1101507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

For the complete classification, see [[#References|[a4]]].

The Liouville equation (a1) is invariant under the infinite group of point transformations

<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/l/l110/l110150/l1101508.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

with arbitrary invertible differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l1101509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015010.png" />. The infinitesimal generator of this group is:

<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/l/l110/l110150/l11015011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015013.png" /> are arbitrary functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015015.png" /> are their first derivatives. It is shown in [[#References|[a2]]] that the equation (a4), and in particular the Liouville equation, does not admit non-trivial (i.e. non-point) Lie tangent transformations.

In addition to the transformations (a3), it is known (see, e.g., [[#References|[a3]]]) that the Liouville equation is related with the wave equation (a2) by the following Bäcklund transformation:

<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/l/l110/l110150/l11015016.png" /></td> </tr></table>

By letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015018.png" /> in (a1), (a2) and (a3), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015019.png" />, one can transform the elliptic Liouville equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015020.png" /> into the [[Laplace equation|Laplace equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015021.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Liouville,   "Sur l'équation aux différences partielles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015022.png" />"  ''J. Math. Pures Appl.'' , '''8'''  (1853)  pp. 71–72</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Lie,   "Discussion der Differentialgleichung <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l11015023.png" />"  ''Arch. for Math.'' , '''6'''  (1881)  pp. 112–124  (Reprinted as: S. Lie: Gesammelte Abhandlungen, Vol. 3, pp. 469–478)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.H. Ibragimov,   "Transformation groups applied to mathematical physics" , Reidel  (1985)  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  "CRC Handbook of Lie group analysis of differential equations"  N.H. Ibragimov (ed.) , '''1''' , CRC  (1994)  pp. Chapt. 12.3</TD></TR></table>