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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110150/l1101502.png" />''
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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]]
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The Liouville equation ''$\def\phi{\varphi}\partial_t\partial_\tau\phi(t,\tau) = e^{\phi(t,\tau)}$'' or
 +
\begin{equation}\label{a1}\phi_{t\tau} = e^\phi\tag{a1}\end{equation}
 +
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]]  
 +
\begin{equation}\label{a2}u_{t\tau} = 0\tag{a2}\end{equation}
 +
by any of the following two
 +
differential substitutions (see
 +
{{Cite|Li}}, formulas (4) and (2)):
 +
\begin{equation}\label{a3}\def\ln{\mathrm{ln\;}}\phi = \ln\big(\frac{2u_t u_\tau}{u^2}\big),\quad \phi = \ln\big(\frac{2u_t u_\tau}{\cos^2 u}\big).\tag{a3}\end{equation}
 +
In other words, the
 +
formulas \eqref{a3} provide the general solution to the Liouville equation,
 +
in terms of the well-known general solution $u=f(t)+g(\tau)$ of the wave equation
 +
\eqref{a2}.
  
<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>
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The Liouville equation appears also in Lie's classification
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{{Cite|Li2}} of second-order differential equations of the
 +
form
 +
\begin{equation}\label{a4}z_{xy} = F(z).\tag{a4}\end{equation}
 +
For the complete classification, see
 +
{{Cite|Ib2}}.
  
by any of the following two differential substitutions (see [[#References|[a1]]], formulas (4) and (2)):
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The Liouville equation \eqref{a1} is invariant under the infinite group of
 +
point transformations
 +
$$\bar t = \alpha(t),\ \bar\tau = \beta(\tau), \ \bar\phi = \phi - \ln \alpha'(t) - \ln \beta'(\tau)\tag{a5}$$
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with arbitrary invertible differentiable
 +
functions $\alpha(t) $ and $\beta(\tau)$. 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/l1101505.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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$$X=\xi(t)\frac{\partial}{\partial t} + \eta(\tau)\frac{\partial}{\partial\tau} - (\xi'(t)+\eta'(\tau))\frac{\partial}{\partial\phi},$$
 +
where $\xi(t)$, $\eta(\tau)$ are arbitrary functions and $\xi'(t)$, $\eta'(\tau)$ are their
 +
first derivatives. It is shown in
 +
{{Cite|Li2}} that the equation \eqref{a4}, and in particular the
 +
Liouville equation, does not admit non-trivial (i.e. non-point) Lie
 +
tangent transformations.
  
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).
+
In addition to the transformations \eqref{a3}, it is known (see, e.g.,
 
+
{{Cite|Ib}}) that the Liouville equation is related with the
The Liouville equation appears also in Lie's classification [[#References|[a2]]] of second-order differential equations of the form
+
wave equation \eqref{a2} by the following Bäcklund transformation:  
 
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$$\phi_t - u_t+ a e^{(\phi+u)/2} = 0,\quad \phi_\tau + u_\tau + \frac{2}{a} e^{(\phi-u)/2} = 0.$$
<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>
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By
 
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letting $x=t+\tau$, $y=i(t-\tau)$ in \eqref{a1}, \eqref{a2} and \eqref{a3}, where $i = \sqrt{-1}$, one can transform
For the complete classification, see [[#References|[a4]]].
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the elliptic Liouville equation $\phi_{xx}+\phi_{yy} = e^\phi$ into the
 
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[[Laplace equation|Laplace equation]] $u_{xx}+u_{yy} = 0$.
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====
 
====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>
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{|
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|-
 +
|valign="top"|{{Ref|Ib}}||valign="top"| N.H. Ibragimov, "Transformation groups applied to mathematical physics", Reidel (1985) (In Russian) {{MR|0785566}} {{ZBL|0558.53040}}
 +
|-
 +
|valign="top"|{{Ref|Ib2}}||valign="top"| "CRC Handbook of Lie group analysis of differential equations" N.H. Ibragimov (ed.), '''1''', CRC (1994) pp. Chapt. 12.3 {{MR|1278257}} {{ZBL|0864.35001}}
 +
|-
 +
||{{Ref|Li}}||| J. Liouville, "Sur l'équation aux différences partielles $\frac{d^2\log\lambda}{du\; dv} \pm \frac{\lambda}{2\alpha^2} = 0\;$" ''J. Math. Pures Appl.'', '''8''' (1853) pp. 71–72
 +
|-
 +
||{{Ref|Li2}}||| S. Lie, "Discussion der Differentialgleichung $\frac{d^2z}{dx\; dy} = F(z)$" ''Lie Arch. VI'', '''6''' (1881) pp. 112–124 (Reprinted as: S. Lie: Gesammelte Abhandlungen, Vol. 3, pp. 469–478) {{ZBL|13.0297.01}}
 +
|-
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|}

Latest revision as of 12:51, 30 March 2024

2020 Mathematics Subject Classification: Primary: 35-XX [MSN][ZBL]


The Liouville equation $\def\phi{\varphi}\partial_t\partial_\tau\phi(t,\tau) = e^{\phi(t,\tau)}$ or \begin{equation}\label{a1}\phi_{t\tau} = e^\phi\tag{a1}\end{equation} is a non-linear partial differential equation (cf. Differential equation, partial) that can be linearized and subsequently solved. Namely, it can be transformed into the linear wave equation \begin{equation}\label{a2}u_{t\tau} = 0\tag{a2}\end{equation} by any of the following two differential substitutions (see [Li], formulas (4) and (2)): \begin{equation}\label{a3}\def\ln{\mathrm{ln\;}}\phi = \ln\big(\frac{2u_t u_\tau}{u^2}\big),\quad \phi = \ln\big(\frac{2u_t u_\tau}{\cos^2 u}\big).\tag{a3}\end{equation} In other words, the formulas \eqref{a3} provide the general solution to the Liouville equation, in terms of the well-known general solution $u=f(t)+g(\tau)$ of the wave equation \eqref{a2}.

The Liouville equation appears also in Lie's classification [Li2] of second-order differential equations of the form \begin{equation}\label{a4}z_{xy} = F(z).\tag{a4}\end{equation} For the complete classification, see [Ib2].

The Liouville equation \eqref{a1} is invariant under the infinite group of point transformations $$\bar t = \alpha(t),\ \bar\tau = \beta(\tau), \ \bar\phi = \phi - \ln \alpha'(t) - \ln \beta'(\tau)\tag{a5}$$ with arbitrary invertible differentiable functions $\alpha(t) $ and $\beta(\tau)$. The infinitesimal generator of this group is:

$$X=\xi(t)\frac{\partial}{\partial t} + \eta(\tau)\frac{\partial}{\partial\tau} - (\xi'(t)+\eta'(\tau))\frac{\partial}{\partial\phi},$$ where $\xi(t)$, $\eta(\tau)$ are arbitrary functions and $\xi'(t)$, $\eta'(\tau)$ are their first derivatives. It is shown in [Li2] that the equation \eqref{a4}, and in particular the Liouville equation, does not admit non-trivial (i.e. non-point) Lie tangent transformations.

In addition to the transformations \eqref{a3}, it is known (see, e.g., [Ib]) that the Liouville equation is related with the wave equation \eqref{a2} by the following Bäcklund transformation: $$\phi_t - u_t+ a e^{(\phi+u)/2} = 0,\quad \phi_\tau + u_\tau + \frac{2}{a} e^{(\phi-u)/2} = 0.$$ By letting $x=t+\tau$, $y=i(t-\tau)$ in \eqref{a1}, \eqref{a2} and \eqref{a3}, where $i = \sqrt{-1}$, one can transform the elliptic Liouville equation $\phi_{xx}+\phi_{yy} = e^\phi$ into the Laplace equation $u_{xx}+u_{yy} = 0$.

References

[Ib] N.H. Ibragimov, "Transformation groups applied to mathematical physics", Reidel (1985) (In Russian) MR0785566 Zbl 0558.53040
[Ib2] "CRC Handbook of Lie group analysis of differential equations" N.H. Ibragimov (ed.), 1, CRC (1994) pp. Chapt. 12.3 MR1278257 Zbl 0864.35001
[Li] J. Liouville, "Sur l'équation aux différences partielles $\frac{d^2\log\lambda}{du\; dv} \pm \frac{\lambda}{2\alpha^2} = 0\;$" J. Math. Pures Appl., 8 (1853) pp. 71–72
[Li2] S. Lie, "Discussion der Differentialgleichung $\frac{d^2z}{dx\; dy} = F(z)$" Lie Arch. VI, 6 (1881) pp. 112–124 (Reprinted as: S. Lie: Gesammelte Abhandlungen, Vol. 3, pp. 469–478) Zbl 13.0297.01
How to Cite This Entry:
Liouville equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville_equation&oldid=18774
This article was adapted from an original article by N.H. Ibragimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article