Liouville equation

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 $$\phi_{t\tau} = e^\phi\tag{a1}$$ 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 $$u_{t\tau} = 0\tag{a2}$$ by any of the following two differential substitutions (see [Li], formulas (4) and (2)): $$\def\ln{\mathrm{ln\;}}\phi = \ln\big(\frac{2u_t u_\tau}{u^2}\big),\quad \phi = \ln\big(\frac{2u_t u_\tau}{\cos^2 u}\big).\tag{a3}$$ In other words, the formulas (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 (a2).

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

The Liouville equation (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 (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., [Ib]) that the Liouville equation is related with the wave equation (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 (a1), (a2) and (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