# Liouville equation

The Liouville equation

(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

(a2) |

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

(a3) |

In other words, the formulas (a3) provide the general solution to the Liouville equation, in terms of the well-known general solution of the wave equation (a2).

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

(a4) |

For the complete classification, see [a4].

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

(a5) |

with arbitrary invertible differentiable functions and . The infinitesimal generator of this group is:

where , are arbitrary functions and , are their first derivatives. It is shown in [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., [a3]) that the Liouville equation is related with the wave equation (a2) by the following Bäcklund transformation:

By letting , in (a1), (a2) and (a3), where , one can transform the elliptic Liouville equation into the Laplace equation .

#### References

[a1] | J. Liouville, "Sur l'équation aux différences partielles " J. Math. Pures Appl. , 8 (1853) pp. 71–72 |

[a2] | S. Lie, "Discussion der Differentialgleichung " Arch. for Math. , 6 (1881) pp. 112–124 (Reprinted as: S. Lie: Gesammelte Abhandlungen, Vol. 3, pp. 469–478) |

[a3] | N.H. Ibragimov, "Transformation groups applied to mathematical physics" , Reidel (1985) (In Russian) |

[a4] | "CRC Handbook of Lie group analysis of differential equations" N.H. Ibragimov (ed.) , 1 , CRC (1994) pp. Chapt. 12.3 |

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Liouville equation.

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