Liouville's Equation

The second-order ordinary differential equation

(1)

is called Liouville's equation (Goldstein and Braun 1973; Zwillinger 1997, p. 124), as are the partial differential equations

(2)

(Matsumo 1987; Zwillinger 1997, p. 133) and

(3)

(Calogero and Degasperis 1982, p. 60; Zwillinger 1997, p. 133).

REFERENCES:

Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations.New York: North-Holland, p. 60, 1982.

Goldstein, M. E. and Braun, W. H. Advanced Methods for the Solution of Differential Equations. NASA SP-316. Washington, DC: U.S. Government Printing Office, p. 98, 1973.

Matsumo, Y. "Exact Solution for the Nonlinear Klein-Gordon and Liouville Equations in Four-Dimensional Euclidean Space." J. Math. Phys. 28, 2317-2322, 1987.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 124 and 133, 1997.