Geodesic
Exact solutions of the Liouville equation in multidimensional spaces
Teoretičeskaâ i matematičeskaâ fizika, Tome 120 (1999) no. 1, pp. 3-19

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Exact solutions of the multidimensional Liouville equation, which are sought in the class of functional forms of the degree $n$ equal to the coordinate space dimension, are constructed based on a special representation of the Laplace and D'Alembert equations. The corresponding Liouville equation solutions are completely described in the coordinate space dimensions $d=3,4$. For $d>4$, the general solution form and the method for obtaining algebraic equations for the coefficients of functional $n$-forms is presented. 
@article{TMF_1999_120_1_a0,
     author = {V. M. Zhuravlev},
     title = {Exact solutions of the {Liouville} equation in multidimensional spaces},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {3--19},
     publisher = {mathdoc},
     volume = {120},
     number = {1},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1999_120_1_a0/}
}
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V. M. Zhuravlev. Exact solutions of the Liouville equation in multidimensional spaces. Teoretičeskaâ i matematičeskaâ fizika, Tome 120 (1999) no. 1, pp. 3-19. http://geodesic.mathdoc.fr/item/TMF_1999_120_1_a0/