Geodesic
B\"acklund transformation for the Liouville equation and gauge conditions in the theory of a~relativistic string
Teoretičeskaâ i matematičeskaâ fizika, Tome 56 (1983) no. 2, pp. 180-191

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It is shown that the gauge conditions in the theory of a relativistic string, which make it possible to replace the nonlinear Liouville equation by the d'Alembert equation, are a direct consequence of the Bäcklund transformation relating the solutions of these equations. A purely geometrical derivation is given of the Bäcklund transformations for the Liouville equation. A classical theory of a relativistic string is constructed in the $t=\tau$ gauge using the moving frame formalism and exterior differential forms in the theory of surfaces. The moving frame on the string trajectory is chosen in a special way. As a result, the theory of a string in fourdimensional space-time reduces to the d'Alembert equation for a single scalar function. 
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     author = {B. M. Barbashov and V. V. Nesterenko},
     title = {B\"acklund transformation for the {Liouville} equation and gauge conditions in the theory of a~relativistic string},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {180--191},
     publisher = {mathdoc},
     volume = {56},
     number = {2},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1983_56_2_a2/}
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B. M. Barbashov; V. V. Nesterenko. B\"acklund transformation for the Liouville equation and gauge conditions in the theory of a~relativistic string. Teoretičeskaâ i matematičeskaâ fizika, Tome 56 (1983) no. 2, pp. 180-191. http://geodesic.mathdoc.fr/item/TMF_1983_56_2_a2/