Geodesic
Generalization of the model of a relativistic string in a geometrical approach
Teoretičeskaâ i matematičeskaâ fizika, Tome 45 (1980) no. 3, pp. 365-376 
B. M. Barbashov; V. V. Nesterenko; A. M. Chervyakov. Generalization of the model of a relativistic string in a geometrical approach. Teoretičeskaâ i matematičeskaâ fizika, Tome 45 (1980) no. 3, pp. 365-376. http://geodesic.mathdoc.fr/item/TMF_1980_45_3_a6/
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     title = {Generalization of the model of a~relativistic string in a~geometrical approach},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     url = {http://geodesic.mathdoc.fr/item/TMF_1980_45_3_a6/}
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A model of a one-dimensionally extended relativistic object is proposed. Its dynamics is determined by the requirement that its covering surface in Minkowski space have constant mean curvature $h$ with respect to each normal direction. A special case of such surfaces is the world surface of a relativistic string (minimal surface with $h=0$). The methods of differential geometry are used to investigate the most interesting cases when the enveloping pseudo-Euclidean space-time has dimensions $D=3,4$. In the ease $D=3$, the proposed model is described by the single nonlinear equation $\square\varphi=h\sh\varphi$. In fourdimensional space-time, the dynamics of the model is determined by the system of two equations $$ \square\varphi=\frac{1}{2}h(e^\varphi-e^{-\varphi}\cos\theta), \quad \square\theta=\frac{1}{2}he^{-\varphi}\sin\theta. $$ A Lax representation for this system is obtained in a geometrical approach, and the use of the inverse scattering technique is briefly discussed.

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