Initial value problems of the sine-Gordon equation and geometric solutions. (English) Zbl 1077.53010
As is well known, by using inverse scattering techniques, every solution \(\varphi (x,y)\) of the sine-Gordon equation \(u_{xy}=\sin u\) can be interpreted as a nonlinear superposition of solutions along the axes \(x=0\) and \(y=0\), of which the geometric interpretation is that every weakly regular surface of Gauss curvature \(K=-1\), in arc length asymptotic line parametrization, is uniquely determined by the values \(\varphi (x,0)\) and \(\varphi (0,y)\) of its coordinate angle along the axes. In this paper, the author introduces a generalized Weierstrass representation of pseudo-spherical surfaces that depends only on these values. Moreover, the author explicitly construct the associated family of pseudo-spherical immersions corresponding to it.
Reviewer: Shen Yi-Bing (Hangzhou)
MSC:
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 58E20 | Harmonic maps, etc. |
References:
| [1] | Kricever, I. M.: An analogue of D’Alembert’s Formula for the equations of the principal chiral field and for the sine-Gordon Equation, Soviet Math. Dokl. 22(1) (1980), 79–84. · Zbl 0496.35073 |
| [2] | Dorfmeister, J., Pedit, F. and Wu, H.: Weierstrass type representations of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), 633–668. · Zbl 0932.58018 |
| [3] | Bobenko, A. I. and Kitaev, A. V.: On asymptotic cones of surfaces with constant curvature and the third Painleve equation, Manuscripta Math. 97 (1998), 489–516; SFB288, Preprint {315} (1998). · Zbl 0965.53010 |
| [4] | Bobenko, A. I., Matthes, D. and Suris, Y. B.: Nonlinear hyperbolic equations in surface theory: Integrable discretizations and approximations results, arXiv:Math.NA/0208042 v1 Aug 2002. · Zbl 1098.53005 |
| [5] | Bobenko, A. I.: Surfaces in terms of 2 by 2 matrices, In: Harmonic Maps and Integrable Systems, Vieweg, 1994, pp. 83–128. · Zbl 0841.53003 |
| [6] | Eisenhart, L. P.: A Treatise in the Differential Geometry of Curves and Surfaces, Dover, New York, 1909. · JFM 40.0657.01 |
| [7] | Wu, H.: Non-linear partial differential equations via vector fields on homogeneous Banach manifolds, Ann. Global Anal. Geom. 10 (1992), 151–170. · Zbl 0756.58023 · doi:10.1007/BF00130917 |
| [8] | Terng, C. L. and Uhlenbeck, K.: Bäcklund transformation and loop group actions, Comm. Pure Appl. Math. 53 (2000), 1–75, math.dg/9805074. · Zbl 1031.37064 · doi:10.1002/(SICI)1097-0312(200001)53:1<1::AID-CPA1>3.0.CO;2-U |
| [9] | Dorfmeister, J. and Sterling, I.: Finite type Lorentz harmonic maps and the method of Symes, J. Differential Geom. Appl. 17 (2002), 43–53. · Zbl 1027.58012 · doi:10.1016/S0926-2245(02)00091-8 |
| [10] | Sym, A.: Soliton surfaces and their applications (Soliton geometry from spectral problems), In: {Lecture Notes in Physics} 239, Springer, New York, 1985, pp. 154–231. · Zbl 0583.53017 |
| [11] | Amsler, M. H.: Des surfaces a courbure negative constante dans l’espace a trois dimensions et de leurs singularites, Math. Ann. 130 (1955), 234–256. · Zbl 0068.35102 · doi:10.1007/BF01343351 |
| [12] | Melko, M. and Sterling, I.: Integrable systems, harmonic maps and the classical theory of surfaces, In: Harmonic Maps and Integrable Systems, Vieweg, 1994, pp. 129–144. · Zbl 0814.58014 |
| [13] | Melko, M. and Sterling, I.: Applications of Soliton theory to the construction of pseudospherical surfaces in R3, Ann. Global Anal. Geom. 11 (1993), 65–107. · Zbl 0810.53003 · doi:10.1007/BF00773365 |
| [14] | Toda, M.: Weierstrass representation of weakly regular pseudospherical surfaces in euclidean space, Balkan J. Geom. Appl. 7(2) (2002), 87–136. · Zbl 1028.53006 |
| [15] | Pressley, A. and Segal, G.: Loop Groups, Oxford Math. Monogr., Oxford University Press, 1986. · Zbl 0618.22011 |
| [16] | Bobenko, A. I.: Constant mean curvature surfaces and integrable equations, Russ. Math. Surveys 46(4) (1991), 1–45. · Zbl 0779.62004 · doi:10.1070/RM1991v046n04ABEH002826 |
| [17] | Chern, S. S. and Terng, C. L.: An analogue of Bäcklund’s Theorem in affine geometry, Rocky Mountain J. Math. 10(1) (1980), 439–458. · Zbl 0407.53002 · doi:10.1216/RMJ-1980-10-1-105 |
| [18] | Dorfmeister, J. and Haak, G.: Meromorphic potentials and smooth surfaces of constant mean curvature, Math. Z. 224 (1997), 603–640. · Zbl 0898.53008 · doi:10.1007/PL00004295 |
| [19] | Lund, F.: Soliton and geometry, In: A. O. Barut (ed.), Proceedings of the NATO ASI on Nonlinear Equations in Physics and Mathematics, Reidel, Dordrecht, 1978. · Zbl 0398.35077 |
| [20] | Toda, M.: Pseudospherical surfaces via moving frames and loop groups, PhD Dissertation, hardbound, University of Kansas, 2000, 114 pp. |
| [21] | Wu, H.: A simple way for determining the normalized potentials for harmonic maps, Ann. Global Anal. Geom. 17 (1999), 189–199. · Zbl 0954.58017 · doi:10.1023/A:1006556302766 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
