Taimanov's surface evolution and Bäcklund transformations for curves
Authors:
Oscar Garay and Joel Langer
Journal:
Conform. Geom. Dyn. 3 (1999), 3749
MSC (1991):
Primary 35Q51, 35Q53, 53A05, 53A35, 53A30
Published electronically:
March 25, 1999
MathSciNet review:
1684040
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Taimanov's evolution of conformally parametrized surfaces in Euclidean space by the modified NovikovVeselov equation is interpreted here (in the revolution case) using hyperbolic geometry and Bäcklund transformations for curves.
 [CI]
A. Calini and T. Ivey, Bäcklund transformations and knots of constant torsion, J. Knot Theory and its Ramifications 7 (1998), p. 719. CMP 99:01
 [Ch]
Bangyen
Chen, Some conformal invariants of submanifolds and their
applications, Boll. Un. Mat. Ital. (4) 10 (1974),
380–385 (English, with Italian summary). MR 0370436
(51 #6663)
 [GP]
Raymond
E. Goldstein and Dean
M. Petrich, The Kortewegde Vries hierarchy as dynamics of closed
curves in the plane, Phys. Rev. Lett. 67 (1991),
no. 23, 3203–3206. MR 1135964
(92g:58050), http://dx.doi.org/10.1103/PhysRevLett.67.3203
 [Iv]
T. Ivey, Helices, Hasimoto surfaces and Bäcklund transformations, Preprint (1998).
 [Ko]
B.
G. Konopelchenko, Induced surfaces and their integrable
dynamics, Stud. Appl. Math. 96 (1996), no. 1,
9–51. MR
1365273 (96i:53011)
 [La]
G.
L. Lamb Jr., Solitons and the motion of helical curves, Phys.
Rev. Lett. 37 (1976), no. 5, 235–237. MR 0473584
(57 #13250)
 [LP 1]
Joel
Langer and Ron
Perline, Poisson geometry of the filament equation, J.
Nonlinear Sci. 1 (1991), no. 1, 71–93. MR 1102831
(92k:58118), http://dx.doi.org/10.1007/BF01209148
 [LP 2]
Joel
Langer and Ron
Perline, Local geometric invariants of integrable evolution
equations, J. Math. Phys. 35 (1994), no. 4,
1732–1737. MR 1267918
(95c:58095), http://dx.doi.org/10.1063/1.530567
 [LP 3]
J. Langer and R. Perline, Curve motion inducing modified Kortewegde Vries systems, Phys. Lett. A 239 (1998), pp. 3749. CMP 98:10
 [LS 1]
Joel
Langer and David
A. Singer, The total squared curvature of closed curves, J.
Differential Geom. 20 (1984), no. 1, 1–22. MR 772124
(86i:58030)
 [LS 2]
Joel
Langer and David
Singer, Curves in the hyperbolic plane and mean curvature of tori
in 3space, Bull. London Math. Soc. 16 (1984),
no. 5, 531–534. MR 751827
(85k:53006), http://dx.doi.org/10.1112/blms/16.5.531
 [Ro]
C. Rogers, Bäcklund transformations in soliton theory, in Soliton theory: a survey of results, ed. A. P. Fordy, St. Martin's Press, 1990. CMP 91:07
 [Ta 1]
Iskander
A. Taimanov, Modified NovikovVeselov equation and differential
geometry of surfaces, Solitons, geometry, and topology: on the
crossroad, Amer. Math. Soc. Transl. Ser. 2, vol. 179, Amer. Math.
Soc., Providence, RI, 1997, pp. 133–151. MR 1437161
(98c:53071)
 [Ta 2]
I. Taimanov, Surfaces of revolution in terms of solitons, Ann. Global Analysis and Geom. 15 (1997), pp. 3749. CMP 98:04
 [We]
Joel
L. Weiner, On a problem of Chen, Willmore, et al, Indiana
Univ. Math. J. 27 (1978), no. 1, 19–35. MR 0467610
(57 #7466)
 [CI]
 A. Calini and T. Ivey, Bäcklund transformations and knots of constant torsion, J. Knot Theory and its Ramifications 7 (1998), p. 719. CMP 99:01
 [Ch]
 B. Y. Chen, Some conformal invariants of submanifolds and their applications, Bol. Un. Mat. Ital. (4) 10 (1974). MR 51:6663
 [GP]
 R. Goldstein and D. Petrich, The Kortewegde Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 67 (1991), p. 3203. MR 92g:58050
 [Iv]
 T. Ivey, Helices, Hasimoto surfaces and Bäcklund transformations, Preprint (1998).
 [Ko]
 B. Konopelchenko, Induced surfaces and their integrable dynamics, Studies in Appl. Math. 96 (1996), p. 9. MR 96i:53011
 [La]
 G. Lamb, Solitons and the motion of helical curves, 37 (1976), p. 235. MR 57:13250
 [LP 1]
 J. Langer and R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991), p. 71. MR 92k:58118
 [LP 2]
 J. Langer and R. Perline, Local geometric invariants of integrable evolution equations, J. Math. Phys. 35 (1994), p. 1732. MR 95c:58095
 [LP 3]
 J. Langer and R. Perline, Curve motion inducing modified Kortewegde Vries systems, Phys. Lett. A 239 (1998), pp. 3749. CMP 98:10
 [LS 1]
 J. Langer and D. Singer, The total squared curvature of closed curves, J. Diff. Geom. 20 (1984), pp. 3749. MR 86i:58030
 [LS 2]
 J. Langer and D. Singer, Curves in the hyperbolic plane and mean curvature of tori in 3space, Bull. London Math Soc. 16 (1984), pp. 3749. MR 85k:53006
 [Ro]
 C. Rogers, Bäcklund transformations in soliton theory, in Soliton theory: a survey of results, ed. A. P. Fordy, St. Martin's Press, 1990. CMP 91:07
 [Ta 1]
 I. Taimanov, Modified NovikovVeselov equation and differential geometry of surfaces, Preprint November 1995 (daga 9511005), Translations Amer. Math. Soc., Ser. 2, 179, 1997. MR 98c:53071
 [Ta 2]
 I. Taimanov, Surfaces of revolution in terms of solitons, Ann. Global Analysis and Geom. 15 (1997), pp. 3749. CMP 98:04
 [We]
 J. Weiner, On a problem of Chen, Willmore, et al., Indiana Univ. Math. J. 27 (1978), pp. 3749. MR 57:7466
Similar Articles
Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society
with MSC (1991):
35Q51,
35Q53,
53A05,
53A35,
53A30
Retrieve articles in all journals
with MSC (1991):
35Q51,
35Q53,
53A05,
53A35,
53A30
Additional Information
Oscar Garay
Affiliation:
Department of Mathematics, Universidad Pais Vasco, Bilbao, Spain
Email:
mtpgabeo@lg.ehu.es
Joel Langer
Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106
Email:
jxl6@po.cwru.edu
DOI:
http://dx.doi.org/10.1090/S1088417399000430
PII:
S 10884173(99)000430
Received by editor(s):
October 28, 1998
Published electronically:
March 25, 1999
Additional Notes:
We wish to acknowledge the support of the Departamento De Educacion, Universidades E Investigacion, Gobierno Vasco, for J. Langer’s visit.
Article copyright:
© Copyright 1999
American Mathematical Society
