Deformations of surfaces associated with integrable Gauss–Mainardi–Codazzi equations

Using the formulation of the immersion of a two-dimensional surface into the three-dimensional Euclidean space proposed recently, a mapping from each symmetry of integrable equations to surfaces in R3 can be established. We show that among these surfaces the sphere plays a unique role. Indeed, under the rigid SU(2) rotations all integrable equations are mapped to a sphere. Furthermore we prove that all compact surfaces generated by the infinitely many generalized symmetries of the sine-Gordon equation are homeomorphic to a sphere. We also find some new Weingarten surfaces arising from the deformations of the modified Kurteweg–de Vries and of the nonlinear Schrodinger equations. Surfaces can also be associated with the motion of curves. We study curve motions on a sphere and we identify a new integrable equation characterizing such a motion for a particular choice of the curve velocity.

[1]  L. Eisenhart.  Transformation of Surfaces. , 1915, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Metin Gürses,et al.  New nonlinear evolution equations from surface theory , 1981 .

[3]  John C. Wood,et al.  Harmonic maps and integrable systems , 1994 .

[4]  D. Levi,et al.  Soliton surfaces , 1985 .

[5]  J. Cieśliński.  A generalized formula for integrable classes of surfaces in Lie algebras , 1997 .

[6]  A. Sym.  Soliton surfaces , 1983 .

[7]  The elliptic Sinh Gordon equation and the construction of toroidal soap bubbles , 1988 .

[8]  Ulrich Pinkall,et al.  On the classification of constant mean curvature tori , 1989 .

[9]  Athanassios S. Fokas,et al.  Surfaces on Lie groups, on Lie algebras, and their integrability , 1996 .

[10]  A. Sym.  Soliton surfaces , 1984 .

[11]  A. Sym.  Soliton surfaces , 1984 .

[12]  Kazuaki Nakayama,et al.  Motion of Curves in the Plane , 1993 .

[13]  D. Levi,et al.  Integrable systems describing surfaces of non-constant curvature , 1990 .

[14]  A. Sym.  Soliton surfaces and their applications (soliton geometry from spectral problems) , 1985 .

[15]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[16]  Weingarten surfaces and nonlinear partial differential equations , 1993 .

[17]  Wolfgang K. Schief,et al.  On the geometry of an integrable (2+1)–dimensional sine–Gordon system , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[18]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[20]  P. Santini,et al.  An elementary geometric characterization of the integrable motions of a curve , 1994 .

[21]  H. Hasimoto,et al.  A soliton on a vortex filament , 1972, Journal of Fluid Mechanics.

[22]  Athanassios S. Fokas,et al.  Important developments in soliton theory , 1993 .

[23]  G. Lamb,et al.  Solitons on moving space curves , 1977 .

[24]  Athanassios S. Fokas,et al.  Symmetries and Integrability , 1987 .

[25]  Rolf Walter,et al.  Explicit examples to the H-problem of heinz hopf , 1987 .

[26]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[27]  Alexander I. Bobenko,et al.  All constant mean curvature tori inR3,S3,H3 in terms of theta-functions , 1991 .

[28]  I. Gel'fand,et al.  A formula for constructing infinitely many surfaces on Lie algebras and integrable equations , 2000 .

[29]  A. Sym.  Soliton surfaces , 1984 .

[30]  Henry C. Wente.  Twisted Tori of Constant Mean Curvature in R3 , 1987 .

[31]  Boris Konopelchenko,et al.  Induced Surfaces and Their Integrable Dynamics , 1996 .

[32]  J. Tafel.  Surfaces in R3 with prescribed curvature , 1995 .

[33]  M. Gürses.  Motion of curves on two-dimensional surfaces and soliton equations , 1998 .

[34]  Alan C. Newell,et al.  Solitons in mathematics and physics , 1987 .

[35]  A. Sym.  Soliton surfaces , 1984 .

[36]  D. Korotkin.  On some integrable cases in surface theory , 1999 .

[37]  Henry C. Wente.  Counterexample to a conjecture of H. Hopf , 1986 .

[38]  Petrich,et al.  The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane. , 1991, Physical review letters.