BÄCKLUND TRANSFORMATIONS AND KNOTS OF CONSTANT TORSION

The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vortex filament flow (also known as the Localized Induction Equation). In doing so, we obtain analytic constant-torsion representatives for a large number of knot types.

[1]  G. M.,et al.  A Treatise on the Differential Geometry of Curves and Surfaces , 1910, Nature.

[2]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[3]  P. Byrd,et al.  Handbook of Elliptic Integrals for Engineers and Physicists , 2014 .

[4]  James H. White.  Self-Linking and the Gauss Integral in Higher Dimensions , 1969 .

[5]  An analogue of Bäcklund's theorem in affine geometry , 1980 .

[6]  伊達 悦朗.  Multi-soliton solutions and quasi-periodic solutions of nonlinear equations of sine-Gordon type , 1983 .

[7]  Joel Langer,et al.  Knotted Elastic Curves in R3 , 1984 .

[8]  J. Pöschel,et al.  Inverse spectral theory , 1986 .

[9]  Allan P. Fordy,et al.  Soliton Theory: A Survey of Results , 1990 .

[10]  David W. McLaughlin,et al.  Geometry of the modulational instability III. Homoclinic orbits for the periodic sine-Gordon equation , 1990 .

[11]  Joel Langer,et al.  Poisson geometry of the filament equation , 1991 .

[12]  H. K. Moffatt,et al.  Interpretation of Invariants of the Betchov-Da Rios Equations and of the Euler Equations , 1991 .

[13]  Renzo L. Ricca,et al.  Rediscovery of Da Rios equations , 1991, Nature.

[14]  H. K. Moffatt.  Topological aspects of the dynamics of fluids and plasmas , 1992 .

[15]  Petrich,et al.  Solitons, Euler's equation, and vortex patch dynamics. , 1992, Physical review letters.

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

[17]  Joel Langer,et al.  Liouville integrability of geometric variational problems , 1994 .

[18]  Michael H. Freedman,et al.  On the Mobius Energy of Knots and Unknots , 1994 .

[19]  D. Mumford.  Elastica and Computer Vision , 1994 .

[20]  Joel Langer,et al.  Local geometric invariants of integrable evolution equations , 1994, solv-int/9401001.

[21]  A. Calini.  A note on a Bäcklund transformation for the continuous Heisenberg model , 1995 .

[22]  Joel Langer,et al.  Lagrangian Aspects of the Kirchhoff Elastic Rod , 1996, SIAM Rev..

[23]  M. Gromov.  Carnot-Carathéodory spaces seen from within , 1996 .