Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 027, 30 pages      arXiv:2308.10125      https://doi.org/10.3842/SIGMA.2024.027
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere

Annalisa Calini a, Thomas Ivey a and Emilio Musso b
a) Department of Mathematics, College of Charleston, Charleston, SC 29424, USA
b) Department of Mathematical Sciences, Politecnico di Torino, Italy

Received September 26, 2023, in final form March 13, 2024; Published online April 02, 2024

Abstract
We investigate geometric evolution equations for Legendrian curves in the 3-sphere which are invariant under the action of the unitary group ${\rm U}(2)$. We define a natural symplectic structure on the space of Legendrian loops and show that the modified Korteweg-de Vries equation, along with its associated hierarchy, are realized as curvature evolutions induced by a sequence of Hamiltonian flows. For the flow among these that induces the mKdV equation, we investigate the geometry of solutions which evolve by rigid motions in ${\rm U}(2)$. Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed.

Key words: mKdV; Legendrian curves; geometric flows; pseudohermitian CR geometry.

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References

  1. Byrd P.F., Friedman M.D., Handbook of elliptic integrals for engineers and physicists, Grundlehren Math. Wiss., Vol. 67, Springer, Berlin, 1954.
  2. Calini A., Ivey T., Topology and sine-Gordon evolution of constant torsion curves, Phys. Lett. A 254 (1999), 170-178.
  3. Calini A., Ivey T., Knot types, Floquet spectra, and finite-gap solutions of the vortex filament equation, Math. Comput. Simulation 55 (2001), 341-350.
  4. Calini A., Ivey T., Integrable geometric flows for curves in pseudoconformal $S^3$, J. Geom. Phys. 166 (2021), 104249, 17 pages, arXiv:1908.02722.
  5. Calini A., Ivey T., Marí-Beffa G., Remarks on KdV-type flows on star-shaped curves, Phys. D 238 (2009), 788-797, arXiv:0808.3593.
  6. Calini A., Ivey T., Marí Beffa G., Integrable flows for starlike curves in centroaffine space, SIGMA 9 (2013), 022, 21 pages, arXiv:1303.1259.
  7. Doliwa A., Santini P.M., An elementary geometric characterization of the integrable motions of a curve, Phys. Lett. A 185 (1994), 373-384.
  8. Etnyre J.B., Legendrian and transversal knots, in Handbook of Knot Theory, Elsevier, Amsterdam, 2005, 105-185, arXiv:math.SG/0306256.
  9. Etnyre J.B., Honda K., Knots and contact geometry I: Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001), 63-120, arXiv:math.GT/0006112.
  10. Goldstein R.E., Petrich D.M., The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 67 (1991), 3203-3206.
  11. Goldstein R.E., Petrich D.M., Solitons, Euler's equation, and vortex patch dynamics, Phys. Rev. Lett. 69 (1992), 555-558.
  12. Haller S., Vizman C., A dual pair for the contact group, Math. Z. 301 (2022), 2937-2973, arXiv:1909.11014.
  13. Heller L., Constrained Willmore tori and elastic curves in 2-dimensional space forms, Comm. Anal. Geom. 22 (2014), 343-369, arXiv:1303.1445.
  14. Konno K., Wadati M., Simple derivation of Bäcklund transformation from Riccati form of inverse method, Progr. Theoret. Phys. 53 (1975), 1652-1656.
  15. Kriegl A., Michor P.W., The convenient setting of global analysis, Math. Surveys Monogr., Vol. 53, American Mathematical Society, Providence, RI, 1997.
  16. Langer J., Perline R., Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991), 71-93.
  17. Langer J., Perline R., Curve motion inducing modified Korteweg-de Vries systems, Phys. Lett. A 239 (1998), 36-40.
  18. Lawden D.F., Elliptic functions and applications, Appl. Math. Sci., Vol. 80, Springer, New York, 1989.
  19. Lerario A., Mondino A., Homotopy properties of horizontal loop spaces and applications to closed sub-Riemannian geodesics, Trans. Amer. Math. Soc. Ser. B 6 (2019), 187-214, arXiv:1509.07000.
  20. Miura R.M., Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9 (1968), 1202-1204.
  21. Musso E., Liouville integrability of a variational problem for Legendrian curves in the three-dimensional sphere, in Selected Topics in Cauchy-Riemann Geometry, Quad. Mat., Vol. 9, Seconda Università di Napoli, Caserta, 2001, 281-306.
  22. Musso E., Motions of curves in the projective plane inducing the Kaup-Kupershmidt hierarchy, SIGMA 8 (2012), 030, 20 pages, arXiv:1205.5329.
  23. Musso E., Nicolodi L., Hamiltonian flows on null curves, Nonlinearity 23 (2010), 2117-2129, arXiv:0911.4467.
  24. Musso E., Nicolodi L., Salis F., On the Cauchy-Riemann geometry of transversal curves in the 3-sphere, J. Math. Phys. Anal. Geom. 16 (2020), 312-363, arXiv:2004.11350.
  25. Musso E., Salis F., The Cauchy-Riemann strain functional for Legendrian curves in the 3-sphere, Ann. Mat. Pura Appl. 199 (2020), 2395-2434, arXiv:2003.01713.
  26. Olver P.J., Applications of Lie groups to differential equations, Grad. Texts in Math., Vol. 107, Springer, New York, 1993.
  27. Pinkall U., Hamiltonian flows on the space of star-shaped curves, Results Math. 27 (1995), 328-332.
  28. Vizman C., Induced differential forms on manifolds of functions, Arch. Math. (Brno) 47 (2011), 201-215, arXiv:1111.3889.
  29. Wadati M., Bäcklund transformation for solutions of the modified Korteweg-de Vries equation, J. Phys. Soc. Japan 36 (1974), 1498-1498.
  30. Webster S.M., Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), 25-41.

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