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

Annalisa Calini ; Thomas Ivey ; Emilio Musso

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

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.

Détail

Keywords: mKdV, Legendrian curves, geometric flows, pseudohermitian CR geometry.

Bibliographie

[1] Byrd P.F., Friedman M.D., Handbook of elliptic integrals for engineers and physicists, Grundlehren Math. Wiss., 67, Springer, Berlin, 1954 | DOI | MR | Zbl

[2] Calini A., Ivey T., “Topology and sine-Gordon evolution of constant torsion curves”, Phys. Lett. A, 254 (1999), 170–178 | DOI

[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 | DOI | MR | Zbl

[4] Calini A., Ivey T., “Integrable geometric flows for curves in pseudoconformal $S^3$”, J. Geom. Phys., 166 (2021), 104249, 17 pp., arXiv: 1908.02722 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[6] Calini A., Ivey T., Marí Beffa G., “Integrable flows for starlike curves in centroaffine space”, SIGMA, 9 (2013), 022, 21 pp., arXiv: 1303.1259 | DOI | MR | Zbl

[7] Doliwa A., Santini P.M., “An elementary geometric characterization of the integrable motions of a curve”, Phys. Lett. A, 185 (1994), 373–384 | DOI | MR | Zbl

[8] Etnyre J.B., “Legendrian and transversal knots”, Handbook of Knot Theory, Elsevier, Amsterdam, 2005, 105–185, arXiv: math.SG/0306256 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[11] Goldstein R.E., Petrich D.M., “Solitons, Euler's equation, and vortex patch dynamics”, Phys. Rev. Lett., 69 (1992), 555–558 | DOI | MR | Zbl

[12] Haller S., Vizman C., “A dual pair for the contact group”, Math. Z., 301 (2022), 2937–2973, arXiv: 1909.11014 | DOI | MR | Zbl

[13] Heller L., “Constrained Willmore tori and elastic curves in 2-dimensional space forms”, Comm. Anal. Geom., 22 (2014), 343–369, arXiv: 1303.1445 | DOI | MR | Zbl

[14] Konno K., Wadati M., “Simple derivation of Bäcklund transformation from Riccati form of inverse method”, Progr. Theoret. Phys., 53 (1975), 1652–1656 | DOI | MR | Zbl

[15] Kriegl A., Michor P.W., The convenient setting of global analysis, Math. Surveys Monogr., 53, American Mathematical Society, Providence, RI, 1997 | DOI | MR | Zbl

[16] Langer J., Perline R., “Poisson geometry of the filament equation”, J. Nonlinear Sci., 1 (1991), 71–93 | DOI | MR | Zbl

[17] Langer J., Perline R., “Curve motion inducing modified Korteweg-de Vries systems”, Phys. Lett. A, 239 (1998), 36–40 | DOI | MR | Zbl

[18] Lawden D.F., Elliptic functions and applications, Appl. Math. Sci., 80, Springer, New York, 1989 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[20] Miura R.M., “Korteweg–de Vries equation and generalizations. I A remarkable explicit nonlinear transformation”, J. Math. Phys., 9 (1968), 1202–1204 | DOI | MR | Zbl

[21] Musso E., “Liouville integrability of a variational problem for Legendrian curves in the three-dimensional sphere”, Selected Topics in Cauchy–Riemann Geometry, Quad. Mat., 9, Seconda Università di Napoli, Caserta, 2001, 281–306 | MR

[22] Musso E., “Motions of curves in the projective plane inducing the Kaup–Kupershmidt hierarchy”, SIGMA, 8 (2012), 030, 20 pp., arXiv: 1205.5329 | DOI | MR | Zbl

[23] Musso E., Nicolodi L., “Hamiltonian flows on null curves”, Nonlinearity, 23 (2010), 2117–2129, arXiv: 0911.4467 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[26] Olver P.J., Applications of Lie groups to differential equations, Grad. Texts in Math., 107, Springer, New York, 1993 | DOI | MR | Zbl

[27] Pinkall U., “Hamiltonian flows on the space of star-shaped curves”, Results Math., 27 (1995), 328–332 | DOI | MR | Zbl

[28] Vizman C., “Induced differential forms on manifolds of functions”, Arch. Math. (Brno), 47 (2011), 201–215, arXiv: 1111.3889 | MR | Zbl

[29] Wadati M., “Bäcklund transformation for solutions of the modified Korteweg–de Vries equation”, J. Phys. Soc. Japan, 36 (1974), 1498–1498 | DOI | MR

[30] Webster S.M., “Pseudo-Hermitian structures on a real hypersurface”, J. Differential Geometry, 13 (1978), 25–41 | DOI | MR | Zbl