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

SIGMA 21 (2025), 009, 18 pages      arXiv:2312.10765      https://doi.org/10.3842/SIGMA.2025.009

Geometric Transformations on Null Curves in the Anti-de Sitter 3-Space

Emilio Musso ^a and Álvaro Pámpano ^b
a) Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
^b) Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409, USA

Received September 24, 2024, in final form February 05, 2025; Published online February 12, 2025

Abstract
We provide a geometric transformation on null curves in the anti-de Sitter 3-space (AdS) which induces the Bäcklund transformation for the KdV equation. In addition, we show that this geometric transformation satisfies a suitable permutability theorem. We also illustrate how to implement it when the original null curve has constant bending.

Key words: anti-de Sitter space; geometric transformations; KdV equation; null curves.

pdf (8974 kb)   tex (8184 kb)  

References

1. Bobenko A.I., Pavlyukevich T.V., Springborn B.A., Hyperbolic constant mean curvature one surfaces: spinor representation and trinoids in hypergeometric functions, Math. Z. 245 (2003), 63-91, arXiv:math.DG/0206021.
2. Bryant R.L., Surfaces of mean curvature one in hyperbolic space, Astérisque 154-155 (1987), 321-347.
3. Bryant R.L., Notes on projective, contact and null curves, arXiv:1905.06117.
4. del Amor J., Giménez A., Lucas P., Hamiltonian structure for null curve evolution, Nonlinearity 27 (2014), 2627-2641.
5. Korteweg D.J., de Vries G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 39 (1895), 422-443.
6. Musso E., Nicolodi L., Hamiltonian flows on null curves, Nonlinearity 23 (2010), 2117-2129, arXiv:0911.4467.
7. Musso E., Nicolodi L., Conformal geometry of isotropic curves in the complex quadric, Internat. J. Math. 33 (2022), 2250054, 32 pages, arXiv:2110.02838.
8. Musso E., Pámpano Á., Integrable flows on null curves in the anti-de Sitter 3-space, Nonlinearity 37 (2024), 115015, 38 pages, arXiv:2311.11137.
9. Pinkall U., Hamiltonian flows on the space of star-shaped curves, Results Math. 27 (1995), 328-332.
10. Tabachnikov S., On centro-affine curves and Bäcklund transformations of the KdV equation, Arnold Math. J. 4 (2018), 445-458, arXiv:1808.08454.
11. Terng C.-L., Uhlenbeck K., Bäcklund transformations and loop group actions, 3.3.CO;2-L">Comm. Pure Appl. Math. 53 (2000), 1-75, arXiv:math.DG/9805074.
12. Terng C.-L., Wu Z., Central affine curve flow on the plane, J. Fixed Point Theory Appl. 14 (2013), 375-396, arXiv:1405.4046.
13. Terng C.-L., Wu Z., Darboux transforms for the $\hat B_n^{(1)}$-hierarchy, J. Geom. Anal. 31 (2021), 4721-4753, arXiv:1912.07046.
14. Wahlquist H.D., Estabrook F.B., Bäcklund transformation for solutions of the Korteweg-de Vries equation, Phys. Rev. Lett. 31 (1973), 1386-1390.
15. Zakharov V.E., Faddeev L.D., Korteweg-de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl. 5 (1971), 280-287.