Geometry of solutions of the geometric curve flows in space

Authors

Zehui Zhao School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 431000, P.R. China
Shiping Zhong School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 431000, P.R. China
Xinjie Wan School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 431000, P.R. China

https://doi.org/10.61383/ejam.20231340

Keywords:

the Hasimoto surface, the shortening trajectory surface, the minimal trajectory surface, the $\sqrt{\tau}$-normal trajectory surface

Abstract

In this study, we aim at investigating the geometry of surfaces corresponding to the geometry of solutions of the geometric curve flows in Euclidean 3-space $\mathbb R^3$ considering the Frenet frame. In particular, we express some geometric properties and some characterizations of $u$-parameter curves and $t$-parameter curves of some trajectory surfaces including the Hasimoto surface, the shortening trajectory surface, the minimal trajectory surface, the $\sqrt{\tau}$-normal trajectory surface in $\mathbb R^3$.

References

N.H. Abdel-All, R.A. Hussien, and T. Youssef, Hasimoto surfaces, Life Science Journal 9 (2012), no. 3, 556--560.

M.E. Aydin, A. Mihai, A.O. Ogrenmis, and M. Ergut, Geometry of the solutions of localized induction equation in the pseudo-Galilean space, Advances in Mathematical Physics 2015 (2015), no. Article ID 905978, 1--7. DOI: https://doi.org/10.1155/2015/905978

Q. Ding and Y.D. Wang, Geometric KdV flows, motions of curves and the third order system of the AKNS hierarchy, International Journal of Mathematics 22 (2011), no. 7, 1013--1029. DOI: https://doi.org/10.1142/S0129167X11007100

Q. Ding, W. Wang, and Y.D. Wang, The Fukumoto-Moffatt's model in the vortex filament and generalized bi-Schr"odinger maps, Physics Letters A 375 (2011), no. 12. DOI: https://doi.org/10.1016/j.physleta.2011.02.035

M. Erdogdu and M "Ozdemir, Geometry of Hasimoto surfaces in Minkowski 3-space, Mathematical Physics Analysis and Geometry 17 (2014), 169--181. DOI: https://doi.org/10.1007/s11040-014-9148-3

M. Elzawy, Hasimoto surfaces in Galilean space $G_3$, Journal of the Egyptian Mathematical Society 29 (2021), no. Article ID 5, 1--9. DOI: https://doi.org/10.1186/s42787-021-00113-y

M. Erdogdu and A. Yavuz, Differential geometric aspects of nonlinear Schr"odinger equation, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 (2021), no. 1, 510--521. DOI: https://doi.org/10.31801/cfsuasmas.724634

K. Eren, New representation of Hasimoto surfaces with the modified orthogonal frame, Konuralp Journal of Mathematics 10 (2022), no. 1, 69--72.

N. G"urb"uz, Hasimoto surfaces according to three classes of curve evolution with Darboux frame in Euclidean space, Gece Publishing (2018).

H. Hasimoto, A soliton on a vortex filament, Journal of Fluid Mechanics 51 (1972), no. 3, 477--485. DOI: https://doi.org/10.1017/S0022112072002307

E. Hamouda, O. Moaaz, C. Cesarano, S. Askar, and A. Elsharkawy, Geometry of solutions of the quasi-vortex filament equation in Euclidean 3-space $E^3$, Mathematics 10 (2022), no. 6, 1--11. DOI: https://doi.org/10.3390/math10060891

T.A. Ivey, Helices, Hasimoto surfaces and B"acklund transformations, Canadian Mathematical Bulletin 43 (2000), no. 4, 427–439. DOI: https://doi.org/10.4153/CMB-2000-051-9

A. Kelleci, M. Bektas, and M. Erg"ut, The Hasimoto surface according to bishop frame, Adiyaman "Universitesi Fen Bilimleri Dergisi 9 (2019), no. 1, 13–22.

S.G. Mohamed, Binormal motions of inextensible curves in de-sitter space $S^2,1$, Journal of the Egyptian Mathematical Society 25 (2017), no. 3, 313--318. DOI: https://doi.org/10.1016/j.joems.2017.04.002

N. G"urb"uz and D.W. Yoon, Hasimoto surfaces for two classes of curve evolution in Minkowski 3-space, Demonstratio Mathematica 53 (2020), no. 1, 277–284. DOI: https://doi.org/10.1515/dema-2020-0019

N.E. G"urb"uz, R. Myrzakulov, and Z. Myrzakulova, Three anholonomy densities for three formulations with anholonomic coordinates with hybrid frame in $R_1^3$, Optik-International Journal for Light and Electron Optics 261 (2022), no. Article ID 169161, 1--17. DOI: https://doi.org/10.1016/j.ijleo.2022.169161

F. Mart'in, A. Savas-Halilaj, and K. Smoczyk, On the topology of translating solitons of the mean curvature flow, Calculus of Variations and Partial Differential Equations 54 (2015), no. 3, 2853--2882. DOI: https://doi.org/10.1007/s00526-015-0886-2

J. Minarvc'ik and M. Benev s, Nondegenerate homotopy and geometric flows, Homology, Homotopy and Applications 24 (2022), no. 2, 255--264. DOI: https://doi.org/10.4310/HHA.2022.v24.n2.a12

R.L. Ricca, Rediscovery of Da Rios equations, Nature 352 (2022), 561--562. DOI: https://doi.org/10.1038/352561a0

C. Yu, H. Schumacher, and K. Crane, Repulsive curves, ACM Transactions on Graphics 40 (2021), no. 2, 1--21. DOI: https://doi.org/10.1145/3439429

R.A. Hussien and S.G. Mohamed, Generated surfaces via inextensible flows of curves in $R^3$, Journal of Applied Mathematics 2016 (2016), no. Article ID 6178961, 1--8. DOI: https://doi.org/10.1155/2016/6178961

M. Bergou, M. Wardetzky, S. Robinson, B. Audoly, and E. Grinspun, Discrete elastic rods, ACM Transactions on Graphics 27 (2008), no. 3, 1--12. DOI: https://doi.org/10.1145/1360612.1360662

J. Minarvc'ik and M. Benev s, Long-term behavior of curve shortening flow in $R^3$, SIAM Journal on Mathematical Analysis 52 (2020), no. 2, 1221--1231. DOI: https://doi.org/10.1137/19M1248522

S.J. Altschuler, Singularities of the curve shrinking flow for space curves, Journal of Differential Geometry 34 (1991), no. 2, 491--514. DOI: https://doi.org/10.4310/jdg/1214447218

S.J. Altschuler and M. Grayson, Shortening space curves and flow through singularities, Journal of Differential Geometry 35 (1992), no. 2, 283--298. DOI: https://doi.org/10.4310/jdg/1214448076

J. Minarvc'ik and M. Benev s, Minimal surface generating flow for space curves of non vanishing torsion, Discrete and Continuous Dynamical Systems Series B 27 (2022), no. 11, 6605--6617. DOI: https://doi.org/10.3934/dcdsb.2022011

M. Benev s, Miroslav Kolav r, and Daniel v sevv coviv c, Qualitative and numerical aspects of a motion of a family of interacting curves in space, SIAM Journal on Applied Mathematics 82 (2022), no. 2, 549--575. DOI: https://doi.org/10.1137/21M1417181

A.T. Ali, H.S.A. Aziz, and A.H. Sorour, Ruled surfaces generated by some special curves in Euclidean 3-space, Journal of the Egyptian Mathematical Society 21 (2013), no. 3, 285--294. DOI: https://doi.org/10.1016/j.joems.2013.02.004

Y. Fukumoto and T. Miyazaki, Three-dimensional distortions of a vortex filament with axial velocity, Journal of Fluid Mechanics 222 (1991), 369--416. DOI: https://doi.org/10.1017/S0022112091001143

Y. Fukumoto and H.K. Moffatt, Motion and expansion of a visous vortex ring, I. a higer-order asymptotic formula for the velocity, Journal of Fluid Mechanics 417 (2000), 1--45. DOI: https://doi.org/10.1017/S0022112000008995

Z.H. Zhao, S.P. Zhong, and X.J. Wan, Geometry of solutions of the geometric curve flows in space, Authorea (2022). DOI: https://doi.org/10.22541/au.169945438.82526877/v1

Downloads

Published

2023 Oct 29

How to Cite

[1]

Z. Zhao, S. Zhong, and X. Wan, “Geometry of solutions of the geometric curve flows in space”, Electron. J. Appl. Math., vol. 1, no. 3, pp. 16–25, Oct. 2023.