Integrable dynamics and compatibility conditions in geometric curve flows

Sameh Shenawy; Bang-Yen Chen; Elsayed I. Mahmoud; Nasser Bin Turki; Abdallah A. Syied; Sameh Shenawy; Bang-Yen Chen; Elsayed I. Mahmoud; Nasser Bin Turki; Abdallah A. Syied

doi:10.3934/math.20261020

AIMS Mathematics

2026, Volume 11, Issue 8: 25404-25423. doi: 10.3934/math.20261020

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Research article

Integrable dynamics and compatibility conditions in geometric curve flows

1.
Basic Science Department, Modern Academy for Engineering and Technology, Maadi 4411602, Egypt
2.
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
3.
Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt
4.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
5.
Faculty of Computers and Information Systems, Egyptian Chinese University, Cairo 4541312, Egypt

Received: 14 June 2026 Revised: 25 July 2026 Accepted: 05 August 2026 Published: 17 August 2026
MSC : 53C25, 53C50, 53C80, 53B20

This paper studies inextensible flows of planar and spatial curves by means of the Frenet frame. The velocity field is decomposed into its tangential, normal, and binormal components, so that the metric variation and the evolution of the intrinsic invariants can be computed explicitly. For planar curves, the tangent, normal, and curvature equations are recorded together with a metric-preserving condition, which relates the tangential velocity to the curvature-weighted normal velocity. For space curves in $ \mathbb{R}^{3} $, the Frenet frame evolution and the coupled curvature-torsion system are derived, and the compatibility of the Frenet and temporal matrices is obtained in a general form retaining the metric-variation term; the zero-curvature equation then appears as the inextensible reduction rather than as an initial assumption. Several special motions are examined, including purely tangential and purely binormal flows, tangent-binormal constant motion, and the curvature-dependent flow $ r_{t} = \frac{1}{2}\kappa^{2}T+\kappa_{s}N+\kappa B $. Explicit solutions of the curvature-torsion system are obtained in closed form, including a travelling wave along which curvature and torsion are linearly related, and a localized travelling-wave curvature profile with constant torsion. The results give a unified geometric formulation for inextensible curve motion and clarify the roles of the velocity components in metric preservation, bending, twisting, and integrability.
- evolution of curves,
- inextendable curves
Citation: Sameh Shenawy, Bang-Yen Chen, Elsayed I. Mahmoud, Nasser Bin Turki, Abdallah A. Syied. Integrable dynamics and compatibility conditions in geometric curve flows[J]. AIMS Mathematics, 2026, 11(8): 25404-25423. doi: 10.3934/math.20261020

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Abstract

This paper studies inextensible flows of planar and spatial curves by means of the Frenet frame. The velocity field is decomposed into its tangential, normal, and binormal components, so that the metric variation and the evolution of the intrinsic invariants can be computed explicitly. For planar curves, the tangent, normal, and curvature equations are recorded together with a metric-preserving condition, which relates the tangential velocity to the curvature-weighted normal velocity. For space curves in $ \mathbb{R}^{3} $, the Frenet frame evolution and the coupled curvature-torsion system are derived, and the compatibility of the Frenet and temporal matrices is obtained in a general form retaining the metric-variation term; the zero-curvature equation then appears as the inextensible reduction rather than as an initial assumption. Several special motions are examined, including purely tangential and purely binormal flows, tangent-binormal constant motion, and the curvature-dependent flow $ r_{t} = \frac{1}{2}\kappa^{2}T+\kappa_{s}N+\kappa B $. Explicit solutions of the curvature-torsion system are obtained in closed form, including a travelling wave along which curvature and torsion are linearly related, and a localized travelling-wave curvature profile with constant torsion. The results give a unified geometric formulation for inextensible curve motion and clarify the roles of the velocity components in metric preservation, bending, twisting, and integrability.

References

[1]	H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477–485. https://doi.org/10.1017/S0022112072002307 doi: 10.1017/S0022112072002307
[2]	G. L. Lamb Jr., Solitons on moving space curves, J. Math. Phys., 18 (1977), 1654–1661. https://doi.org/10.1063/1.523453 doi: 10.1063/1.523453
[3]	H. Öztekin, H. Gün Bozok, Inextensible flows of curves in 4-dimensional Galilean space $G_{4}$, Math. Sci. Appl. E-Notes, 1 (2013), 28–34.
[4]	D. Ševčovič, S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Math. Meth. Appl. Sci., 35 (2012), 1784–1798. https://doi.org/10.1002/mma.2554 doi: 10.1002/mma.2554
[5]	A. T. Yakut, A. Kızılay, On the curve evolution with a new modified orthogonal Sabban frame, AIMS Mathematics, 9 (2024), 29573–29586. https://doi.org/10.3934/math.20241432 doi: 10.3934/math.20241432
[6]	T. Körpınar, S. Baş, M. Yeneroğlu, R. C. Demirkol, On characterization of inextensible flows with modified orthogonal frame, Proceedings of the 16th International Geometry Symposium, 2018,254–254.
[7]	H. Gün Bozok, A. Elsharkawy, Evolution of curves with modified orthogonal frame in $G^{3}$, Int. J. Geom. Methods M., 23 (2026), 2550311. https://doi.org/10.1142/S0219887825503116 doi: 10.1142/S0219887825503116
[8]	S. Gaber, A. H. Sorour, The geometry of the inextensible flows of timelike curves according to the quasi-frame in Minkowski space $\mathbb{R}^{2, 1}$, Symmetry, 15 (2023), 654. https://doi.org/10.3390/sym15030654 doi: 10.3390/sym15030654
[9]	M. Turan, H. Gün Bozok, M. Ergüt, Inextensible flows of curves with quasi-frame in $G^{3}$, International Journal of Maps in Mathematics, 8 (2025), 150–159.
[10]	A. Kızılay, A. T. Yakut, Inextensible flows of space curves according to a new orthogonal frame with curvature in $\mathbb{E}_{1}^{3}$, Int. Electron. J. Geom., 16 (2023), 577–593. https://doi.org/10.36890/iejg.1274663 doi: 10.36890/iejg.1274663
[11]	M. Gürses, A. Pekcan, Moving null curves and integrability, Math. Meth. Appl. Sci., 49 (2026), 6129–6142. https://doi.org/10.1002/mma.70381 doi: 10.1002/mma.70381
[12]	N. H. Abdel-All, S. G. Mohamed, M. T. Al-Dossary, Evolution of generalized space curve as a function of its local geometry, Appl. Math., 5 (2014), 2381–2392. https://doi.org/10.4236/am.2014.515230 doi: 10.4236/am.2014.515230
[13]	H. Liu, S. Huang, Y. Gai, Robot continuous trajectory planning based on Frenet-Serret formulas, Proceedings of the 2010 International Conference on Computer, Mechatronics, Control and Electronic Engineering, 2010,258–261. https://doi.org/10.1109/CMCE.2010.5610238 doi: 10.1109/CMCE.2010.5610238
[14]	J. Inoguchi, Plane curves and solitons (Japanese), Tokyo: Asakura Publishing, 2010.
[15]	K. Kajiwara, Lecture notes on curves and solitons (Japanese), Tokyo: Rikkyo University Center for Theoretical Physics and Mathematics, 2013.
[16]	R. Mukherjee, R. Balakrishnan, Moving curves of the sine-Gordon equation: new links, Phys. Lett. A, 372 (2008), 6347–6362. https://doi.org/10.1016/j.physleta.2008.08.070 doi: 10.1016/j.physleta.2008.08.070
[17]	K. Nakayama, M. Wadati, Motion of curves in the plane, J. Phys. Soc. Jpn., 62 (1993), 473–479. https://doi.org/10.1143/JPSJ.62.473 doi: 10.1143/JPSJ.62.473

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