On Noether's conservation laws of the Sine-Gordon equation using moving frames

Masoudi, Yousef; Nadjafikhah, Mehdi; Toomanian, Megerdich

doi:10.22075/ijnaa.2022.24650.2791

On Noether's conservation laws of the Sine-Gordon equation using moving frames

Document Type : Research Paper

Authors

Yousef Masoudi ¹

Mehdi Nadjafikhah ²

Megerdich Toomanian ³

¹ Department of Mathematics, Naghadeh Branch, Islamic Azad University, Naghadeh, Iran

² Department of Pure Mathematics, School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran

³ Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

10.22075/ijnaa.2022.24650.2791

Abstract

The wide significance of the problem of finding conservation laws in a great number of applications in mechanics and physics is beyond any doubt. The aim of this paper is to obtain conservation laws of the Sine-Gordon equation via the concept of moving frames and the variational principle. For this purpose, we first present a Lagrangian whose Euler-Lagrange equation is the Sine-Gordon equation, and then by Noether’s First Theorem and Mansfield’s method, we obtain the space of conservation laws in terms of invariants and the adjoint representation of a moving frame, for that Lagrangian, which is invariant under HRT group action.

Keywords

Sine-Gordon equation

moving frame

conservation laws

normalized differential invariants

syzygy

[1] M. Fels and P.J. Olver, Moving coframes: II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), no. 2, 127–208.
[2] T.M.N. Gon¸calves and E.L. Mansfield, Moving frames and conservation laws for Euclidean invariant Lagrangians, Stud. Appl. Math. 130 (2013), no. 2, 134–166.

[3] T. M. N. Gon¸calves and E. L. Mansfield, Moving Frames and Noether’s Conservation Laws the General Case, Vol. 4. Forum of Mathematics, Sigma. Cambridge University Press, 2016.
[4] T.M.N. Gon¸calves and E.L. Mansfield, On moving frames and Noether’s conservation laws, Stud. Appl. Math. 128 (2012), no. 1, 1–29.
[5] E.L. Mansfield, A practical guide to the invariant calculus, Vol. 26. Cambridge University Press, 2010.
[6] Y. Masoudi and M. Nadjafikhah, Moving frames and conservation laws of a Lagrangian invariant under the Hyperbolic Rotation-Translation group, Hokkaido Math. J. 47 (2018), no. 3, 557–579.
[7] E. Noether, Invariant variation problems, Transport Theory Statistic. Phys. 1 (1971), no. 3, 186–207.
[8] P.J. Olver, Applications of Lie groups to differential equations, Vol. 107. Springer Science and Business Media, 2000.