10.30495/ijm2c.2023.1982280.1271

Investigating the New Conservation Laws of Hunter-Saxton Equation via Lie Symmetries

PDF
  • Mehdi Jafari*1,
  • Somayeh Sadat Mahdion1,
  1. Department of Mathematics, Payame Noor University, PO BOX 19395-4697, Tehran, Iran

Received: 02-03-2023

Accepted: 15-06-2023

Published in Issue 01-06-2023

Copyright (c) 2024 International Journal of Mathematical Modeling & Computations

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Jafari, M., & Mahdion, S. S. (2023). Investigating the New Conservation Laws of Hunter-Saxton Equation via Lie Symmetries. International Journal of Mathematical Modelling & Computations, 13(2), 0-0. https://doi.org/10.30495/ijm2c.2023.1982280.1271
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

‎In this research‎, ‎using the multiplier method and the 2-dimensional‎ homotopy operator‎, ‎higher order conservation laws for the‎ ‎Hunter-Saxton equation are computed‎. ‎Also‎, ‎in order to construct new‎ ‎conservation laws‎, ‎the invariance properties of the multipliers are‎ ‎studied using Lie classical symmetries‎.

Keywords

  • Lie Symmetries,
  • ‎ Conversation laws,
  • ‎ Multiplier‎ method‎,
  • Homotopy operator,
  • ‎ Hunter-Saxton equation
PDF

References

  1. R. Abedian, The comparison of two high-order semi-discrete central schemes for solving hyperbolic
  2. conservation laws, International Journal of Mathematical Modelling & Computations, 7 (1) (2017)
  3. –54.
  4. S. C. Anco and G. W. Bluman, Direct construction method for conservation laws of partial differential
  5. equations, Part II: General treatment, Eur. J. Appl. Math., 13 (2002) 567–585.
  6. Y. AryaNejad and R. Mirzavand, Conservation laws and invariant solutions of time-dependent
  7. Calogero-Bogoyavlenskii-Schiff equation, Journal of Linear and Topological Algebra, 11 (04) (2022)
  8. –252.
  9. G. W. Bluman, New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl., 322 (2006) 233–250.
  10. A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal.,
  11. (3) (2005) 996–1026.
  12. R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev.
  13. Lett., 71 (11) (1993) 1661–1664.
  14. Y. Hou, E. G. Fan and P. Zhao, Algebro-geometric solutions for the Hunter-Saxton hierarchy,
  15. Zeitschrift fur angewandte Mathematik und Physik, 65 (3) (2014) 487–520.
  16. J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math, 51 (1991) 1498–1521.
  17. J. K. Hunter and Y. X. Zheng, On a completely integrable nonlinear hyperbolic variational equation,
  18. Phys. D., 79 (2-4) (1994) 361–386.
  19. R. I. Ivanov, Algebraic discretization of the Camassa-Holm and Hunter-Saxton equations, J. Nonlinear
  20. Math. Phys., 15 (2) (2008) 1–12.
  21. M. Jafari, S. Mahdion, A. Akg¨ul and S. M. Eldin, New conservation laws of the Boussinesq and
  22. generalized Kadomtsev-Petviashvili equations via homotopy operator, Results in Physics, 47 (2023)
  23. M. Jafari and A. Tanhaeivash, Symmetry group analysis and similarity reductions of the thin film
  24. equation, Journal of Linear and Topological Algebra, 10 (04) (2021) 287–293.
  25. M. Jafari, A. Zaeim and S. Mahdion, Scaling symmetry and a new conservation law of the Harry
  26. Dym equation, Mathematics Interdisciplinary Research, 6 (2) (2021) 151–158.
  27. M. Jafari, A. Zaeim and A. Tanhaeivash, Symmetry group analysis and conservation laws of the potential modified KdV equation using the scaling method, International Journal of Geometric Methods
  28. in Modern Physics, 19 (07) (2022) 2250098.
  29. M. Nadjafikhah and F. Ahangari, Symmetry analysis and conservation laws for the Hunter-Saxton
  30. equation, Communications in Theoretical Physics, 59 (3) (2013) 335.
  31. M. Nadjafikhah, M. Jafari, Computation of partially invariant solutions for the Einstein Walker
  32. manifolds identifying equations, Communications in Nonlinear Science and Numerical Simulation, 18
  33. (12) (2013) 3317–3324.
  34. M. Nadjafikhah and M. Jafari, Symmetry reduction of the twodimensional Ricci flow equation, Geometry, 2013 (1) (2013) 373701.
  35. P. J. Olver, Applications of Lie Groups to Differential Equations, In: Graduate Texts in Mathematics,
  36. , Springer, New York, (1993).
  37. D. Poole and W. Hereman, The homotopy operator method for symbolic integration by parts and
  38. inversion of divergences with applications, Appl. Anal., 87 (2010) 433–455.
  39. K. Tian and Q. P. Liu, Conservation laws and symmetries of Hunter-Saxton equation: revisited,
  40. Nonlinearity, 29 (3) (2016) 737.
  41. H. Yazdani, M. Nadjafikhah and M. Toomanian, Apply new optimized MRA & invariant solutions on
  42. the generalized-FKPP equation, International Journal of Mathematical Modelling & Computations,
  43. (4) (2021) 1–12.
  44. Z. Zhao, Conservation laws and nonlocally related systems of the Hunter-Saxton equation for liquid

Most read articles by the same author(s)