Higher-Order Nonlinear Dynamical Systems and Invariant Lagrangians on a Lie Group: The Case of Nonlocal Hunter–Saxton Type Peakons

• Rami Ahmad El-Nabulsi ^[1] ; Waranont Anukool ^[1]
  1. [1] Chiang Mai University

     Chiang Mai University

     Tailandia

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 4, 2024
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • A G-strand is an evolutionary map g(t, s) : R×R → G into a Lie group G that follows from the Hamilton’s principle for a certain class of G-invariant Lagrangians defined on the Lie algebra g of the group G. t and s are independent variables associated to a Ginvariant Lagrangian. The G-strand equations comprises a system of integrable partial differential equations obtained from the Euler–Poincaré variational equations coupled to an auxiliary zero curvature equation. Some of these integrable partial differential equations include the Hunter–Saxton equation that arises in the study of nematic liquid crystals and the Camassa–Holm equation that arises in modeling waves in shallow water including solitons and peakons. However, nonlocal integrable systems have attracted significant attention in recent years. In this study, we use a higher-order nonlocal operator approach to study nonlocal Hunter–Saxton type peakons. Peakonsantipeakons collision on Lie group is also analyzed and discussed. It was observed that the system of "two-peakon" collisions exhibits a kind of disordered behavior which is observed in various integrable and non-integrable nonlinear evolution dynamical systems.

• Referencias bibliográficas
  • 1. Guha, P., Olver, P.J.: Geodesic flow and two (super) component analog of the Camassa–Holm equation. SIGMA 2, 054 (2006)
  • 2. Holm, D.D., Ivanov, R.I., Percival, J.R.: G-strands. J. Nonlinear Sci. 22, 517–551 (2012)
  • 3. Arnaudon, A., Holm, D.D., Ivanov, R.I.: G-strands on symmetric spaces. Proc. R. Soc. A473, 20160795 (2017)
  • 4. Chang, X., Szmigielski, J.: Lax integrability and the peakon problem for the modified Camassa–Holm equation. Commun. Math. Phys. 358, 295–341...
  • 5. Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal....
  • 6. Anco, S.C., Recio, E.: A general family of multi-peakon equations and their properties J. . Phys. A: Math. Theor. 52, 125203 (2019)
  • 7. Constantin, A., McKean, H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949–982 (1999)
  • 8. Zhang, G.: New periodic exact traveling wave solutions of Camassa–Holm equation. Part. Differ. Equ. Appl. Math. 6, 100426 (2022)
  • 9. Camassa, R., Holm, D., Hyman, J.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)
  • 10. Lenells, J.: The Hunter–Saxton equation describes the geodesic flow on a sphere. J. Geom. Phys. 57, 2049–2064 (2007)
  • 11. Hunter, J.K., Zheng, Y.: On a completely integrable nonlinear hyperbolic variational equation. Phys. D 79, 361–386 (1994)
  • 12. Khesin, B., Misiolek, G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176, 116–144 (2003)
  • 13. Khesin, B., Lenells, J., Misiolek, G.: Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms. Math....
  • 14. Lenells, J.: The Hunter–Saxton equation: a geometric approach. SIAM J. Math. Anal. 40, 266–277 (2008)
  • 15. Anco, S.C., Recio, E.: Accelerating dynamical peakons and their behavior. Dis. Cont. Dyn. Syst. A 39, 6131–6148 (2019)
  • 16. Anco, S.C., Kraus, D.: Hamiltonian structure of peakons as weak solutions for the modified Camassa–Holm equation. Dis. Cont. Dyn. Syst....
  • 17. Anco, S.C., Recio, E., Gandarias, M.L., Bruzon, M.S.: A nonlinear generalization of the Camassa–Holm equation with peakon solutions. Dynamical...
  • 18. Anco, S.C., da Silva, P.L., Freire, I.L.: A family of wave-breaking equations generalizing the Camassa–Holm and Novikov equations. J....
  • 19. Anco, S.C., Mobasheramini, F.: Integrable U(1)-invariant peak on equations from the NLS hierarchy. Phys. D 355, 1–23 (2017)
  • 20. McLachlan, R., Zhang, X.: Well-posedness of a modified Camassa–Holm equation. J. Differ. Equ. 246, 3241–3259 (2009)
  • 21. McLachlan, R., Zhang, X.: Asymptotic blowup profiles for modified Camassa–Holm equations. SIAM J. Appl. Dyn. Syst. 10, 452–468 (2011)
  • 22. Mu, C., Zhou, S., Zeng, R.: Well-posedness and blow-up phenomena for a higher order shallow water equation. J. Differ. Equ. 251, 3488–3499...
  • 23. Wang, F., Li, F., Qiao, Z.: Well-posedness and peakons for a higher-order µ-Camassa–Holm equation, arXiv:1712.07996
  • 24. Hayashi, M., Shigemoto, K., Tsukioka, T.: Elliptic solutions for higher-order KdV equations. J. Phys. Comm. 4, 045013 (2020)
  • 25. Beyer, H.R., Aksoylu, B., Celiker, F.: On a class of nonlocal wave equations from applications. J. Math. Phys. 57, 062902 (2016)
  • 26. El-Nabulsi, R.A., Anukool, W.: A family of nonlinear Schrodinger equations and their solitons solutions. Chaos Solitons Fractals 166,...
  • 27. El-Nabulsi, R.A., Anukool, W.: A generalized nonlinear cubic-quartic Schrodinger equation and its implications in quantum wire. Eur. Phys....
  • 28. El-Nabulsi, R.A.: Complex backward-forward derivative operator in non-local-in-time Lagrangians mechanics. Qual. Theor. Dyn. Sys. 16,...
  • 29. Sakkaravarthi, K., Singh, S., Karjanto, N.: Exploring the dynamics of nonlocal nonlinear waves: analytical insights into the extended...
  • 30. Wu, H.-Y., Fei, J.-X., Ma, Z.-Y., Chen, J.-C., Ma, M.-X.: Symmetry breaking soliton, breather, and lump solutions of a nonlocal Kadomtsev–Petviashvili...
  • 31. Lou, S.Y., Huang, F.: Alice–Bob physics: coherent solutions of nonlocal KdV systems. Sci. Rep. 7(1), 869–911 (2017)
  • 32. El-Nabulsi, R.A.: On nonlocal complex Maxwell equations and wave motion in electrodynamics and dielectric media. Opt. Quant. Elect. 50,...
  • 33. El-Nabulsi, R.A.: Non-standard higher-order G-strand partial differential equations on matrix Lie algebra. J. Niger. Math. Soc. 36, 101–112...
  • 34. Wang, X., Li, C.: Solitons, breathers and rogue waves in the coupled nonlocal reverse-time nonlinear Schrödinger equations. J. Geom. Phys....
  • 35. Yang, J.K.: General N-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations. Phys. Lett. A 383, 328–337 (2019)
  • 36. Ma, W.X.: Integrable nonlocal nonlinear Schrödinger equations associated with so(3, R). Proc. Amer. Math. Soc. B 9, 1–11 (2022)
  • 37. Zhang, X., Chen, Y., Zhang, Y.: Breather, lump and X soliton solutions to nonlinear KP equation. Comput. Appl. Math. 74, 2341–2347 (2017)
  • 38. Khare, A., Saxena, A.: Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations. J. Math. Phys. 56, 032104...
  • 39. Wen, X.Y., Yan, Z., Yang, Y.: Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced...
  • 40. Gay-Balmaz, F., Ratiu, T.S.: The geometric structure of complex fluids. Adv. Appl. Math. 42, 176–275 (2009)
  • 41. Gay-Balmaz, F., Holm, D., Ratiu, T.: Higher order Lagrange–Poincaré, and Hamilton–Poincaré, reductions. Bull. Braz. Math. Soc. 42, 579–606...
  • 42. Gay-Balmaz, F., Holm, D.D., Meier, D.M., Ratiu, T.S., Vialard, F.-X.: Invariant higher-order variational problems. Commun. Math. Phys....
  • 43. Gay-Balmaz, F., Holm, D.D., Meier, D.M., Ratiu, T.S., Vialard, F.-X.: Invariant higher-order variational problems II. J. Nonlinear Sci....
  • 44. Kenneth, S., Miller, S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience...
  • 45. Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079–1085 (1966)
  • 46. Suykens, J.A.K.: Extending Newton’s law from nonlocal-in-time kinetic energy. Phys. Lett. A 373, 1201–1211 (2009)
  • 47. El-Nabulsi, R.A.: On nonlocal fractal laminar steady and unsteady flows. Acta Mech. 232, 1413–1424 (2021)
  • 48. Li, Z.-Y., Fu, J.-L., Chen, L.-Q.: Euler–Lagrange equation from nonlocal-in-time kinetic energy of nonconservative system. Phys. Lett....
  • 49. El-Nabulsi, R.A.: Nonlocal tidal effects of the moon and numerical estimation of the secular drift rate for a GPS satellite. Adv. Space...
  • 50. El-Nabulsi, R.A.: Nonlinear wave equations from a non-local complex backward–forward derivative operator. Waves Rand. Complex Med. 31,...
  • 51. El-Nabulsi, R.A., Anukool, W.: Fermat’s principle and the effect of jerk on light motion near the Sun. Mod. Phys. Lett. A 36, 2150277...
  • 52. Holm, D.D., Kupershmidt, B.A.: Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity....
  • 53. Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler–Poincaré equations and semidirect products with applications to continuum theories....
  • 54. Holm, D.D.: Euler-Poincaré Dynamics of Perfect Complex Fluids. Springer, New York (2002)
  • 55. Holm, D.D., Ivanov, R.I.: Gstrands and peakon collisions on Diff(R). SIGMA 9, 027–041 (2013)
  • 56. Holm, D.D., Ivanov, R.I.: Euler–Poincaré equations for G-strands. J. Phys. Conf. Ser. 482, 012018 (2014)
  • 57. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied...
  • 58. Crampin, M.: Lagrangian submanifolds and the Euler–Lagrange equations in higher-order mechanics. Lett. Math. Phys. 19, 53–58 (1990)
  • 59. Arnold, V.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides...
  • 60. Gesztesy, F., Holden, H.: Algebro-geometric solutions of the Camassa Holm hierarchy. Rev. Mat. Iberoam. 19(1), 73–142 (2003)
  • 61. Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
  • 62. Hunter, J.K., Saxton, R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498–1521 (1991)
  • 63. Greenberg, M.D.: Applications of Green’s Functions in Science and Engineering. Prentice Hall, Englewood Cliffs (1971)
  • 64. Szmigielski, J., Zhou, L.: Colliding Peakons and the formation of shocks in the Degasperis-Procesi equation. Proc. R. Soc. Lond. Ser....
  • 65. Szmigielski, J., Zhou, L.: Peakon-antipeakon interactions in the Degasperis–Procesi equation. In: Algebraic and geometric Aspects of Integrable...
  • 66. Maccari, A.: Chaotic and fractal patterns for interacting nonlinear waves. Chaos Solitons Fractals 43, 86–95 (2010)
  • 67. Wunsch, M.: The generalized Hunter–Saxton system. SIAM J. Math. Anal. 42, 1286–1304 (2010)
  • 68. Wunsch, M.: On the Hunter–Saxton system. Disc. Cont. Dyn. Syst. B 12, 647–656 (2009)
  • 69. Morozov, O.I.: Integrability structures of the generalized Hunter–Saxton equation. Anal. Math. Phys. 11, 50 (2021)
  • 70. Cotter, C.J., Deasy, J., Pryer, T.: The r-Hunter–Saxton equation, smooth and singular solutions and their approximation. Nonlinearity...
  • 71. Peng, L., Li, J., Mei, M., Zhang, K.: Convergence rate of the vanishing viscosity limit for the Hunter–Saxton equation in the half space....
  • 72. del Mar Gonzalez, M., Ignat, L.I., Manea, D., Moroianu, S.: Concentration limit for non-local dissipative convection–diffusion kernels...
  • 73. Shi, Z., Li, Y.: Generalized nonlocal symmetries of two-component Camassa–Holm and Hunter–Saxton systems. Symmetry 14, 528 (2022)
  • 74. Bogdaanov, R.I.: Nonlocal integrals and conservation laws in the theory of nonlinear solitons. J. Math. Sci. 149, 1400–1416 (2008)
  • 75. Abdelouhab, L., Bona, J., Felland, M., Saut, J.C.: Nonlocal models for nonlinear, dispersive waves. Physica D 40, 360–392 (1989)
  • 76. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)
  • 77. Guha, P., Olver, P.: Geodesic flow and two (super) component analog of the Camassa–Holm equation. In: SIGMA2 (2006) 054
  • 78. Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139, 7–59 (2017)
  • 79. Cen, J., Carrea, F., Fring, A., Taira, T.: Stability in integrable nonlocal nonlinear equations. Phys. Lett. A 435, 128060 (2022)