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Orbital dynamics satisfying the 4th-order stationary extended Fisher-Kolmogorov equation

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Abstract

In this study, we discuss the central force problem by using the nonlocal-in-time kinetic energy approach. At low length scales, the system is dominated by the generalized 4th-order extended Fisher-Kolmogorov stationary equation and by the 4th-order stationary Swift-Hohenberg differential equation under explicit conditions. The energy is a conserved quantity along orbits of the extended Fisher-Kolmogorov stationary equation. The system is quantized, the system is stable, and the ground energy problem is solved.

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References

  1. Bertrand, J. Théorème relatif du mouvement d’un point attire vers un centre fixe. C. R. Acad. Sci, 1873, 77: 849–853.

    MATH  Google Scholar 

  2. Fitzpatrick, R. An Introduction to Celestial Mechanics. Cambridge: Cambridge University Press, 2009.

    MATH  Google Scholar 

  3. Higgs, W. Dynamical symmetries in a spherical geometry. I. Journal of Physics A: Mathematical and General, 1979, 12(3): 309–323.

    MathSciNet  MATH  Google Scholar 

  4. Cooper, F., Khare, A., Sukhatme, U. Supersymmetry and quantum mechanics. Physics Reports, 1995, 251 (5–6): 267–385.

    MathSciNet  MATH  Google Scholar 

  5. Gurappa, N., Panigrahi, P. K., Raju, T. S., Srinivasan, V. Quantum equivalent of the bertrand’s theorem. Modern Physics Letters A, 2000, 15(30): 1851–1857.

    MathSciNet  MATH  Google Scholar 

  6. Suykens, J. A. K. Extending Newton’s law from nonlocal-in-time kinetic energy. Physics Letters A, 2009, 373(14): 1201–1211.

    MATH  Google Scholar 

  7. Feynman, R. P. Space-time approach to non-relativistic quantum mechanics. Reviews of Modern Physics, 1948, 20(2): 367.

    MathSciNet  MATH  Google Scholar 

  8. Moares, E. M. Time varying heat conduction in solids. In: Heat Conduction-Basic Research. INTECH, 2011.

  9. El-Nabulsi, R. A. Non-standard non-local-in-time lagrangians in classical mechanics. Qualitative Theory of Dynamical Systems, 2014, 13(1): 149–160.

    MathSciNet  MATH  Google Scholar 

  10. El-Nabulsi, R. A. Complex backward-forward derivative operator in non-local-in-time lagrangians mechanics. Qualitative Theory of Dynamical Systems, 2017, 16(2): 223–234.

    MathSciNet  MATH  Google Scholar 

  11. Li, Z. Y., Fu, J. L., Chen, L. Q. Euler-Lagrange equation from nonlocal-in-time kinetic energy of nonconservative system. Physics Letters A, 2009, 374(2): 106–109.

    MathSciNet  MATH  Google Scholar 

  12. Stecki, J. On the kinetic equation nonlocal in time for the generalized self-diffusion process. Journal of Computational Physics, 1971, 7(3): 547–553.

    MathSciNet  Google Scholar 

  13. Gomis, J., Kamimura, K., Llosa, J. Hamiltonian formalism for space-time noncommutative theories. Physical Review D, 2001, 63(4): 045003.

    MathSciNet  Google Scholar 

  14. El-Nabulsi, R. A. On nonlocal complexified Schrödinger equation and emergence of discrete quantum mechanics. Quantum Studies: Mathematics and Foundations, 2016, 3(4): 327–335.

    MathSciNet  MATH  Google Scholar 

  15. El-Nabulsi, R. A. Generalized Klein-gordon and Dirac equations from nonlocal kinetic approach. Zeitschrift Für Naturforschung A, 2016, 71(9): 817–821.

    Google Scholar 

  16. Nelson, E. Derivation of the Schrödinger equation from Newtonian mechanics. Physical Review, 1966, 150(4): 1079.

    Google Scholar 

  17. Gordeziani, D. G. On some initial conditions for parabolic equations. Reports of the Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, 1989, 4: 57–60.

    Google Scholar 

  18. Gordeziani, D. G. On one problem for the Navier-Stokes equation. Abstracts, Contin. Mech. Related Probl. Anal., Tbilisi, 1991: 83.

  19. Gordeziani, D. G. On solution of in time nonlocal problems for some equations of mathematical physics. ICM-94, Abstracts, Short Comm, 1994: 240.

  20. Gordeziani, D. G., Grigalashvili, Z. Non-local problems in time for some equations of mathematical physics. Dokl. Semin. Inst. Prikl. Mat. im. I. N. Vekua, 1993, 22: 108–114.

    MathSciNet  Google Scholar 

  21. Hu, H. P., Wu, M. X. Evidence of non-local physical, chemical and biological effects supports quantum brain. Neuro Quantology, 2007, 4(4): 291–306.

    Google Scholar 

  22. Cushman, J. H., Ginn, T. R. Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Transport in Porous Media, 1993, 13(1): 123–138.

    Google Scholar 

  23. Carmichael, H. An Open Systems Approach to Quantum Optics. Berlin, Heidelberg: Springer Berlin Heidelberg, 1993.

    MATH  Google Scholar 

  24. El-Nabulsi, R. A. Massive photons in magnetic materials from nonlocal quantization. Journal of Magnetism and Magnetic Materials, 2018, 458: 213–216.

    Google Scholar 

  25. El-Nabulsi, R. A. Nonlocal approach to nonequilibrium thermodynamics and nonlocal heat diffusion processes. Continuum Mechanics and Thermodynamics, 2018, 30(4): 889–915.

    MathSciNet  MATH  Google Scholar 

  26. El-Nabulsi, R. A. On nonlocal complex Maxwell equations and wave motion in electrodynamics and dielectric media. Optical and Quantum Electronics, 2018, 50(4): 170.

    Google Scholar 

  27. El-Nabulsi, R. A. Modeling of electrical and mesoscopic circuits at quantum nanoscale from heat momentum operator. Physica E: Low-Dimensional Systems and Nanostructures, 2018, 98: 90–104.

    Google Scholar 

  28. El-Nabulsi, R. A. Nonlocal approach to energy bands in periodic lattices and emergence of electron mass enhancement. Journal of Physics and Chemistry of Solids, 2018, 122: 167–173.

    Google Scholar 

  29. Kamalov, T. Classical and quantum-mechanical axioms with the higher time derivative formalism. Journal of Physics: Conference Series, 2013, 442: 012051.

    Google Scholar 

  30. Nottale, L. Fractal Space-time and Microphysics. World Scientific, 1993.

    MATH  Google Scholar 

  31. Perdang, M., Lejeune, A. Cellular automata. In: Cellular Automata: Prospects in Astrophysical Applications. World Scientific, 1993.

  32. Gelfand, I. M., Fomin, S. V. Calculus of Variations. Prentice-Hall, Inc., 1963.

    MATH  Google Scholar 

  33. Haider, M. Bertrand’s Theorem. Karlstads Universitet, 2013.

    Google Scholar 

  34. Kalies, W. D., Kwapisz, J., Vander Vorst, R. C. A. M. Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria. Communications in Mathematical Physics, 1998, 193(2): 337–371.

    MathSciNet  MATH  Google Scholar 

  35. Kalies, W. D., Vander Vorst, R. C. A. M. Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation. Journal of Differential Equations, 1996, 131(2): 209–228.

    MathSciNet  MATH  Google Scholar 

  36. Champneys, A. R. Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Physica D: Nonlinear Phenomena, 1998, 112(1–2): 158–186.

    MathSciNet  MATH  Google Scholar 

  37. Swift, J., Hohenberg, P. C. Hydrodynamic fluctuations at the convective instability. Physical Review A, 1977, 15(1): 319.

    Google Scholar 

  38. Van den Berg, G. J. B. Dynamics and equilibria of fourth order differential equations. Ph.D. Thesis. Leiden University, 2000.

    MATH  Google Scholar 

  39. Pais, A., Uhlenbeck, G. E. On field theories with non-localized action. Physical Review, 1950, 79(1): 145.

    MathSciNet  MATH  Google Scholar 

  40. Nesterenko, V. V. Instability of classical dynamics in theories with higher derivatives. Physical Review D, 2007, 75(8): 087703.

    MathSciNet  Google Scholar 

  41. Simon, J. Z. Higher-derivative Lagrangians, nonlocality, problems, and solutions. Physical Review D, 1990, 41(12): 3720.

    MathSciNet  Google Scholar 

  42. Simon, J. Z. Higher derivative expansions and non-locality. Ph.D. Thesis. University of California, Santa Barbara, 1990.

    Google Scholar 

  43. Gribov, V. N. The Theory of Complex Angular Momenta. Cambridge: Cambridge University Press, 2003.

    Google Scholar 

  44. Andersson, N., Thylwe, K. E. Complex angular momentum approach to black-hole scattering. Classical and Quantum Gravity, 1994, 11(12): 2991–3001.

    MathSciNet  Google Scholar 

  45. Regge, T. Introduction to complex orbital momenta. Il Nuovo Cimento, 1959, 14(5): 951–976.

    MathSciNet  MATH  Google Scholar 

  46. Boulware, D. G., Deser, S., Stelle, K. S. In Quantum Field Theory and Quantum Statistics: Essays in Honor of the Sixtieth Birthday of E S. Fradkin, edited by Batalin, I. A., Isham, C. J., Vilkovisky, C. A. Hilger, Bristol, England, 1987.

  47. Appell, P. Quelques remarques sur La théorie des potentiels multiformes. Mathematische Annalen, 1887, 30(1): 155–156.

    MathSciNet  MATH  Google Scholar 

  48. Carter, B. Hamilton-Jacobi and Schrodinger separable solutions of Einstein’s equations. Communications in Mathematical Physics, 1968, 10(4): 280–310.

    MathSciNet  MATH  Google Scholar 

  49. Ciotti, L., Bertin, G. A simple method to construct exact density-potential pairs from a homeoidal expansion. Astronomy & Astrophysics, 2005, 437(2): 419–427.

    MATH  Google Scholar 

  50. Venezia, F. The Complex Potential Technique in Stellar Dynamics. SCUOLA DI SCIENZE Corso di Laurea in Astrofisica e Cosmologia, Universita de Bologna, Sessione II-2, seduta Anno Accademico: 2016/2017.

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Acknowledgements

I would like to thank the group of anonymous referees for their useful comments and valuable suggestions.

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Authors and Affiliations

  1. Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, Kolonaki, 10671, Athens, Greece

    Rami Ahmad El-Nabulsi

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  1. Rami Ahmad El-Nabulsi
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Correspondence to Rami Ahmad El-Nabulsi.

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Rami Ahmad El-Nabulsi holds a Ph.D. degree in particle physics, mathematical physics and modeling from Provence University, France, and a diploma of advanced studies in plasma physics and radiation astrophysics from the same institution. He worked with different worldwide research departments in UK, Republic of Korea, China, Greece and he is actually affiliated to ATINER-Athens, Mathematics and Physics Departments. He is the author of more than 190 peer-reviewed papers in peer-refereed journals and a reviewer for more than 65 scientific journals where he has reviewed more than 750 manuscripts till the present moment. His research ranges from applied mathematics to theoretical physics including nonlinear dynamical systems, space physics, celestial dynamics, high-energy astrophysics and cosmology, geophysical flows, physics and chemistry of solids, magnetic materials, quantum mechanics, and physics at nanoscales among others.

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El-Nabulsi, R.A. Orbital dynamics satisfying the 4th-order stationary extended Fisher-Kolmogorov equation. Astrodyn 4, 31–39 (2020). https://doi.org/10.1007/s42064-019-0058-9

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