Skip to main content
SpringerLink
Log in

On q-Gaussian Integrable Hamiltonian Reductions in Anisentropic Magneto-gasdynamics

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Integrable substructure in 2+1-dimensional anisentropic magneto-gasdynamics is investigated via a general elliptic vortex ansatz. The procedure involves introduction of a q-Gaussian density representation. Thermodynamically consistent relations are isolated associated with certain integrable Hamiltonian reductions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Neukirch, T.: Quasi-equilibria: a special class of time-dependent solutions for two-dimensional magnetohydrodynamics. Phys. Plasmas 2, 4389–4399 (1995)

    Article  MathSciNet  Google Scholar 

  2. Neukirch, T., Priest, E.R.: Generalization of a special class of time-dependent solutions of the two-dimensional magnetohydrodynamic equations to arbitrary pressure profiles. Phys. Plasmas 7, 3105–3107 (2000)

    Article  MathSciNet  Google Scholar 

  3. Neukirch, T., Cheung, D.L.: A class of accelerated solutions of the two-dimensional ideal magnetohydrodynamic equations. Proc. R. Soc. Lond. A 457, 2547–2566 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Rogers, C.: Elliptic warm core theory: the pulsrodon. Phys. Lett. A 138, 267–273 (1989)

    Article  MathSciNet  Google Scholar 

  5. Cushman-Roisin, B., Heil, W.H., Nov, D.: Oscilliation and rotations of elliptical warm core rings. J. Geophys. Res. 90, 11756–11764 (1985)

    Article  Google Scholar 

  6. Csanady, G.T.: The birth and death of a warm core ring. J. Geophys. Res. 84, 777–780 (1979)

    Article  Google Scholar 

  7. Joyce, T.M., et al.: Rapid evolution of a Gulf stream warm core ring. Nature 308, 837–840 (1984)

    Article  Google Scholar 

  8. Joyce, T.M.: Velocity and hydrographic structure of a Gulf stream warm-core ring. J. Phys. Oceanogr. 14, 936–947 (1984)

    Article  Google Scholar 

  9. Ball, F.: Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. J. Fluid Mech. 17, 240–256 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  10. Holm, D.: Elliptical vortices and integrable Hamiltonian dynamics of the rotating shallow water equations. J. Fluid Mech. 227, 393–406 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  11. Rubino, A., Brandt, P.: Warm-core eddies studied by laboratory experiments and numerical modelling. J. Phys. Oceanogr. 33, 431–435 (2003)

    Article  MathSciNet  Google Scholar 

  12. Rogers, C., An, H.: Ermakov-Ray-Reid systems in 2+1-dimensional rotating shallow water theory. Stud. Appl. Math. 125, 275–299 (2010)

    MATH  MathSciNet  Google Scholar 

  13. Rogers, C.: Generation of invariance theorems for nonlinear boundary-value problems in shallow water theory: an application of MACSYMA. In: Numerical and Applied Mathematics, IMACS Meeting Proceedings, Paris, pp. 69–74 (1989)

    Google Scholar 

  14. Levi, D., Nucci, M.C., Rogers, C., Winternitz, P.: Group theoretical analysis of rotating shallow liquid in a rigid container. J. Phys. A, Math. Gen. 22, 4743–4767 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ray, J.R.: Nonlinear superposition law for generalised Ermakov systems. Phys. Lett. A 78, 4–6 (1980)

    Article  MathSciNet  Google Scholar 

  16. Reid, J.L., Ray, J.R.: Ermakov systems, nonlinear superposition, and solutions of nonlinear equations of motion. J. Math. Phys. 21, 1583–1587 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rogers, C., Hoenselaers, C., Ray, J.R.: On 2+1-dimensional Ermakov systems. J. Phys. A, Math. Gen. 26, 2625–2633 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Rogers, C., Schief, W.K.: Multi-component Ermakov systems: structure and linearization. J. Math. Anal. Appl. 198, 194–220 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schief, W.K., Rogers, C., Bassom, A.: Ermakov systems of arbitrary order and dimension. Structure and linearization. J. Phys. A, Math. Gen. 29, 903–911 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rogers, C., Malomed, B., Chow, K.W., An, H.: Ermakov-Ray-Reid systems in nonlinear optics. J. Phys. A, Math. Theor. 43, 455214 (2010). (15 pp.)

    Article  MathSciNet  Google Scholar 

  21. Rogers, C., Malomed, B., An, H.: Ermakov-Ray-Reid reductions of variational approximations in nonlinear optics. Stud. Appl. Math. 129, 389–413 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rogers, C., Schief, W.K.: The pulsrodon in 2+1-dimensional magnetogasdynamics. Hamiltonian structure and integrability. J. Math. Phys. 52, 083701 (2011) (20 pp.)

    Article  MathSciNet  Google Scholar 

  23. Rogers, C., Schief, W.K.: On the integrability of a Hamiltonian reduction of a 2+1-dimensional non-isothermal rotating gas cloud system. Nonlinearity 24, 3165–3178 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ovsiannikov, L.V.: New solutions of equations of hydrodynamics. Dokl. Akad. Nauk 111, 47–49 (1956)

    Google Scholar 

  25. Dyson, F.J.: Dynamics of a spinning gas cloud. J. Math. Mech. 18, 91–101 (1968)

    MATH  Google Scholar 

  26. Gell-Mann, M., Tsallis, C. (eds.): Nonextensive Entropy: Interdisciplinary Applications. Oxford University Press, Oxford (2004)

    Google Scholar 

  27. Tsallis, C.: Introduction to Non-extensive Statistical Mechanics—Approaching a Complex World. Springer, New York (2009)

    Google Scholar 

  28. Naudts, J.: Generalised Thermostatistics. Springer, New York (2011)

    Book  MATH  Google Scholar 

  29. Gaffet, B.: Spinning gas clouds with precession: a new formulation. J. Phys. A, Math. Theor. 43, 165207 (2010)

    Article  MathSciNet  Google Scholar 

  30. Schief, W.K., An, H., Rogers, C.: Universal and integrable aspects of an elliptic vortex representation in 2+1-dimensional magnetogasdynamics. Stud. Appl. Math. 130, 49–79 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. Steen, A.: Om Formen for Integralet af den lineaere Differentialligning af anden Orden. Overs. over d. K. Danske Vidensk. Selsk. Forh 1–12 (1874)

  32. Ermakov, V.P.: Second-order differential equations. Conditions for complete integrability. Univ. Izv. Kiev 20, 1–25 (1880)

    Google Scholar 

  33. Pinney, E.: The nonlinear differential equation y″(x)+p(x)y+cy −3=0. Proc. Am. Math. Soc. 1, 681 (1950)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia

    C. Rogers & W. K. Schief

Authors
  1. C. Rogers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. K. Schief
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Rogers.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rogers, C., Schief, W.K. On q-Gaussian Integrable Hamiltonian Reductions in Anisentropic Magneto-gasdynamics. Acta Appl Math 132, 515–525 (2014). https://doi.org/10.1007/s10440-014-9926-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-014-9926-8

Keywords

Search

Navigation