Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 017, 17 pages      math-ph/0602008

Applications of Symmetry Methods to the Theory of Plasma Physics

Giampaolo Cicogna a, Francesco Ceccherini b and Francesco Pegoraro b
a) Dip. di Fisica and INFN, Largo B. Pontecorvo 3, Ed. B-C, 56127 - Pisa, Italy
b) Dip. di Fisica, INFM and CNISM, Largo B. Pontecorvo 3, Ed. B-C, 56127 - Pisa, Italy

Received October 17, 2005, in final form January 20, 2006; Published online February 02, 2006

The theory of plasma physics offers a number of nontrivial examples of partial differential equations, which can be successfully treated with symmetry methods. We propose three different examples which may illustrate the reciprocal advantage of this "interaction" between plasma physics and symmetry techniques. The examples include, in particular, the complete symmetry analysis of system of two PDE's, with the determination of some conditional and partial symmetries, the construction of group-invariant solutions, and the symmetry classification of a nonlinear PDE.

Key words: Lie point symmetries; partial differential equations; plasma physics.

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  1. Ovsjannikov L.V., Group properties of differential equations, Novosibirsk, Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, 1962 (English transl.: Group analysis of differential equations, New York, Academic Press, 1982).
  2. Olver P.J., Application of Lie groups to differential equations, 2nd ed., Berlin, Springer, 1988.
  3. Stephani H., Differential equations. Their solution using symmetries, Cambridge, University Press, 1989.
  4. Bluman G.W., Kumei S., Symmetries and differential equations, Berlin, Springer, 1989.
  5. Gaeta G., Nonlinear symmetries and nonlinear equations, Dordrecht, Kluwer, 1994.
  6. Ibragimov N.H. (Editor), CRC Handbook of Lie group analysis of differential equations, Boca Raton, CRC Press, 1994, Vol. 1; 1995, Vol. 2; 1996, Vol. 3.
  7. Bluman G.W., Anco S.C., Symmetry and integration methods for differential equations, New York, Springer, 2002.
  8. Harrison B.K., The differential form method for finding symmetries, SIGMA, 2005, V.1, paper 001, 12 pages, math-ph/0510068.
  9. Ceccherini F., Cicogna G., Pegoraro F., Symmetry properties of a system of Euler-type equations for magnetized plasmas, J. Phys. A: Math. Gen., 2005, V.38, 4597-4610.
  10. Ceccherini F., Montagna C., Pegoraro F., Cicogna G., Two-dimensional Harris-Liouville plasma kinetic equilibria, Phys. Plasmas, 2005, V.12, 052506-1/8.
  11. Gusyatnikova V.N., Samokhin A.V., Titov V.S., Vinogradov A.M., Yamaguzhin V.A., Symmetries and conservation laws of Kadomtsev-Pogutse equations, Acta Appl. Math., 1989, V.15, 23-64.
  12. Ibragimov N.H., Elementary Lie group analysis and ordinary differential equations, Chichester, J. Wiley & Sons, 1999.
  13. Kiselev A.V., On the geometry of the Liouville equation: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math., 2002, V.72, 33-49.
  14. Fushchych W.I., Serov N.I., The symmetry and some exact solutions of the nonlinear many-dimensional Liouville, d'Alembert and eikonal equations, J. Phys. A: Math. Gen., 1983, V.16, 3545-3658.
  15. Pucci E., Salvatori M.C., Group properties of a class of semilinear hyperbolic equations, Internat. J. Non-Linear Mech., 1986, V.21, N 2, 147-155.
  16. Liouville J., Sur l'équation aux differences partielles 2logl/uvl/2a2=0, J. Math. Pure Appl., 1853, V.36, 71-72.
  17. Bateman H., Partial differential equations of mathematical physics, New York, Dover, 1944.
  18. Brito F., Leite M.L., Souza-Neto V., Liouville formula under the viewpoint of minimal surfaces, Comm. Pure Appl. Analysis, 2004, V.3, 41-51.
  19. Chen W., Li C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 1991, V.63, 615-622.
  20. Crowdy D.G., General solutions to the 2D Liouville equation, Int. J. Engn. Sci., 1997, V.35, N 2, 141-149.
  21. Anderson R.L., Ibragimov N.H., Lie-Bäcklund transformations in applications, Philadelphia, SIAM, 1979.
  22. Khater A.H., Callebaut D.K., El-Kalaawy O.H., Bäcklund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magnetostatic atmosphere, IMA J. Appl. Math., 2000, V. 65, N 1, 97-108.
  23. Champagne B., Hereman W., Winternitz P., The computer calculation of Lie point symmetries of large systems of differential equations, Comput. Phys. Comm., 1991, V.66, 319-340.
  24. Hereman W., Review of symbolic software for Lie symmetry analysis, Math. Comp. Modelling, 1997, V.25, 115-132 (see also [6 ,Vol. 3, Chapter 13]).
  25. Baumann G., Symmetry analysis of differential equations with Mathematica, Berlin, Springer, 2000.
  26. Samokhin A.V., Nonlinear M.H.D. equations: symmetries, solutions and conservation laws, Dokl. Akad. Nauk. SSSR, 1985, V.285, N 5, 1101-1106 (English transl.: Sov. Phys. Dokl., 1985, V.30, N 12, 1020-1022).
  27. Bluman G.W., Cole J.D., The general similarity solution of the heat equation, J. Math. Mech., 1969, V.18, 1025-1042.
  28. Bluman G.W., Cole J.D., Similarity methods for differential equations, Berlin, Springer, 1974.
  29. Fushchych W.I., Tsyfra I.M., On a reduction and solutions of the nonlinear wave equations with broken symmetry, J. Phys. A: Math. Gen., 1987, V.20, L45-L48.
  30. Levi D., Winternitz P., Non-classical symmetry reduction: example of the Boussinesq equation, J. Phys. A: Math. Gen., 1989, V.22, 2915-2924.
  31. Winternitz P., Lie groups and solutions of nonlinear partial differential equations, in Integrable Systems, Quantum Groups, and Quantum Field Theories, Proc. XXIII GIFT Intern. Seminar (1992, Salamanca), Editors L.A. Ibort and M.A. Rodríguez, NATO ASI Ser. C, V.409, Dordrecht, Kluwer, 1993, 429-495.
  32. Fushchych W.I. (Editor), Symmetry analysis of equations of mathematical physics, Kyiv, Institute of Mathematics of Acad. Sci. of Ukraine, 1992.
    Fushchych W.I., Conditional symmetries of the equations of mathematical physics, in Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, Proc. Intern. Workshop (1992, Acireale), Editors N.H. Ibragimov, M. Torrisi and A. Valenti, Dordrecht, Kluwer, 1993, 231-239.
  33. Cicogna G., Gaeta G., Partial Lie-point symmetries of differential equations, J. Phys. A: Math. Gen., 2001, V.34, N 3, 491-512.
  34. Cicogna G., Partial symmetries and symmetric sets of solutions to PDE's, in Symmetry and Perturbation Theory, Proc. 2002 SPT Conference (2002, Cala Gonone), Editors S. Abenda, G. Gaeta and S. Walcher, Singapore, World Scientific, 2002, 26-33.
    Cicogna G., Symmetric sets of solutions to differential problems, in Proceedinds of Fourth International Conference "Symmetry in Nonlinear Mathematical Physics" (July 9-15, 2003, Kyiv), Editors A.G. Nikitin, V.M. Boyko and R.O. Popovych, Proceedings of Institute of Mathematics, Kyiv, 2004, V.43, Part 1, 120-127.
  35. Olver P.J., Rosenau Ph., The construction of special solutions to partial differential equations, Phys. Lett. A, 1986, V.114, N 3, 107-112.
    Olver P.J., Rosenau Ph., Group-invariant solutions of differential equations, SIAM J. Appl. Math., 1987, V.47, N 2, 263-278.
  36. Olver P.J., Symmetry and explicit solutions of partial differential equations, Appl. Num. Math., 1992, V.10, 307-324.
  37. Pucci E., Similarity reductions of partial differential equations, J. Phys. A: Math. Gen., 1992, V.25, 2631-2640.
  38. Pucci E., Saccomandi G., On the weak symmetry group of partial differential equations, J. Math. Anal. Appl., 1992, V.163, N 2, 588-598.
    Pucci E., Saccomandi G., Evolution equations, invariant surface conditions and functional separation of variables, Phys. D, 2000, V.139, 28-47.
  39. Olver P.J., Direct reduction and differential constraints, Proc. Roy. Soc. London Ser. A, 1994, V.444, 509-523.
  40. Zhdanov R.Z., Tsyfra I.M., Reduction of differential equations and conditional symmetries, Ukr. Math. J., 1996, V.48, 595-602.
  41. Popovych R.O., On reduction and Q-conditional (nonclassical) symmetry, in Proceedings of the Second International Conference "Symmetry in Nonlinear Mathematical Physics" (July 7-13, 1997, Kyiv), Editors M.I. Shkil', A.G. Nikitin and V.M. Boyko, Kyiv, Institute of Mathematics, 1997, V.2, 437-443, math-ph/0207015.
  42. Zhdanov R.Z., Tsyfra I.M., Popovych R.O., Precise definition of reduction of partial diferential equations, J. Math. Anal. Appl., 1999, V.238, 101-123, math-ph/0207023.
  43. Yehorchenko I.A., Differential invariants and construction of conditionally invariant equations, math-ph/0304029.
  44. Cicogna G., `Weak' symmetries and adapted variables for differential equations, Int. J. Geom. Meth. Mod. Phys., 2004, V.1, N 1-2, 23-31.
    Cicogna G., A discussion on the different notions of symmetry of differential equations, in Proceedinds of Fifth International Conference "Symmetry in Nonlinear Mathematical Physics" (June 23-29, 2003, Kyiv), Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A. Yehorchenko, Proceedings of Institute of Mathematics, Kyiv, 2004, V.50, Part 1, 77-84.
  45. Cicogna G., Laino M., On the notion of conditional symmetry of differential equations, Rev. Math. Phys., 2006, to appear.
  46. Grasso D., Califano F., Pegoraro F., Porcelli F., Phase mixing and island saturation in Hamiltonian reconnection, Phys. Rev. Lett., 2001, V.86, N 22, 5051-5054.
  47. Del Sarto D., Califano F., Pegoraro F., Secondary instabilities and vortex formation in collisionless fluid megnetic reconnection, Phys. Rev. Lett., V.91, 2003, N 23, 235001-1/4.
  48. Kadomtsev B.B., Pogutse O.P., Theory of electron transport in a strong magnetic field, Soviet Phys. JETP Lett., 1984, V.39, N 5, 269-272.
  49. Bogoyavlenskij O.I., Infinite symmetries of the ideal MHD equilibrium equations, Phys. Lett. A, 2001, V.291, 256-264.
  50. Nucci M.C., Group analysis for M.H.D. equations, Atti Sem. Mat. Fis. Univ. Modena, 1984, V.33, N 1, 21-34.
  51. Nikitin A.G., Popovych R.O., Group classification of nonlinear Schrödinger equations, Ukr. Math. J., 2001, V.53, N 8, 1255-1265, math-ph./0301009.
  52. Popovych R.O., Yehorchenko I.A., Group classification of generalised eikonal equations, Ukr. Math. J., 2001, V.53, N 11, 1841-1850, math-ph/0112055.
  53. Popovych R.O., Ivanova N.M., New results on group classification of nonlinear diffusion-convection equations, J. Phys. A: Math. Gen., 2004, V.37, 7547-7565, math-ph/0306035.
  54. Güngör F., Lahno V.I., Zhdanov R.Z., Symmetry classification of KdV-type nonlinear evolution equations, J. Math. Phys., 2004, V.45, 2280-2313, nlin.SI/0201063.
  55. Zhdanov R.Z., Lahno V.I., Group classification of the general evolution equation: local and quasilocal symmetries, SIGMA, 2005, V.1, paper 009, 7 pages, nlin.SI/0510003.
  56. Wesson J., Tokamaks, 2nd ed., The Oxford Engineering Series, V.48, Oxford, Clarendon, 1997.
  57. Cicogna G., "Symmetry classification" of a class of PDE's containing several arbitrary functions, in preparation.
  58. Lovelace R.V.E., Mehanian C., Mobarry C.M., Sulkanen M.E., Theory of axisymmetric magnetohydrodynamic flows-disks, Astrophys. J. Suppl., 1986, V.62, 1-37.
  59. Euler N., Steeb W.-H., Mulser P., Symmetries of a nonlinear equation in plasma physics, J. Phys. A: Math. Gen., 1991, V.24, L785-L787.
  60. Kovalev V.F., Krivenko S.V., Pustovalov V.V., Symmetry group of Vlasov-Maxwell equations in plasma theory, J. Nonlinear Math. Phys., 1996, V.3, N 1-2, 175-180.
  61. Kovalev V.F., Bychenkov V.Yu., Tikhonchuk V.T., Particle dynamics during adiabatic expansion of a plasma bunch, J. Exp. Theor. Phys., 2002, V.95, N 2, 226-241.
  62. Kovalev V.F., Bychenkov V.Yu., Analytic solutions to Vlasov equations for expanding plasmas, Phys. Rev. Lett., 2003, V.90, N 18, 185004-1/4.

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