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


SIGMA 2 (2006), 068, 17 pages      nlin.SI/0610011      http://dx.doi.org/10.3842/SIGMA.2006.068

Painlevé Analysis and Similarity Reductions for the Magma Equation

Shirley E. Harris a and Peter A. Clarkson b
a) Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford, OX1 3LB, UK
b) Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7NF, UK

Received September 27, 2006; Published online October 05, 2006

Abstract
In this paper, we examine a generalized magma equation for rational values of two parameters, m and n. Firstly, the similarity reductions are found using the Lie group method of infinitesimal transformations. The Painlevé ODE test is then applied to the travelling wave reduction, and the pairs of m and n which pass the test are identified. These particular pairs are further subjected to the ODE test on their other symmetry reductions. Only two cases remain which pass the ODE test for all such symmetry reductions and these are completely integrable. The case when m = 0, n = −1 is related to the Hirota-Satsuma equation and for m = ½, n = −½, it is a real, generalized, pumped Maxwell-Bloch equation.

Key words: Painlevé analysis; similarity reductions; magma equation.

pdf (273 kb)   ps (230 kb)   tex (148 kb)

References

  1. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H., The inverse scattering transform - Fourier analysis for nonlinear problems, Stud. Appl. Math., 1974, V.53, 249-315.
  2. Ablowitz M.J., Ramani A., Segur H., A connection between nonlinear evolution equations and ordinary differential equations of P-type. I, J. Math. Phys., 1980, V.21, 715-721.
  3. Barcilon V., Richter F.M., Nonlinear waves in compacting media, J. Fluid Mech., 1986, V.164, 429-448.
  4. Bluman G.W., Cole J.D., The general similarity solution of the heat equation, J. Math. Mech., 1969, V.18, 1025-1042.
  5. 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.
  6. Clarkson P.A., Mansfield E.L., On a shallow water wave equation, Nonlinearity, 1994, V.7, 975-1000, solv-int/9401003.
  7. Clarkson P.A., Mansfield E.L., Milne A.E., Symmetries and exact solutions of a 2+1-dimensional Sine-Gordon system, Phil. Trans. R. Soc. Lond. A, 1996, V.354, 1807-1835, solv-int/9412003.
  8. Conte R., Musette M., Grundland A.M., Bäcklund transformation of partial differential equations from the Painlevé-Gambier classification. II. Tzitzeica equation, J. Math. Phys., 1999, V.40, 2092-2106.
  9. Deift P., Tomei C., Trubowitz E., Inverse scattering and the Boussinesq equation, Commun. Pure Appl. Math., 1982, V.35, 567-628.
  10. Drazin P.G., Johnson R.S., Solitons: an introduction, Cambridge, Cambridge University Press, 1989.
  11. Harris S.E., Conservation laws for a nonlinear wave equation, Nonlinearity, 1996, V.9, 187-208.
  12. Hereman W., Symbolic software for Lie symmetry analysis, in CRC Handbook of Lie Group Analysis of Differential Equations. III. New Trends in Theoretical Developments an Computational Methods, Editor N.H. Ibragimov, Boca Raton, CRC Press, 1996, Chapter XII, 367-413.
  13. Hereman W., Review of symbolic software for Lie symmetry analysis, Math. Comput. Modelling, 1997, V.25, 115-132.
  14. Hirota R., Satsuma J., N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Japan, 1976, V.40, 611-612.
  15. Jeffrey A., Kakutani T., Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation, SIAM Rev., 1972, V.14, 582-643.
  16. Konopelchenko B.G., Rogers C., On 2+1-dimensional nonlinear systems of Loewner type, Phys. Lett. A, 1991, V.158, 391-397.
  17. Konopelchenko B.G., Rogers C., On generalized Loewner systems: novel integrable equations in 2+1-dimensional, J. Math. Phys., 1993, V.34, 214-242.
  18. Marchant T.R., Smyth N.F., Approximate solutions for magmon propagation from a reservoir IMA J. Appl. Math., 2005, V.70, 796-813.
  19. McKenzie D.P., The generation and compaction of partially molten rock, J. Petrol., 1984, V.25, 713-765.
  20. Nakayama M., Mason D.P., Compressive solitary waves in compacting media, Internat. J. Non-Linear Mech., 1991, V.26, 631-640.
  21. Nakayama M., Mason D.P., Rarefactive solitary waves in two-phase fluid flow of compacting media, Wave Motion, 1992, V.15, 357-392.
  22. Nakayama M., Mason D.P., On the existence of compressive solitary waves in compacting media, J. Phys. A: Math. Gen., 1994, V.27, 4589-4599.
  23. Nakayama M., Mason D.P., Perturbation solution for small amplitude solitary waves in two-phase fluid flow of compacting media, J. Phys. A: Math. Gen., 1999, V.32, 6309-6320.
  24. Olson P., Christensen U., Solitary wave propagation in a fluid conduit within a viscous matrix, J. Geophys. Res., 1986, V.91, 6367-6374.
  25. Olver P.J., Applications of Lie groups to differential equations, 2nd ed., Graduate Texts Math., Vol. 107, New York, Springer, 1993.
  26. Scott D.R., Stevenson D.J., Magma solitons, Geophys. Res. Lett., 1984, V.11, 1161-1164.
  27. Scott D.R., Stevenson D.J., Whitehead J.A., Observations of solitary waves in a viscously deformable pipe, Nature, 1986, V.319, 759-761.
  28. Takahashi D., Sachs J.R., Satsuma J., Properties of the magma and modified magma equations, J. Phys. Soc. Japan, 1990, V.59, 1941-1953.
  29. Takahashi D., Satsuma J., Explicit solutions of magma equation, J. Phys. Soc. Japan, 1988, V.57, 417-421.
  30. Weiss J., Tabor M., Carnevale G., The Painlevé property for partial differential equations, J. Math. Phys., 1983, V.24, 522-526.
  31. Whitehead J.A., A laboratory demonstration of solitons using a vertical watery conduit in syrup, Amer. J. Phys., 1987, V.55, 998-1003.
  32. Whitehead J.A., Helfrich K.R., The Korteweg-de Vries equation from conduit and magma migration equations, Geophys. Res. Lett., 1986, V.13, 545-546.
  33. Whitehead J.A., Helfrich K.R., Magma waves and diapiric dynamics, in Magma Transport and Storage, Editor M.P. Ryan, Wiley & Sons, 1990, 53-76.
  34. Zabusky N.J., A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, in Proc. Symp. Nonlinear Partial Differential Equations, Editor W. Ames, New York, Academic Press, 1967, 223-258.

Previous article   Next article   Contents of Volume 2 (2006)