Acta Applicandae Mathematica

, Volume 39, Issue 1–3, pp 245–276 | Cite as

Symmetry reductions and exact solutions of shallow water wave equations

  • Petter A. Clarkson
  • Elizabeth L. Mansfield
Part II: Invited Contributions

Abstract

In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation
$$u_{xxxt} + \alpha u_x u_{xt} + \beta u_t u_{xx} - u_{xt} - u_{xx} = 0,$$
whereα andβ are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. Two special cases of this equation, or the equivalent nonlocal equation obtained by settingux=U, have been discussed in the literature. The caseα=2β was discussed by Ablowitz, Kaup, Newell and Segur (Stud. Appl. Math.,53 (1974), 249), who showed that this case was solvable by inverse scattering through a second-order linear problem. This case and the caseα=β were studied by Hirota and Satsuma (J. Phys. Soc. Japan,40 (1976), 611) using Hirota's bi-linear technique. Further, the caseα=β is solvable by inverse scattering through a third-order linear problem.

In this paper, a catalogue of symmetry reductions is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole (J. Math. Mech,18 (1969), 1025). The classical Lie method yields symmetry reductions of (1) expressible in terms of the first, third and fifth Painlevé transcendents and Weierstrass elliptic functions. The nonclassical method yields a plethora of exact solutions of (1) withα=β which possess a rich variety of qualitative behaviours. These solutions all like a two-soliton solution fort < 0 but differ radically fort > 0 and may be viewed as a nonlinear superposition of two solitons, one travelling to the left with arbitrary speed and the other to the right with equal and opposite speed. These families of solutions have important implications with regard to the numerical analysis of SWW and suggests that solving (1) numerically could pose some fundamental difficulties. In particular, one would not be able to distinguish the solutions in an initial-value problem since an exponentially small change in the initial conditions can result in completely different qualitative behaviours.

We compare the two-soliton solutions obtained using the nonclassical method to those obtained using the singular manifold method and Hirota's bi-linear method.

Further, we show that there is an analogous nonlinear superposition of solutions for two (2+1)dimensional generalisations of the SWW Equation (1) withα=β. This yields solutions expressible as the sum of two solutions of the Korteweg-de Vries equation.

Mathematics subject classification (1991)

58F07 

Key words

symmetry reductions exact solutions shallow water wave equation 

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References

  1. 1.
    Ablowitz, M. J. and Clarkson, P. A.:Solitons, Nonlinear Evolution Equations and Inverse Scattering, Lect. Notes Math., Vol. 149, C.U.P., Cambridge, 1991.Google Scholar
  2. 2.
    Ablowitz, M. J., Kaup, D. J., Newell, A. C., and Segur, H.:Stud. Appl. Math. 53 (1974), 249–315.Google Scholar
  3. 3.
    Ablowitz, M. J., Ramani, A., and Segur, H.:Phys. Rev. Lett. 23 (1978), 333–338.Google Scholar
  4. 4.
    Ablowitz, M. J., Ramani, A., and Segur, H.:J. Math. Phys. 21 (1980), 715–721.Google Scholar
  5. 5.
    Ablowitz, M. J., Schober, C., and Herbst, B. M.:Phys. Rev. Lett. 71 (1993), 2683–2686.Google Scholar
  6. 6.
    Ablowitz, M. J. and Villarroel, J.:Stud. Appl. Math. 85 (1991), 195–213.Google Scholar
  7. 7.
    Anderson, R. L. and Ibragimov, N. H.:Lie-Bäcklund Transformations in Applications, SIAM, Philadelphia, 1979.Google Scholar
  8. 8.
    Benjamin, T. B., Bona, J. L., and Mahoney, J.:Phil. Trans. R. Soc. Land. Ser. A 272 (1972), 47–78.Google Scholar
  9. 9.
    Bluman, G. W. and Cole, J. D.:J. Math. Mech. 18 (1969), 1025–1042.Google Scholar
  10. 10.
    Bluman, G. W. and Kumei, S.:Symmetries and Differential Equations, inAppl. Math. Sci., Vol. 81, Springer-Verlag, Berlin, 1989.Google Scholar
  11. 11.
    Bogoyavlenskii, O. I.:Math. USSR Izves. 34 (1990), 245–259.Google Scholar
  12. 12.
    Bogoyavlenskii, O. I.:Russ. Math. Surv. 45 (1990), 1–86.Google Scholar
  13. 13.
    Boiti, M., Leon, J. J-P, Manna, M., and Pempinelli, F.:Inverse Problems 2 (1986), 271–279.Google Scholar
  14. 14.
    Buchberger, B.: in J. Rice (ed.),Mathematical Aspects of Scientific Software, Springer-Verlag, 1988, pp. 59–87.Google Scholar
  15. 15.
    Champagne, B., Hereman, W., and Winternitz, P.:Comp. Phys. Comm. 66 (1991), 319–340.Google Scholar
  16. 16.
    Clarkson, P. A.: Nonclassical symmetry reductions for the Boussinesq equation, inChaos, Solitons and Fractals, 1994, to appear.Google Scholar
  17. 17.
    Clarkson, P. A. and Kruskal, M. D.:J. Math. Phys. 30 (1989), 2201–2213.Google Scholar
  18. 18.
    Clarkson, P. A. and Mansfield, E. L.:Physica D 70 (1994), 250–288.Google Scholar
  19. 19.
    Clarkson, P. A. and Mansfield, E. L.:Nonlinearity 7 (1994), 975–1000.Google Scholar
  20. 20.
    Clarkson, P. A. and Mansfield, E. L.: Algorithms for the nonclassical method of symmetry reductions,SIAM J. Appl. Math., 1994, to appear.Google Scholar
  21. 21.
    Clarkson, P. A. and Mansfield, E. L.: Exact solutions for some (2+1)-dimensional shallow water wave equations, Preprint, Department of Mathematics, University of Exeter, 1994.Google Scholar
  22. 22.
    Cole, J. D.:Quart. Appl. Math. 9 (1951), 225–236.Google Scholar
  23. 23.
    Conte, R. and Musette, M.:J. Math. Phys. 32 (1991), 1450–1457.Google Scholar
  24. 24.
    Deift, P., Tomei, C., Trubowitz, E.:Comm. Pure Appl. Math. 35 (1982), 567–628.Google Scholar
  25. 25.
    Dorizzi, B., Grammaticos, B., Ramani, A., and Winternitz, P.:J. Math. Phys. 27 (1986), 2848–2852.Google Scholar
  26. 26.
    Espinosa, A. and Fujioka, J.:J. Phys. Soc. Japan 63 (1994), 1289–1294.Google Scholar
  27. 27.
    Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R.:Phys. Rev. Lett. 19 (1967), 1095–1097.Google Scholar
  28. 28.
    Gilson, C. R., Nimmo, J. J. C., and Willox, R.:Phys. Lett. 180A (1993), 337–345.Google Scholar
  29. 29.
    Fushchich, W. I.:Ukrain. Math. J. 43 (1991), 1456–1470.Google Scholar
  30. 30.
    Hereman, W.:Euromath Bull. 1(2) (1994), 45–79.Google Scholar
  31. 31.
    Hietarinta, J.: in R. Conte and N. Boccara (eds),Partially Integrable Evolution Equations in Physics, NATO ASI Series C: Mathematical and Physical Sciences, Vol. 310, Kluwer, Dordrecht, 1990, pp. 459–478.Google Scholar
  32. 32.
    Hirota, R.: in R. K. Bullough and P. J. Caudrey (eds),Solitons, Topics in Current Physics, Vol. 17, Springer-Verlag, Berlin, 1980, pp. 157–176.Google Scholar
  33. 33.
    Hirota, R. and Itô, M.:J. Phys. Soc. Japan 52 (1983), 744–748.Google Scholar
  34. 34.
    Hirota, E. and Satsuma, J.:J. Phys. Soc. Japan 40 (1976), 611–612.Google Scholar
  35. 35.
    Hopf, E.:Comm. Pure Appl. Math. 3 (1950), 201–250.Google Scholar
  36. 36.
    Ince, E. L.:Ordinary Differential Equations, Dover, New York, 1956.Google Scholar
  37. 37.
    Jimbo, M. and Miwa, T.:Publ. R.I.M.S. 19 (1983), 943–1001.Google Scholar
  38. 38.
    Leble, S. B. and Ustinov, N. V.:Inverse Problems 210 (1994), 617–633.Google Scholar
  39. 39.
    Levi, D. and Winternitz, P.:J. Phys. A: Math. Gen. 22 (1989), 2915–2924.Google Scholar
  40. 40.
    Mansfield, E. L.:Diffgrob: A symbolic algebra package for analysing systems of PDE using Maple, ftp euclid.exeter.ac.uk, login: anonymous, password: your email address, directory: pub/liz, 1993.Google Scholar
  41. 41.
    Mansfield, E. L. and Fackerell, E. D.: Differential Gröbner Bases, Preprint 92/108, Macquarie University, Sydney, Australia, 1992.Google Scholar
  42. 42.
    McLeod, J. B. and Olver, P. J.:SIAM J. Math. Anal. 14 (1983), 488–506.Google Scholar
  43. 43.
    Musette, M., Lambert, F., and Decuyper, J. C.:J. Phys. A: Math. Gen. 20 (1987), 6223–6235.Google Scholar
  44. 44.
    Olver, P. J.:Applications of Lie Groups to Differential Equations, 2nd edn, Graduate Texts Math., Vol. 107, Springer-Verlag, New York, 1993.Google Scholar
  45. 45.
    Olver, P. J. and Rosenau, P.:Phys. Lett. 114A (1986), 107–112.Google Scholar
  46. 46.
    Olver, P. J. and Rosenau, P.:SIAM J. Appl. Math. 47 (1987), 263–275.Google Scholar
  47. 47.
    Peregrine, H.:J. Fluid Mech. 25 (1966), 321–330.Google Scholar
  48. 48.
    Reid, G. J.:J. Phys. A: Math. Gen. 23 (1990), L853-L859.Google Scholar
  49. 49.
    Reid, G. J.:Europ. J. Appl. Math. 2 (1991), 293–318.Google Scholar
  50. 50.
    Reid, G. J. and Wittkopf, A.: A Differential Algebra Package for Maple, ftp 137.82.36.21 login: anonymous, password: your email address, directory: pub/standardform, 1993.Google Scholar
  51. 51.
    Schwarz, F.:Computing 49 (1992), 95–115.Google Scholar
  52. 52.
    Tamizhmani, K. M. and Punithavathi, P.:J. Phys. Soc. Japan 59 (1990), 843–847.Google Scholar
  53. 53.
    Topunov, V. L.:Acta Appl. Math. 16 (1989), 191–206.Google Scholar
  54. 54.
    Weiss, J.:J. Math. Phys. 24 (1983), 1405–1413.Google Scholar
  55. 55.
    Weiss, J., Tabor, M., and Carnevale, G.:J. Math. Phys. 24 (1983), 522–526.Google Scholar
  56. 56.
    Whittaker, E. E. and Watson, G. M.:Modern Analysis, 4th edn, C.U.P., Cambridge, 1927.Google Scholar
  57. 57.
    Winternitz, P.: Lie groups and solutions of nonlinear partial differential equations, in L. A. Ibort and M. A. Rodriguez (eds),Integrable Systems, Quantum Groups, and Quantum Field Theories, NATO ASI Series C., Vol. 409, Kluwer, Dordrecht, 1993, pp. 429–495.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Petter A. Clarkson
    • 1
  • Elizabeth L. Mansfield
    • 1
  1. 1.Department of MathematicsUniversity of ExeterExeterUK

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