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New exact solutions of the Boussinesq equation

Published online by Cambridge University Press:  16 July 2009

Peter A. Clarkson
Peter A. Clarkson
Affiliation:
Department of Mathematics, University of Exeter, Exeter, EX4 4QE, England
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Abstract

In this paper new exact solutions are derived for the physically and mathematically significant Boussinesq equation. These are obtained in two different ways: first, by generating exact solutions to the ordinary differential equations which arise from (classical and nonclassical) similarity reductions of the Boussinesq equation (these ordinary differential equations are solvable in terms of the first, second and fourth Painlevé equations); and second, by deriving new space-independent similarity reductions of the Boussinesq equation. Extensive sets of exact solutions for both the second and fourth Painlevé equations are also generated. The symbolic manipulation language MACSYMA is employed to facilitate the calculations involved.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 1 , Issue 3 , September 1990 , pp. 279 - 300
Copyright
Copyright © Cambridge University Press 1990

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References

Ablowitz, M. J. & Haberman, R. 1975 J. Math. Phys. 16, 2301–2305.CrossRefGoogle Scholar
Ablowitz, M. J. & Segur, H. 1977a Stud. Appl. Math. 57, 13–44.CrossRefGoogle Scholar
Ablowitz, M. J. & Segur, H. 1977b Phys. Rev. Lett. 38, 1103–1106.CrossRefGoogle Scholar
Ablowitz, M. J., Baryaacov, D. & Fokas, A. S. 1983 Stud. Appl. Math. 69, 135–143.CrossRefGoogle Scholar
Ablowitz, M. J., Ramani, A. & Segur, H. 1980 J. Math. Phys. 21, 1006–1015.CrossRefGoogle Scholar
Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions, Dover, New York.Google Scholar
Airault, H. 1979 Stud. Appl. Math. 61, 31–53.CrossRefGoogle Scholar
Bassom, A. P. & Hall, P. 1989 Stud. Appl. Math. 81, 185–219.CrossRefGoogle Scholar
Boussinesq, J. 1871 Comptes Rendus 72, 755–759.Google Scholar
Boussinesq, J. 1872 J. Pure Appl. 7, 55–108.Google Scholar
Bluman, G. W. & Cole, J. D. 1969 J. Math. Mech. 18, 1025–1042.Google Scholar
Bluman, G. W. & Cole, J. D. 1974 Similarity Methods for Differential Equations, Appl. Math. Sci. 13, Springer-Verlag, BerlinCrossRefGoogle Scholar
Bluman, G. W. & Kumei, S. 1989 Symmetries and Differential Equations, Appl. Math. Sci. 81, Springer-Verlag, Berlin.CrossRefGoogle Scholar
De Boer, P. C. T. & Luoford, G. S. S. 1975 Plasma Phys. 17, 29–43.CrossRefGoogle Scholar
Caudrey, P. J. 1980 Phys. Lett. 79A, 264–268;CrossRefGoogle Scholar
1982 Physica 6D, 51–77.Google Scholar
Champagne, B. & Winternitz, P. 1985 Preprint CRM-1278, Montreal.Google Scholar
Clarkson, P. A. 1989a J. Phys. A: Math. Gen. 22, 2355–2367.CrossRefGoogle Scholar
Clarkson, P. A. 1989b J. Phys. A: Math. Gen. 22, 3821–3848.Google Scholar
Clarkson, P. A. & Kruskal, M. D. 1989 J. Math. Phys. 30, 2201–2213.CrossRefGoogle Scholar
Clarkson, P. A. & McLeod, J. B. 1988 Arch. Rat. Mech. Anal. 103, 97–138.Google Scholar
Larkson, P. A. & Winternitz, P. 1989 ‘Nonclassical symmetry reductions for the Kadometsev- Petviashvili equation’, preprint.Google Scholar
Davis, H. T. 1962 Introduction to Nonlinear Dfferential and Integral Equations, Dover, New York.Google Scholar
Deift, P., Tomei, C. & Trubowitz, E. 1982 Comm. Pure Appl. Math. 35, 567–628.CrossRefGoogle Scholar
Erdélyl, A., Magnus, W. & Oberhettinger, F. & Tricomi, F. G. 1953 Higher Transcendental Functions II, McGraw-Hill, New York.Google Scholar
Flaschka, H. & Newell, A. C. 1980 Comm. Math. Phys. 76, 65–116.CrossRefGoogle Scholar
Fokas, A. S. & Ablowitz, M. J. 1981 Phys. Rev. Lett. 47, 1096–1100.CrossRefGoogle Scholar
Fokas, A. S. & Ablowitz, M. J. 1982 J. Math. Phys. 23, 2033–2042.CrossRefGoogle Scholar
Fokas, A. S. & Ablowitz, M. J. 1983a Stud. Appl. Math. 69, 211–228.CrossRefGoogle Scholar
Fokas, A. S. & Ablowitz, M. J. 1983b Comm. Math. Phys. 91, 381–403.CrossRefGoogle Scholar
Fokas, A. S., Mugan, U. & Ablowitz, M. J. 1988 Physica 30D, 247–283.Google Scholar
Giannini, J. A. & Joseph, R. I. 1989 Phys. Lett. 141, 417–419.CrossRefGoogle Scholar
Gibbon, J. D., Newell, A. C., Tabor, M. & Zeng, Y. B. 1988 Nonlinearity 1, 481–490.CrossRefGoogle Scholar
Gromak, V. I. 1977 Diff Eqns. 14, 1510–1513.Google Scholar
Gromak, V. I. 1987 Diff Eqns. 23, 506–513.Google Scholar
Hansen, A. G. 1964 Similarity Analyses of Boundary Value Problems in Engineering, Prentice-Hall, Englewood Cliffs.Google Scholar
Hastings, S. P. & McLeod, J. B. 1980 Arch. Rat. Mech. Anal. 73, 31–51.CrossRefGoogle Scholar
Hastings, S. P. & Troy, W. C. 1989 SIAM J. Math. Anal. 20, 634–642.CrossRefGoogle Scholar
Head, A. K. 1989 Private communication.Google Scholar
Holmes, P. & Spence, D. 1984 Q. Jl. Mech. Appl. Math. 37, 525–538.CrossRefGoogle Scholar
Ince, E. L. 1956 Ordinary Differential Equations, Dover, New York.Google Scholar
Its, A. R. & Kapaev, A. A. 1988 Math. USSR Izvestiya 31, 193–207.CrossRefGoogle Scholar
Its, A. R. & Novokshenov, V. Yu. 1986 The Isomonodromic Deformation Method in the Theory of Painlevé Equations, Lect Notes Math. 1191, Springer-Verlag, Berlin.Google Scholar
Joshi, N. & Kruskal, M. D. 1988 Phys. Len. 130, 129–137.CrossRefGoogle Scholar
Kapaev, A. A. 1988a Diff. Eqns. 24, 1107–1115.Google Scholar
Kapaev, A. A. 1988b Theo. Math. Phys. 77, 1227–1234.CrossRefGoogle Scholar
Kitaev, A. A. 1985 Theo. Math. Phys. 64, 878–894.CrossRefGoogle Scholar
Lebeau, G. & Lochak, P. 1987 J. Diff. Eqns. 68, 344–372.CrossRefGoogle Scholar
Lebedev, N. N. 1972 Special Functions and their Applications, Dover, New York.Google Scholar
Levi, D. & Winternitz, P. 1989 J. Phys. A: Math. Gen. 22, 2915–2924.CrossRefGoogle Scholar
Lukashevich, N. A. 1965 Diff. Eqns. 1, 561–564.Google Scholar
Lukashevich, N. A. 1967 Diff. Eqns. 3, 395–399.Google Scholar
Lukashevich, N. A. 1971 Diff. Eqns. 7, 853–854.Google Scholar
Manakov, S. V. 1981 Physica 3D, 420–427.Google Scholar
Miles, J. W. 1978 Proc. Roy. Soc. Lond. A 361, 277–291.Google Scholar
Murato, Y. 1985 Funkcial. Ekvac. 28, 1–32.Google Scholar
Nishitani, T. & Tajiri, M. 1982 Phys. Len. 89A, 379–380.CrossRefGoogle Scholar
Okamoto, K. 1986 Math. Ann. 275, 222–255.CrossRefGoogle Scholar
Olver, P. J. 1986 Applications of Lie Groups to Dfferential Equations, Grad. Texts Math. 107, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Olver, P. J. & Rosenau, P. 1986 Phys. Lett. 114A, 107–112.Google Scholar
Olver, P. J. & Rosenau, P. 1987 SIAM J. Appl. Math. 47, 263–278.CrossRefGoogle Scholar
Quispel, G. R. W., Nijhoff, F. W. & Capel, H. W. 1982 Phys. Lett. 91A, 143–145.CrossRefGoogle Scholar
Rosales, R. R. 1978 Proc. Roy. Soc. Lond. A 361, 265–275.Google Scholar
Rosenau, P. & Schwarzmeier, J. L. 1986 Phys. Len. 115A, 75–77.CrossRefGoogle Scholar
Schwarz, F. 1985 Computing 34, 91–106.CrossRefGoogle Scholar
Schwarz, F. 1988 SIAM Rev. 30, 450–481.CrossRefGoogle Scholar
Segur, H. & Ablowitz, M. J. 1981 Physica 3D, 165–184.Google Scholar
Seyler, C. E. & Fenstermacher, D. L. 1984 Phys. Fluids 27, 4–7.CrossRefGoogle Scholar
Sneddon, I. N. 1980 Special Functions of Mathematical Physics and Chemistry, 3rd Ed, Longman, London.Google Scholar
Suleimanov, B. I. 1987 Diff. Eqns. 23, 569–576.Google Scholar
Toda, M. 1975 Phys. Rep. 18, 1–125.CrossRefGoogle Scholar
Ursell, F. 1953 Proc. Camb. Phil. Soc. 49, 682–694.Google Scholar
Whittaker, E. E. & Watson, G. M. 1927 Modern Analysis, 4th Ed, CUP, Cambridge.Google Scholar
Zabusky, N. J. 1967 In Nonlinear Partial Differential equations (ed. Ames, W. F.), pp. 233–258. Academic, New York.Google Scholar
Zakharov, V. E. 1974 Sov. Phys. JETP 38, 108–110.Google Scholar