Shallow water wave systems

Studies in Applied Mathematics

Volumn 101, Issue 4, 1998, Pages 389-432

  Clarkson, Peter A ^a   Priestley, Thomas J ^b  

^a UNIVERSITY OF KENT   (United Kingdom)

^b NONE

Author keywords

[No Author keywords available]

Indexed keywords

WAVE EQUATIONS;

% REDUCTIONS; CLASSICAL METHODS; DIRECT METHOD; KRUSKAL; NONZERO CONSTANTS; SHALLOW WATER WAVE EQUATION; SHALLOW WATER WAVES; SYMMETRY REDUCTION; SYSTEMS OF EQUATIONS; WAVE SYSTEM;

WATER WAVES;

EID: 0039164075     PISSN: 00222526     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9590.00099     Document Type: Article

References (54)

1. A. ESPINOSA and J. FUJIOKA, Hydrodynamic foundation and Painlevé analysis of the Hirota-Satsuma-type equations, J. Phys. Soc. Jpn. 63:1289-1294 (1994).
2. M. J. ABLOWITZ, D. J. KAUP, A. C. NEWELL, and H. SEGUR, The inverse scattering transform: Fourier analysis for nonlinear problems, Stud. Appl. Math. 53:249-315 (1974).
3. R. HIROTA and J. SATSUMA, N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Jpn. 40:611-612 (1976).
4. M. MUSETTE and R. CONTE, Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations, J. Math. Phys. 32:1450-1457 (1991).
5. P. DEIFT, C. TOMEI, and E. TRUBOWITZ, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math. 35:567-628 (1982).
6. H. PEREGRINE. Calculations of the development of an undular bore, J. Fluid Mech. 25:321-330 (1966).
7. T. B. BENJAMIN, J. L. BONA, and J. MAHONEY, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A 272:47-78 (1972).
8. R. HIROTA, Direct methods in soliton theory, in Solitons (R. K. Bullough and P. J. Caudrey, Eds.), Topics in Current Physics, Vol. 17. pp. 157-176, Springer-Verlag, Berlin, 1980.
9. J. HIETARINTA, Hirota's bilinear method and partial integrability, in Partially Integrable Evolution Equations in Physics (R. Conte and N. Bocarra, Eds.), NATO ASI Series C: Mathematical and Physical Sciences, Vol. 310, pp. 459-478, Kluwer, Dordrecht, 1990.
10. P. A. CLARKSON and E. L. MANSFIELD, On a shallow water wave equation, Nonlinearity 7:975-1000 (1994).
11. P. A. CLARKSON and E. L. MANSFIELD, Symmetry reductions and exact solutions of shallow water wave equations, Acta Appl. Math. 39:245-276 (1995).
12. M. J. ABLOWITZ, A. RAMANI, and H. SEGUR, A connection between nonlinear evolution equations and ordinary differential equations of P-type, I, J. Math. Phys. 21:715-721 (1980).
13. J. WEISS, M. TABOR, and G. CARNEVALE, The Painlevé property for partial differential equations, J. Math. Phys. 24:522-526 (1983).
14. V. A. GALAKTIONOV, On new exact blow-up solutions for nonlinear heat conduction equations with source and applications, Differential Integral Equations 3:863-874 (1990).
15. W. F. AMES, F. V. POSTELL, and E. ADAMS, Optimal numerical algorithms, Appl. Num. Math. 10:235-259 (192).
16. Yu. I. SHOKIN, The Method of Differential Approximation, Springer, New York, 1983.
17. W. HEREMAN, Review of symbolic software for the computation of Lie symmetries of differential equations, Euromath. Bull. 2(1):45-79 (1994).
18. B. CHAMPAGNE, W. HEREMAN, and P. WINTERNITZ, The computer calculation of Lie point symmetries of large systems of differential equations, Comp. Phys. Comm. 66:319-340 (1991).
19. L. V. OVSIANNIKOV, Group Analysis of Differential Equations, Academic Press, New York, 1982 [Trans. W. F. Ames].
20. J. ONDICH, A differential constraints approach to partial invariance, Eur. J. Appl. Math. 6:631-637 (1995).
21. N. N. YANENKO, Theory of consistency and methods of integrating systems of nonlinear partial differential equations, in Proc. 4th All-Union Mathematics Congress, pp. 247-259, Leningrad, 1964 [in Russian].
22. P. J. OLVER and P. ROSENAU, The construction of special solutions to partial differential equations, Phys. Lett. A 114:107-112 (1986).
23. P. J. OLVER and P. ROSENAU, Group-invariant solutions of differential equations, SIAM J. Appl. Math. 47:263-275 (1987).
24. G. W. BLUMAN and J. D. COLE, The general similarity of the heat equation, J. Math. Mech. 18:1025-1042 (1969).
25. D. LEVI and P. WINTERNITZ, Nonclassical symmetry reduction: Example of the Boussinesq equation, J. Phys. A: Math. Gen. 22:2915-2924 (1989).
26. E. M. VOROB'EV, Symmetries of compatibility conditions for systems of differential equations, Acta Appl. Math. 24:1-24 (1991).
27. P. A. CLARKSON and M. D. KRUSKAL, New similarity solutions of the Boussinesq equation, J. Math. Phys. 30:2201-2213 (1989).
28. P. A. CLARKSON, Nonclassical symmetry reductions of the Boussinesq equation, Chaos Solitons Fractals 5:2261-2301 (1995).
29. M. C. NUCCI and P. A. CLARKSON, The nonclassical method is more general than the direct method for symmetry reductions: An example of the Fitzhugh-Nagumo equation, Phys. Lett. A 164:49-56 (1992).
30. P. J. OLVER, Direct reduction and differential constraints, Proc. Roy. Soc. London Ser. A 444:509-523 (1994).
31. D. J. ARRIGO, P. BROADBRIDGE, and J. M. HILL, Nonclassical symmetry solutions and the methods of Bluman-Cole and Clarkson-Kruskal, J. Math. Phys. 34:4692-4703 (1993).
32. E. Pucci, Similarity reductions of partial differential equations, J. Phys. A: Math. Gen. 25:2631-2640 (1992).
33. S. ZIDOWITZ, Conditional symmetries and the direct reduction of partial differential equations, in Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics (N. H. Ibragimov, M. Torrisi, and A. Valenti, Eds.), pp. 387-393, Kluwer, Dordrecht, 1993.
34. G. I. BURDE, The construction of special explicit solutions of the boundary-layer equations: Steady flows, Quart. J. Mech. Appl. Math. 47:247-260 (1994).
35. G. I. BURDE, The construction of special explicit solutions of the boundary-layer equations: Unsteady flows, Quart. J. Mech. Appl. Math. 48:611-633 (1995).
36. S. HOOD, New exact solutions of Burgers's equation - An extension to the direct method of Clarkson and Kruskal, J. Math. Phys. 36:1971-1990 (1995).
37. G. W. BLUMAN and V. SHTELEN, Developments in similarity methods related to the pioneering work of Julian Cole, in Mathematics Is for Solving Problems (L. P. Cook, V. Roytburd, and M. Turin, Eds.), pp. 105-117, SIAM, Philadelphia, 1996.
38. G. I. BURDE, New similarity solutions of the steady-state boundary-layer equations, J. Phys. A: Math. Gen. 29:1665-1983 (1996).
39. P. J. OLVER and E. M. VOROB'EV, Nonclassical and conditional symmetries, in Lie Group Analysis of Differential Equations, III, New Trends in Theoretical Developments and Computational Methods (N. H. Ibragimov, Ed.), Chap. X. pp. 291-328, CRC Press, Boca Raton, 1996.
40. P. A. CLARKSON and E. L. MANSFIELD Symmetry reductions and exact solutions of a class of nonlinear heat equations, Phys. D 70:250-288 (1993).
41. E. L. MANSFIELD and E. D. FACKERELL, Differential Gröbner Bases, Preprint 92/108, Macquarie University, Sydney, Australia, 1992.
42. G. J. REID, A triangularization algorithm which determines the Lie symmetry algebra of any system of PDEs, J. Phys. A: Math. Gen. 23:L853-L859 (1990).
43. G. J. REID, Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution, Eur. J. Appl. Math. 2:293-318 (1991).
44. E. L. MANSFIELD, diffgrob2: A symbolic algebra package for analysing systems of PDE using Maple, ftp ftp.ukc.ac.uk, login: anonymous, password: your email address, directory: pub/maths/liz (1993).
45. P. A. CLARKSON and E. L. MANSFIELD, Algorithms for the nonclassical method of symmetry reductions, SIAM J. Appl. Math. 54:1693-1719 (1994).
46. P. A. CLARKSON, E. L. MANSFIELD, and A. E. MILNE, Symmetries and exact solutions of a 2+1-dimensional Sine-Gordon system, Philos. Trans. Roy. Soc. London Ser. A 354:1807-1835 (1996).
47. P. A. CLARKSON, E. L. MANSFIELD, and T. J. PRIESTLEY, Symmetries of a class of nonlinear third order partial differential equations, Math. Comput. Modelling, 25:195-212 (1997).
48. E. L. MANSFIELD and P. A. CLARKSON, Symmetries and exact solutions for a 2+1-dimensional shallow water wave equation, Math. Comput. Simulation 43:39-55 (1997).
49. M. J. ABLOWITZ, A. RAMANI, and H. SEGUR, Nonlinear evolution equations and ordinary differential equations of Painlevé type, Phys. Rev. Lett. 23:333-338 (1978).
50. P. A. CLARKSON, D. K. LUDLOW. and T. J. PRIESTLEY, The direct and nonclassical methods for symmetry reductions of nonlinear partial differential equations, Meth. Appl. Analysis 4:173-195 (1997).
51. T. J. PRIESTLEY, Methods of Symmetry Reduction and Their Application, Ph.D. Thesis, University of Kent at Canterbury, 1996.
52. S-Y. LOU, Nonclassical symmetry reductions for the dispersive wave equations in shallow water, J. Math. Phys. 33:4300-4305 (1992).
53. S-Y. LOU and H-Y. RUAN, Non-classical analysis and Painlevé property for the Kupershmidt equations, J. Phys. A: Math. Gen. 26:4679-4688 (1993).
54. M. D. KRUSKAL, Speaking at the "Symmetries and Integrability of Difference Equations" meeting, held at the University of Kent at Canterbury, 1996.