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Symmetries of a class of nonlinear third-order partial differential equations. (English) Zbl 0879.35005
Symmetry reductions of the following class of nonlinear third-order partial differential equations \[ u_t-\varepsilon u_{xxt}+2\kappa u_x-u u_{xxx}-\alpha u u_x-\beta u_x u_{xx}=0 \] with four arbitrary constants \(\varepsilon,\kappa,\alpha,\beta\) are considered. This class has previously been studied by C. Gilson and A. Pickering [Phys. A, Math. Gen. 28, 2871-2888 (1995; Zbl 0830.35127)] using Painlevé theory. It contains as special cases the Fornberg-Whitham, the Rosenau-Hyman, and the Camassa-Holm equation. The authors apply besides the standard symmetry approach also the non-classical method of G. W. Bluman and J. D. Cole [J. Math. Mech. 18, 1025-1042, (1969; Zbl 0187.03502)]. Using the so-called differential Gröbner bases developed by one of the authors they obtain a symmetry classification of the parameters \(\varepsilon,\kappa,\alpha,\beta\). The computations are done with the help of the Maple package.

35A25 Other special methods applied to PDEs
58J70 Invariance and symmetry properties for PDEs on manifolds
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
35Q58 Other completely integrable PDE (MSC2000)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
68W30 Symbolic computation and algebraic computation
Full Text: DOI
[1] Fornberg, B.; Whitham, G.B., A numerical and theoretical study of certain nonlinear wave phenomena, Phil. trans. R. soc. lond. A, 289, 373-404, (1978) · Zbl 0384.65049
[2] Whitham, G.B., Variationsl methods and applications to water waves, (), 6-25 · Zbl 0163.21104
[3] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley New York · Zbl 0373.76001
[4] Rosenau, P.; Hyman, J.M., Compactons: solitons with finite wavelength, Phys. rev. lett., 70, 564-567, (1993) · Zbl 0952.35502
[5] Camassa, R.; Holm, D.D., An integrable shallow water equation with peaked solitons, Phys. rev. lett., 71, 1661-1664, (1993) · Zbl 0972.35521
[6] Camassa, R.; Holm, D.D.; Hyman, J.M., A new integrable shallow water equation, Adv. appl. mech., 31, 1-33, (1994) · Zbl 0808.76011
[7] Fuchssteiner, B., The Lie algebra structure of a nonlinear evolution equations admitting infinite dimensional abelian symmetry groups, Progr. theor. phys., 65, 861-876, (1981) · Zbl 1074.58501
[8] Fuchssteiner, B.; Fokas, A.S., Symplectic structure, their Bäcklund transformations and hereditary symmetries, Physica, D4, 47, (1981) · Zbl 1194.37114
[9] Cooper, F.; Shepard, H., Solitons in the Camassa-Holm shallow water equation, Phys. lett., A194, 246-250, (1994) · Zbl 0961.76512
[10] Fokas, A.S., Moderately long water waves of moderable amplitude are integrable, (1994), (preprint)
[11] Fokas, A.S., On a class of physically important integrable equations, Physica, 87D, 145-150, (1994) · Zbl 1194.35363
[12] Fokas, A.S.; Santini, P.M., An inverse acoustic problem and linearization of moderate amplitude dispersive waves, (1994), (preprint)
[13] Fuchssteiner, B., Some tricks from the symmetry-toolbox for nonlinear equations—generalizations of the Camassa-Holm equation, Physica, 95D, 229-243, (1993) · Zbl 0900.35345
[14] Gilson, C.; Pickering, A., Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations, J. phys. A: math. gen., 28, 2871-2888, (1995) · Zbl 0830.35127
[15] Marinakis, V.; Bountis, T.C., On the integrability of a new class of water wave equations, (1995), Department of Mathematics, University of Patras Greece, (preprint) · Zbl 1084.76511
[16] Olver, P.J.; Rosenau, P., Tri-Hamiltonian soliton-compacton duality, Phys. rev E, 53, 1900-1906, (1996)
[17] Benjamin, T.B.; Bona, J.L.; Mahoney, J., Model equations for long waves in nonlinear dispersive systems, Phil. trans. R. soc. lond. ser. A, 272, 47-78, (1972) · Zbl 0229.35013
[18] Peregrine, H., Calculations of the development of an undular bore, J. fluid mech., 25, 321-330, (1966)
[19] McLeod, J.B.; Olver, P.J., The connection between partial differential equations soluble by inverse scat-tering and ordinary differential equations of Painlevé type, SIAM J. math. anal., 14, 488-506, (1983) · Zbl 0518.35075
[20] Makhankov, V.G., Dynamics of classical solitons, Phys. rep., 35, 1-128, (1978)
[21] Olver, P.J., Direct reduction and differential constraints, (), 509-523 · Zbl 0814.35003
[22] Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R., Method for solving the KdV equation, Phys. rev. lett., 19, 1095-1097, (1967) · Zbl 1103.35360
[23] Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H., The inverse scattering transform—fourier analysis for nonlinear problems, Stud. appl. math., 53, 249-315, (1974) · Zbl 0408.35068
[24] Hirota, R.; Satsuma, J., N-soliton solutions of model equations for shallow water waves, J. phys. soc. Japan, 40, 611-612, (1976) · Zbl 1334.76016
[25] Weiss, J.; Tabor, M.; Carnevale, G., The Painlevé property for partial differential equations, J. math. phys., 24, 522-526, (1983) · Zbl 0514.35083
[26] Ablowitz, M.J.; Ramani, A.; Segur, H., Nonlinear evolution equations and ordinary differential equations of Painlevé type, Phys. rev. lett., 23, 333-338, (1978)
[27] Ablowitz, M.J.; Ramani, A.; Segur, H., A connection between nonlinear evolution equations and ordinary differential equations of P-type. I, J. math. phys., 21, 715-721, (1980) · Zbl 0445.35056
[28] Ramani, A.; Dorizzi, B.; Grammaticos, B., Painlevé conjecture revisited, Phys. rev. lett., 49, 539-1541, (1982)
[29] Ranada, A.F.; Ramani, A.; Dorizzi, B.; Grammaticos, B., The weak-Painlevé property as a criterion for the integrability of dynamical systems, J. math. phys., 26, 708-710, (1985) · Zbl 0613.34036
[30] Hereman, W., Review of symbolic software for the computation of Lie symmetries of differential equations, Euromath. bull., 1, 2, 45-79, (1994) · Zbl 0891.65081
[31] Champagne, B.; Hereman, W.; Winternitz, P., The computer calculation of Lie point symmetries of large systems of differential equations, Comp. phys. comm., 66, 319-340, (1991) · Zbl 0875.65079
[32] Bluman, G.W.; Cole, J.D., The general similarity of the heat equation, J. math. mech., 18, 1025-1042, (1969) · Zbl 0187.03502
[33] Vorob’ev, E.M., Symmetries of compatibility conditions for systems of differential equations, Acta appl. math., 24, 1-24, (1991) · Zbl 0734.35002
[34] Levi, D.; Winternitz, P., Nonclassical symmetry reduction: example of the Boussinesq equation, J. phys. A: math. gen., 22, 2915-2924, (1989) · Zbl 0694.35159
[35] Clarkson, P.A.; Kruskal, M.D., New similarity solutions of the Boussinesq equation, J. math. phys., 30, 2201-2213, (1989) · Zbl 0698.35137
[36] Clarkson, P.A., Nonclassical symmetry reductions of the Boussinesq equation, Chaos, solitons and fractals, 5, 2261-2301, (1989) · Zbl 1080.35530
[37] Nucci, C.; Clarkson, P.A., The nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh-Nagumo equation, Phys. lett., A164, 49-56, (1992)
[38] Arrigo, D.; Broadbridge, P.; Hill, J.M., Nonclassical symmetry solutions and the methods of bluman-Cole and clarkson-Kruskal, J. math. phys., 34, 4692-4703, (1993) · Zbl 0784.35097
[39] Pucci, E., Similarity reductions of partial differential equations, J. phys. A: math. gen., 25, 2631-2640, (1992) · Zbl 0754.35148
[40] Galaktionov, V.A., On new exact blow-up solutions for nonlinear heat conduction equations with source and applications, Diff. int. eqns., 3, 863-874, (1990) · Zbl 0735.35074
[41] Galaktionov, V.A.; Dorodnytzin, V.A.; Elenin, G.G.; Kurdjumov, S.P.; Samarskii, A.A., Quasilinear heat equations with source: blow-up, localization, symmetry, exact solutions, asymptotics, structures, J. sov. math., 41, 1222-1292, (1988) · Zbl 0699.35134
[42] Ames, W.F., Optimal numerical algorithms, Appl. num. math., 10, 235-259, (1992) · Zbl 0758.65059
[43] Shokin, Yu.I., The method of differential approximation, (1983), Springer New York · Zbl 0511.65067
[44] Mansfield, E.L.; Clarkson, P.A., Applications of the differential algebra package diffgrob2 to classical symmetries of differential equations, J. symb. comp., (1995), (to appear)
[45] Clarkson, P.A.; Mansfield, E.L., Symmetry reductions and exact solutions of a class of nonlinear heat equations, Physica, D70, 250-288, (1994) · Zbl 0812.35017
[46] Mansfield, E.L.; Fackerell, E.D., Differential Gröbner bases, (1992), Macquarie University Sydney, Australia, (preprint)
[47] Reid, G.J., A triangularization algorithm which determines the Lie symmetry algebra of any system of pdes, J. phys. A: math. gen., 23, L853-L859, (1990) · Zbl 0724.35001
[48] Reid, G.J., 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) · Zbl 0768.35001
[49] Mansfield, E.L., Diffgrob2: A symbolic algebra package for analysing systems of PDE using Maple, (1993), ftpftp.ukc.ac.uk login: Anonymous, password: Your email address, directory: pub/maths/liz
[50] Clarkson, P.A.; Mansfield, E.L., On a shallow water wave equation, Nonlinearity, 7, 975-1000, (1994) · Zbl 0803.35111
[51] Clarkson, P.A.; Mansfield, E.L., Algorithms for the nonclassical method of symmetry reductions, SIAM J. appl. math., 54, 1693-1719, (1994) · Zbl 0823.58036
[52] Clarkson, P.A.; Mansfield, E.L., Symmetry reductions and exact solutions of shallow water wave equations, Acta. appl. math., 39, 245-276, (1995) · Zbl 0835.35006
[53] Olver, P.J., Applications of Lie groups to differential equations, () · Zbl 0591.73024
[54] Clarkson, P.A., New exact solutions for the Boussinesq equation, Europ. J. appl. math., 1, 279-300, (1990) · Zbl 0721.35074
[55] Lou, S.-Y., A note on the new similarity reductions of the Boussinesq equation, Phys. lett., A151, 133-135, (1990)
[56] Conte, R.; Fordy, A.P.; Pickering, A., A perturbative Painlevé approach to nonlinear differential equations, Physica, D69, 33-58, (1993) · Zbl 0794.34011
[57] Clarkson, P.A.; Fokas, A.S.; Ablowitz, M.J., Hodograph transformations on lineraizable partial differential equations, SIAM J. appl. math., 49, 1188-1209, (1989) · Zbl 0694.35005
[58] Ablowitz, M.J.; Clarkson, P.A., Solitons, nonlinear evolution equations and inverse scattering, () · Zbl 0762.35001
[59] Rosenau, P., Nonlinear dispersion and compact structures, Phys. rev. lett., 73, 1737-1741, (1993) · Zbl 0953.35501
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