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Symmetry reductions and exact solutions of shallow water wave equations. (English) Zbl 0835.35006

Summary: We study symmetry reductions and exact solutions of the shallow water wave (SWW) equation \[ u_{xxxt}+ \alpha u_x u_{xt}+ \beta u_t u_{xx}- u_{xt}- u_{xx}= 0,\tag{1} \] where \(\alpha\) and \(\beta\) are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation.
In this paper, a catalogue of symmetry reductions is obtained using the classical Lie method and the nonclassical method due to G. W. Bluman and J. D. Cole [J. Math. Mech. 18, 1025-1042 (1969; Zbl 0187.035)]. The classical Lie method yields symmetry reductions of (1) expressible in terms of the first, third and fifth Painlevé tarnscendents and Weierstrass elliptic functions. The nonclassical method yields a plethora of exact solutions of (1) with \(\alpha= \beta\) which possess a rich variety of qualitative behaviours. These solutions are all like a two-soliton solution for \(t< 0\) but differ radically for \(t> 0\) 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 bilinear method. Further, we show that there is an analogous nonlinear superposition of solutions for two \((2+ 1)\)-dimensional generalizations of the SWW equation (1) with \(\alpha= \beta\). This yields solutions expressible as the sum of two solutions of the Korteweg-de Vries equation.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
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.)
35Q35 PDEs in connection with fluid mechanics
58J70 Invariance and symmetry properties for PDEs on manifolds

Citations:

Zbl 0187.035
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References:

[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. · Zbl 0762.35001
[2] Ablowitz, M. J., Kaup, D. J., Newell, A. C., and Segur, H.:Stud. Appl. Math. 53 (1974), 249-315.
[3] Ablowitz, M. J., Ramani, A., and Segur, H.:Phys. Rev. Lett. 23 (1978), 333-338.
[4] Ablowitz, M. J., Ramani, A., and Segur, H.:J. Math. Phys. 21 (1980), 715-721. · Zbl 0445.35056 · doi:10.1063/1.524491
[5] Ablowitz, M. J., Schober, C., and Herbst, B. M.:Phys. Rev. Lett. 71 (1993), 2683-2686. · doi:10.1103/PhysRevLett.71.2683
[6] Ablowitz, M. J. and Villarroel, J.:Stud. Appl. Math. 85 (1991), 195-213.
[7] Anderson, R. L. and Ibragimov, N. H.:Lie-Bäcklund Transformations in Applications, SIAM, Philadelphia, 1979. · Zbl 0447.58001
[8] Benjamin, T. B., Bona, J. L., and Mahoney, J.:Phil. Trans. R. Soc. Land. Ser. A 272 (1972), 47-78. · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032
[9] Bluman, G. W. and Cole, J. D.:J. Math. Mech. 18 (1969), 1025-1042.
[10] Bluman, G. W. and Kumei, S.:Symmetries and Differential Equations, inAppl. Math. Sci., Vol. 81, Springer-Verlag, Berlin, 1989. · Zbl 0698.35001
[11] Bogoyavlenskii, O. I.:Math. USSR Izves. 34 (1990), 245-259. · Zbl 0712.35083 · doi:10.1070/IM1990v034n02ABEH000628
[12] Bogoyavlenskii, O. I.:Russ. Math. Surv. 45 (1990), 1-86. · Zbl 0754.35127 · doi:10.1070/RM1990v045n04ABEH002377
[13] Boiti, M., Leon, J. J-P, Manna, M., and Pempinelli, F.:Inverse Problems 2 (1986), 271-279. · Zbl 0617.35119 · doi:10.1088/0266-5611/2/3/005
[14] Buchberger, B.: in J. Rice (ed.),Mathematical Aspects of Scientific Software, Springer-Verlag, 1988, pp. 59-87.
[15] Champagne, B., Hereman, W., and Winternitz, P.:Comp. Phys. Comm. 66 (1991), 319-340. · Zbl 0875.65079 · doi:10.1016/0010-4655(91)90080-5
[16] Clarkson, P. A.: Nonclassical symmetry reductions for the Boussinesq equation, inChaos, Solitons and Fractals, 1994, to appear.
[17] Clarkson, P. A. and Kruskal, M. D.:J. Math. Phys. 30 (1989), 2201-2213. · Zbl 0698.35137 · doi:10.1063/1.528613
[18] Clarkson, P. A. and Mansfield, E. L.:Physica D 70 (1994), 250-288. · Zbl 0812.35017 · doi:10.1016/0167-2789(94)90017-5
[19] Clarkson, P. A. and Mansfield, E. L.:Nonlinearity 7 (1994), 975-1000. · Zbl 0803.35111 · doi:10.1088/0951-7715/7/3/012
[20] Clarkson, P. A. and Mansfield, E. L.: Algorithms for the nonclassical method of symmetry reductions,SIAM J. Appl. Math., 1994, to appear. · Zbl 0823.58036
[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. · Zbl 0803.35111
[22] Cole, J. D.:Quart. Appl. Math. 9 (1951), 225-236.
[23] Conte, R. and Musette, M.:J. Math. Phys. 32 (1991), 1450-1457. · Zbl 0734.35086 · doi:10.1063/1.529302
[24] Deift, P., Tomei, C., Trubowitz, E.:Comm. Pure Appl. Math. 35 (1982), 567-628. · Zbl 0489.35073 · doi:10.1002/cpa.3160350502
[25] Dorizzi, B., Grammaticos, B., Ramani, A., and Winternitz, P.:J. Math. Phys. 27 (1986), 2848-2852. · Zbl 0619.35086 · doi:10.1063/1.527260
[26] Espinosa, A. and Fujioka, J.:J. Phys. Soc. Japan 63 (1994), 1289-1294. · Zbl 0972.35514 · doi:10.1143/JPSJ.63.1289
[27] Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R.:Phys. Rev. Lett. 19 (1967), 1095-1097. · Zbl 1103.35360 · doi:10.1103/PhysRevLett.19.1095
[28] Gilson, C. R., Nimmo, J. J. C., and Willox, R.:Phys. Lett. 180A (1993), 337-345.
[29] Fushchich, W. I.:Ukrain. Math. J. 43 (1991), 1456-1470.
[30] Hereman, W.:Euromath Bull. 1(2) (1994), 45-79.
[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.
[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.
[33] Hirota, R. and Itô, M.:J. Phys. Soc. Japan 52 (1983), 744-748. · doi:10.1143/JPSJ.52.744
[34] Hirota, E. and Satsuma, J.:J. Phys. Soc. Japan 40 (1976), 611-612. · Zbl 1334.76016 · doi:10.1143/JPSJ.40.611
[35] Hopf, E.:Comm. Pure Appl. Math. 3 (1950), 201-250. · Zbl 0039.10403 · doi:10.1002/cpa.3160030302
[36] Ince, E. L.:Ordinary Differential Equations, Dover, New York, 1956. · Zbl 0063.02971
[37] Jimbo, M. and Miwa, T.:Publ. R.I.M.S. 19 (1983), 943-1001. · Zbl 0557.35091 · doi:10.2977/prims/1195182017
[38] Leble, S. B. and Ustinov, N. V.:Inverse Problems 210 (1994), 617-633. · Zbl 0806.35170 · doi:10.1088/0266-5611/10/3/008
[39] Levi, D. and Winternitz, P.:J. Phys. A: Math. Gen. 22 (1989), 2915-2924. · Zbl 0694.35159 · doi:10.1088/0305-4470/22/15/010
[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.
[41] Mansfield, E. L. and Fackerell, E. D.: Differential Gröbner Bases, Preprint 92/108, Macquarie University, Sydney, Australia, 1992.
[42] McLeod, J. B. and Olver, P. J.:SIAM J. Math. Anal. 14 (1983), 488-506. · Zbl 0518.35075 · doi:10.1137/0514042
[43] Musette, M., Lambert, F., and Decuyper, J. C.:J. Phys. A: Math. Gen. 20 (1987), 6223-6235. · Zbl 0657.35115 · doi:10.1088/0305-4470/20/18/022
[44] Olver, P. J.:Applications of Lie Groups to Differential Equations, 2nd edn, Graduate Texts Math., Vol. 107, Springer-Verlag, New York, 1993. · Zbl 0785.58003
[45] Olver, P. J. and Rosenau, P.:Phys. Lett. 114A (1986), 107-112.
[46] Olver, P. J. and Rosenau, P.:SIAM J. Appl. Math. 47 (1987), 263-275. · Zbl 0621.35007 · doi:10.1137/0147018
[47] Peregrine, H.:J. Fluid Mech. 25 (1966), 321-330. · doi:10.1017/S0022112066001678
[48] Reid, G. J.:J. Phys. A: Math. Gen. 23 (1990), L853-L859. · Zbl 0724.35001 · doi:10.1088/0305-4470/23/17/001
[49] Reid, G. J.:Europ. J. Appl. Math. 2 (1991), 293-318. · Zbl 0768.35001 · doi:10.1017/S0956792500000577
[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.
[51] Schwarz, F.:Computing 49 (1992), 95-115. · Zbl 0759.68042 · doi:10.1007/BF02238743
[52] Tamizhmani, K. M. and Punithavathi, P.:J. Phys. Soc. Japan 59 (1990), 843-847. · doi:10.1143/JPSJ.59.843
[53] Topunov, V. L.:Acta Appl. Math. 16 (1989), 191-206. · Zbl 0703.35005 · doi:10.1007/BF00046572
[54] Weiss, J.:J. Math. Phys. 24 (1983), 1405-1413. · Zbl 0531.35069 · doi:10.1063/1.525875
[55] Weiss, J., Tabor, M., and Carnevale, G.:J. Math. Phys. 24 (1983), 522-526. · Zbl 0514.35083 · doi:10.1063/1.525721
[56] Whittaker, E. E. and Watson, G. M.:Modern Analysis, 4th edn, C.U.P., Cambridge, 1927. · JFM 53.0180.04
[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. · Zbl 0830.35004
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