zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 +αu x u xt +βu t u xx -u xt -u xx =0,(1)

where α and β 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 α=β 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 α=β. This yields solutions expressible as the sum of two solutions of the Korteweg-de Vries equation.

MSC:
35A30Geometric theory for PDE, characteristics, transformations
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q35PDEs in connection with fluid mechanics
58J70Invariance and symmetry properties
Software:
SYMMGRP
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.
[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.
[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.
[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.
[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.
[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. · 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.
[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.
[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.
[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.