zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. (English) Zbl 1169.76010
Summary: In recent years two nonlinear dispersive partial differential equations have attracted much attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin-Bona-Mahoney and Korteweg-de Vries equations. In particular, they accommodate wave breaking phenomena.

76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q53KdV-like (Korteweg-de Vries) equations
35Q35PDEs in connection with fluid mechanics
[1]Alinhac, S., Gérard, P.: Opérateurs Pseudo-différentiels et Théorème de Nash-Moser. Savoirs Actuels. InterEditions, Paris; Editions du Centre National de la Recherche Scientifique (CNRS), Meudon, 190 pp., 1991
[2]Angulo J., Bona J.L., Linares F., Scialom M.: Scaling, stability and singularities for nonlinear, dispersive wave equations: the critical case. Nonlinearity 15, 759–786 (2002) · Zbl 1034.35116 · doi:10.1088/0951-7715/15/3/315
[3]Alvarez-Samaniego B., Lannes D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171, 485–541 (2008) · Zbl 1131.76012 · doi:10.1007/s00222-007-0088-4
[4]Alvarez-Samaniego B., Lannes D.: A Nash–Moser theorem for singular evolution equations. Application to the Serre and Green–Naghdi equations. Indiana Univ. Math. J. 57, 97–131 (2008)
[5]Benjamin T.B., Bona J.L., Mahoney J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. A 227, 47–78 (1972) · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032
[6]Bressan A., Constantin A.: Global conservative solutions of the Camassa–Holm equation. Arch. Rat. Mech. Anal. 183, 215–239 (2007) · Zbl 1105.76013 · doi:10.1007/s00205-006-0010-z
[7]Camassa R., Holm D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993) · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661
[8]Constantin A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond. A 457, 953–970 (2001) · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[9]Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[10]Constantin A., Gerdjikov V.S., Ivanov R.I.: Inverse scattering transform for the Camassa–Holm equation. Inverse Probl. 22, 2197–2207 (2006) · Zbl 1105.37044 · doi:10.1088/0266-5611/22/6/017
[11]Constantin A., Strauss W.: Stability of the Camassa–Holm solitons. J. Nonlinear Sci 12, 415–422 (2002) · Zbl 1022.35053 · doi:10.1007/s00332-002-0517-x
[12]Craik A.D.D.: The origins of water wave theory. Ann. Rev. Fluid Mech. 36, 1–28 (2004) · Zbl 1076.76011 · doi:10.1146/annurev.fluid.36.050802.122118
[13]Degasperis A., Holm D., Hone A.: A new integrable equation with peakon solutions. Theor. Math. Phys. 133, 1461–1472 (2002)
[14]Degasperis A., Procesi M. Asymptotic integrability. Symmetry and Perturbation Theory (Eds. Degasperis A. and Gaeta G.) World Scientific, Singapore, 23–37, 1999
[15]Drazin P.G., Johnson R.S.: Solitons: An Introduction. Cambridge University Press, Cambridge (1992)
[16]Escher J., Liu Y., Yin Z.: Global weak solutions and blow-up structure for the Degasperis–Procesi equation. J. Funct. Anal. 241, 457–485 (2006) · Zbl 1126.35053 · doi:10.1016/j.jfa.2006.03.022
[17]Fokas A.S., Fuchssteiner B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D 4, 821–831 (1981)
[18]Fornberg B., Whitham G.B.: A numerical and theoretical study of certain nonlinear wave phenomena. Philos. Trans. R. Soc. Lond. A 289, 373–404 (1978) · Zbl 0384.65049 · doi:10.1098/rsta.1978.0064
[19]Green A.E., Naghdi P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246 (1976) · Zbl 0351.76014 · doi:10.1017/S0022112076002425
[20]Ivanov R.I.: On the integrability of a class of nonlinear dispersive wave equations. J. Nonlinear Math. Phys. 12, 462–468 (2005) · Zbl 1089.35522 · doi:10.2991/jnmp.2005.12.4.2
[21]Johnson R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (1997)
[22]Johnson R.S.: Camassa–Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 457, 63–82 (2002)
[23]Korteweg D.J., de Vries G.: On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag. 39, 422 (1895)
[24]Lenells J.: Conservation laws of the Camassa–Holm equation. J. Phys. A 38, 869–880 (2005) · Zbl 1076.35100 · doi:10.1088/0305-4470/38/4/007
[25]Li Y.A.: A shallow-water approximation to the full water wave problem. Commun. Pure Appl. Math. 59, 1225–1285 (2006) · Zbl 1169.76012 · doi:10.1002/cpa.20148
[26]Matsuno Y.: The N-soliton solution of the Degasperis–Procesi equation. Inverse Probl. 21, 2085–2101 (2005) · Zbl 1112.37072 · doi:10.1088/0266-5611/21/6/018
[27]McKean H.P.: Breakdown of the Camassa–Holm equation. Commun. Pure Appl. Math. 57, 416–418 (2004) · Zbl 1052.35130 · doi:10.1002/cpa.20003
[28]Molinet L.: On well-posedness results for the Camassa–Holm equation on the line: a survey. J. Nonlinear Math. Phys. 11, 521–533 (2004) · Zbl 1069.35076 · doi:10.2991/jnmp.2004.11.4.8
[29]Peregrine D.H.: Calculations of the development of an undular bore. J. Fluid Mech. 25, 321–330 (1966) · doi:10.1017/S0022112066001678
[30]Souganidis P.E., Strauss W.A.: Instability of a class of dispersive solitary waves. Proc. R. Soc. Edinburgh Sect. A 114, 195–212 (1990)
[31]Stoker J.J.: Water Waves. Interscience Publ., New York (1957)
[32]Tao T. Low-regularity global solutions to nonlinear dispersive equations. Surveys in Analysis and Operator Theory, Proc. Centre Math. Appl. Austral. Nat. Univ., 19–48, 2002
[33]Whitham G.B. Linear and Nonlinear Waves. Wiley, New York