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)
On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation. (English) Zbl 1080.76016
Summary: In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by R. Dullin, G. Gottwald and D. Holm [Phys. Rev. Lett. 87, No. 9, 4501–4504 (2001)]. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, the convergence of solutions to the DGH equation as α 2 0 is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained for the DGH equation. After giving the condition of existence of a peakon for the DGH equation, it turns out that the peakon is nonlinearly stable.
MSC:
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25Solitary waves (inviscid fluids)
35Q35PDEs in connection with fluid mechanics
35Q51Soliton-like equations
References:
[1]Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993) · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661
[2]Fuchssteiner, B., Fokas, A. S.: Symplectic structures, their Backlund transformation and hereditary symmetries. Physica D 4, 47–66 (1981) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[3]Johnson, R. S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid. Mech. 457, 63–82 (2002)
[4]Dullin, R., Gottwald, G., Holm, D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys.Rev. Lett. 87(9), 4501–4504 (2001)
[5]Camassa, R., Holm, D., Hyman, J.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994) · doi:10.1016/S0065-2156(08)70254-0
[6]Fisher, M., Schiff, J.: The Camassa Holm equations: conserved quantities and the initial value problem. Phy. Lett. A. 259(3), 371–376 (1999) · Zbl 0936.35166 · doi:10.1016/S0375-9601(99)00466-1
[7]Clarkson, P.A., Mansfield, E.L., Priestley, T.J.: Symmetries of a class of nonlinear third-order partial differential equations. Math. Comput. Modelling 25(8–9), 195–212 (1997)
[8]Kraenkel, R.A., Senthilvelan, M., Zenchuk, A.I.: On the integrable perturbations of the Camassa-Holm equation. J. Math. Phys. 41(5), 3160–3169 (2000) · Zbl 1052.37058 · doi:10.1063/1.533298
[9]Cooper, F., Shepard, H.: Solitons in the Camassa–Holm shallow water equation. Phys. Lett. A 194(4), 246–250 (1994) · Zbl 0961.76512 · doi:10.1016/0375-9601(94)91246-7
[10]Tian, L., Xu, G., Liu, Z.: The concave or convex peaked and smooth soliton solutions of Camassa-Holm equation. Appl. Math. Mech. 123(5), 557–567 (2002)
[11]Tian, L., Song, X.: New peaked solitary wave solutions of the generalized Camassa- Holm equation. Chaos, Solitons and Fractals 19(3), 621–637 (2004) · Zbl 1068.35123 · doi:10.1016/S0960-0779(03)00192-9
[12]Tian, L., Yin, J.: New compacton solutions and solitary solutions of fully nonlinear generalized Camassa-Holm equations. Chaos, Soliton and Fractals 20(4), 289–299 (2004) · Zbl 1046.35101 · doi:10.1016/S0960-0779(03)00382-5
[13]Constantin, A., Escher J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Sup. Pisa 26, 303–328 (1998)
[14]Constantin, A.: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann.Inst.Fourier (Grenoble) 50, 321–362 (2000)
[15]Constantin, A.: The Hamitonian structure of the Camassa-Holm equation. Exposition. Math. 15, 53–85 (1997)
[16]Constantin, A., McKean, H. P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52(8), 949–982 (1999) · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[17]Constantin, A., Strauss, W.A.: Stability of the Camassa-Holm solitons. J. Nonlinear Sci. 12, 415–422 (2002) · Zbl 1022.35053 · doi:10.1007/s00332-002-0517-x
[18]Constantin, A.: On the scattering problem for the Camassa-Holm equation. Proc. R.Soc . London A 457, 953–970 (2001) · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[19]Lenells, J.: The Scattering approach for the Camassa-Holm equation. J. Nonliear Math.Phys. 9(4), 389–393 (2002) · Zbl 1014.35082 · doi:10.2991/jnmp.2002.9.4.2
[20]Beals, R., Sattinger, D., Szmigielski, J.: Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. Math. 140, 190–206 (1998) · Zbl 0919.35118 · doi:10.1006/aima.1998.1768
[21]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
[22]Danchin, R.: A Few Remarks on the Camassa–Holm Equation. Differential and Integral Equations 14, 953–988 (2001)
[23]Guo, BL., Liu, ZR.: Peaked wave solutions of CH - γ equation. Sci China (Ser. A) 33(4), 325–337 (2003)
[24]Tang, M., Yang, C.: Extension on peaked wave solutions of CH - γ equation. Chaos, Solitons and Fractals 20, 815–825 (2004) · Zbl 1049.35153 · doi:10.1016/j.chaos.2003.09.018
[25]Bona, J., Smith, R.: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans.Royal Soc. London Series A 278, 555–601 (1975) · Zbl 0306.35027 · doi:10.1098/rsta.1975.0035
[26]Kato, T.: On the Cauchy problem for the (generalized) KdV equation. Studies in Applied Mathematics, Advances in Mathematics Supplementary. Vol.8, NewYork-London: Academic Press, 1983, pp. 93–128
[27]Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. J.Funct.Anal. 74, 160–197 (1987) · Zbl 0656.35122 · doi:10.1016/0022-1236(87)90044-9
[28]Gelfand, I.M., Dorfman, I.Ya.R.: Hamiltonian operators and algebraic structures related to them. Funct. Anal. Appl. 13, 248–262 (1979) · Zbl 0437.58009 · doi:10.1007/BF01078363
[29]Li, Y., Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Eq. 162, 27–63 (2000) · Zbl 0958.35119 · doi:10.1006/jdeq.1999.3683
[30]Rodriguez-Blanco, G.: On the Cauchy problem for the Camassa–Holm equation. Nonlinear Anal. 46, 309–327 ( 2001) · Zbl 0980.35150 · doi:10.1016/S0362-546X(01)00791-X
[31]Constantin, A., Escher, J.: Global weak solutions for a shallow water equation. Indiana Univ. Math. J 47(4), 1527–1545 (1998)
[32]Dunford, N., Schwartz, J.T.: Linear operators. Vol.2, New York: Wiley, 1988
[33]Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988) · Zbl 0671.35066 · doi:10.1002/cpa.3160410704
[34]Ablowitz, M., Clarkson, P.: Soliton, nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press, 1993
[35]Deift, P., Trubowits, E.: Inverse scattering on the line. Comm Pure Appl. Math. 32, 121–251 (1979) · Zbl 0395.34019 · doi:10.1002/cpa.3160320202
[36]Li, Y.A., Olver, P.J.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Eq. 162, 27–63 (2000) · Zbl 0958.35119 · doi:10.1006/jdeq.1999.3683