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 Cauchy problem of the Camassa-Holm equation. (English) Zbl 1222.35169
Summary: The purpose of this paper is to investigate the Cauchy problem of the Camassa-Holm equation. By using the abstract method proposed and studied by T. Kato [in: Spectral Theor. Differ. Equat., Proc. Symp. Dundee 1974, Lect. Notes Math. 448, 25–70 (1975; Zbl 0315.35077)] and priori estimates, the existence and uniqueness of the global solution to the Cauchy problem of the Camassa-Holm equation in L p frame under certain conditions are obtained. In addition, the continuous dependence of the solution of this equation on the linear dispersive coefficient contained in the equation is obtained.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B03Existence, uniqueness, and regularity theory (fluid mechanics)
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
References:
[1]Camassa R. and Holm D., An integrable shallow water equation with peaked solutions, Phys. Rev. Lett., 1993, 71: 1661–1664 · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661
[2]Camassa R., Holm D. and Hyman J., A new integrable shallow water equation, Adv. Appl. Mech., 1994, 31: 1–33 · doi:10.1016/S0065-2156(08)70254-0
[3]Dai H. H., Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 1998, 127: 193–207 · Zbl 0910.73036 · doi:10.1007/BF01170373
[4]Fuchssteiner B., The Lie algebra structure of nonlinear evolution equations admitting infinite dimensional Abelian symmetry group, Prog. Theor. Phys., 1981, 65: 861–876 · Zbl 1074.58501 · doi:10.1143/PTP.65.861
[5]Fuchssteiner B., Some tricks for the symmetry–toolbox for nonlinear equations: generalizations of the Camassa-Holm equations, Physica D, 1996, 95: 229–243 · Zbl 0900.35345 · doi:10.1016/0167-2789(96)00048-6
[6]Schiff J., Zero curvature formulations of dual hierarchies, J. Math. Phys., 1996, 37: 1928–1938 · Zbl 0863.35093 · doi:10.1063/1.531486
[7]Schiff J., The Camassa-Holm equation: a loop group approach, Physica D, 1998, 121: 24–43 · Zbl 0943.37034 · doi:10.1016/S0167-2789(98)00099-2
[8]Constantin A. and Mckean H. P., A shallow water equation on the circles, Commun. Pure Appl. Math., 1999, 52: 949–982 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[9]Alber M. S., Camassa R., Holm D., and Marsden J. E., The geometry of peaked solitons and billiard solutions of a class of integrable PDEs, Lett. Math. Phys., 1994, 32: 137–151 · Zbl 0808.35124 · doi:10.1007/BF00739423
[10]Alber M. S., Camassa R., Fedorov Y. N., Holm D., and Marsden J. E., On billiard solutions of nonlinear PDEs, Phys. Lett. A, 1999, 264: 171–178 · Zbl 0944.37032 · doi:10.1016/S0375-9601(99)00784-7
[11]Cooper F. and Shepard H., Solitons in the Camassa-Holm shallow water equations, Phys. Lett. A, 1994, 194: 246–250 · Zbl 0961.76512 · doi:10.1016/0375-9601(94)91246-7
[12]Alber M. S., Camassa R., Holm D., and Marsden J. E., On the link between umbilic geodesics and soliton solutions of nonlinear PDEs, Proc. R. Soc. Lond. A, 1995, 450: 677–692 · Zbl 0835.35125 · doi:10.1098/rspa.1995.0107
[13]Li Y. A. and Olver P. J., Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I. Compactons and peakon, Discrete Cont. Dyn. Syst., 1997, 3: 419–432 · Zbl 0949.35118 · doi:10.3934/dcds.1997.3.419
[14]Li Y. A. and Olver P. J., Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system II. Complex analytic behavior and convergence to non-analytic solution, Discrete Cont. Dyn. Syst., 1998, 4: 159–191
[15]Constantin A., On the Cauchy problem for the periodic Camassa-Holm equation, J. Differ. Equ., 1997, 141: 218–223 · Zbl 0889.35022 · doi:10.1006/jdeq.1997.3333
[16]Constantin A. and Escher J., Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 1998, 47: 1527–1545
[17]Constantin A. and Escher J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 1998, 181: 229–243 · Zbl 0923.76025 · doi:10.1007/BF02392586
[18]Constantin A. and Escher J., Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1998, 26: 303–328
[19]Kwek K. H., Gao H. J., Zhang W. N., and Qu C.-C., An initial boundary problem of Camassa-Holm equation, J. Math. Phys., 2000, 41: 8279–8285 · Zbl 0972.35134 · doi:10.1063/1.1288498
[20]Zhang P. and Zheng Y. X., On oscillations of an asymptotic equation of a nonlinear variational wave equation, Asymptot. Anal., 1998, 18: 307–327
[21]Zhang P. and Zheng Y. X., On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation, Acta Math. Sin., Engl. Ser., 1999, 15: 115–130 · Zbl 0930.35142 · doi:10.1007/s10114-999-0063-7
[22]Zhang P. and Zheng Y. X., Existence and uniqueness of solutions of an asymptotic equation arising from a variational wave equation with general data, Arch. Ration. Mech. Anal., 2000, 155: 49–83 · doi:10.1007/s205-000-8002-2
[23]Xin Z. P. and Zhang P., On the weak solutions to a shallow water equation, Commun. Pure Appl. Math., 2000, 53: 1411–1433 · doi:10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5
[24]Zhang P. and Zheng Y. X., Singular and rarefactive solutions to a nonlinear variational wave equation, Chin. Ann. Math., Ser. B., 2001, 22: 159–170 · Zbl 0980.35137 · doi:10.1142/S0252959901000152
[25]Hunter J. H. and Zheng Y. X., On a completely integrable nonlinear hyperbolic variational equation, Physica D, 1994, 79: 361–386
[26]Hunter J. H. and Zheng Y. X., On a nonlinear hyperbolic variational equation, I. Global existence of weak solutions, Arch. Ration. Mech. Anal., 1995, 129: 305–335 · Zbl 0834.35085 · doi:10.1007/BF00379259
[27]Himonas A. A. and Misidek G., Well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differ. Equ., 2000, 161: 479–495 · Zbl 0945.35073 · doi:10.1006/jdeq.1999.3695
[28]Li Y. A. and Olver P. J., Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equ., 2000, 162: 27–63 · Zbl 0958.35119 · doi:10.1006/jdeq.1999.3683
[29]Camassa R., Characteristic variables for a completely integrable shallow water equation, Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years after NEEdS’79 (Gallipoli, 1999), River Edge, NJ: World Science Publishing, 2000: 65–74
[30]Kato T., Quasi-linear equations of evolution with applications to partial differential equations, spectral theory and differential equations, Lect. Notes Math., 1975(448): 25–70 (Berlin Heidelberg New York: Springer) · doi:10.1007/BFb0067080
[31]Yosida K., Functional Analysis, Berlin/New York: Springer-Verlag, 1966