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)
Numerical study of traveling-wave solutions for the Camassa--Holm equation. (English) Zbl 1136.35448
Summary: We explore numerically different aspects of periodic traveling-wave solutions of the Camassa-Holm equation. In particular, the time evolution of some recently found new traveling-wave solutions and the interaction of peaked and cusped waves is studied.

35Q53KdV-like (Korteweg-de Vries) equations
35B10Periodic solutions of PDE
65M70Spectral, collocation and related methods (IVP of PDE)
Full Text: DOI
[1] Beals, R.; Sattinger, D.; Szmigielski, J.: Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. math. 40, 190-206 (1998) · Zbl 0919.35118
[2] Camassa, R.; Holm, D.: An integrable shallow water equation with peaked solitons. Phys. rev. Lett. 71, 1661-1664 (1993) · Zbl 0972.35521
[3] Constantin, A.: On the Cauchy problem for the periodic Camassa-Holm equation. J. different. Equat. 141, 218-235 (1997) · Zbl 0889.35022
[4] Constantin, A.: On the inverse spectral problem for the Camassa-Holm equation. J. funct. Anal. 155, 352-363 (1998) · Zbl 0907.35009
[5] Constantin, A.: On the scattering problem for the Camassa-Holm equation. Proc. R. Soc. London 457, 953-970 (2001) · Zbl 0999.35065
[6] Constantin, A.; Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta math. 181, 229-243 (1998) · Zbl 0923.76025
[7] Constantin, A.; Escher, J.: Global existence and blow-up for a shallow water equation. Ann. sci. Norm. sup. Pisa 26, 303-328 (1998) · Zbl 0918.35005
[8] Constantin, A.; Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. pure appl. Math. 51, 475-504 (1998) · Zbl 0934.35153
[9] Constantin, A.; Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233, 75-91 (2000) · Zbl 0954.35136
[10] Constantin, A.; Kolev, B.: On the geometric approach to the motion of inertial mechanical systems. J. phys. A 35, R51-R79 (2002) · Zbl 1039.37068
[11] Constantin, A.; Kolev, B.: Geodesic flow on the diffeomorphism group of the circle. Comment. math. Helv. 78, 787-804 (2003) · Zbl 1037.37032
[12] Constantin, A.; Mckean, H.: A shallow water equation on the circle. Commun. pure appl. Math. 52, 949-982 (1999) · Zbl 0940.35177
[13] Constantin, A.; Molinet, L.: Global weak solutions for a shallow water equation. Commun. math. Phys. 211, 45-61 (2000) · Zbl 1002.35101
[14] Constantin, A.; Molinet, L.: Orbital stability of solitary waves for a shallow water equation. Phys. D 157, 75-89 (2001) · Zbl 0984.35139
[15] Constantin, A.; Strauss, W.: Stability of peakons. Commun. pure appl. Math. 53, 603-610 (2000) · Zbl 1049.35149
[16] Constantin, A.; Strauss, W.: Stability of the Camassa-Holm solitons. J. nonlinear sci. 12, 415-422 (2002) · Zbl 1022.35053
[17] Danchin, R.: A few remarks on the Camassa-Holm equation. Differ. integral equat. 14, 953-988 (2001) · Zbl 1161.35329
[18] Ferreira, M.; Kraenkel, R.; Zenchuk, A.: Soliton-cuspon interaction for the Camassa-Holm equation. J. phys. A: math. Gen. 32, 8665-8670 (1999) · Zbl 0946.35088
[19] Fuchssteiner, B.; Fokas, A.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Phys. D 4, 47-66 (1981) · Zbl 1194.37114
[20] Johnson, R.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. fluid mech. 455, 63-82 (2002) · Zbl 1037.76006
[21] Johnson, R.: On solutions of the Camassa-Holm equation. Proc. R. Soc. London A 459, 1687-1708 (2003) · Zbl 1039.76006
[22] Kraenkel, R.; Zenchuk, A.: Camassa-Holm equation: transformation to deformed sinh-Gordon equations, cuspon and soliton solutions. J. phys. A: math. Gen. 32, 4733-4747 (1999) · Zbl 0941.35094
[23] Lenells, J.: The scattering approach for the Camassa-Holm equation. J. nonlinear math. Phys. 9, 389-393 (2002) · Zbl 1014.35082
[24] Lenells, J.: Stability of periodic peakons. Int. math. Res. notices 10, 485-499 (2004) · Zbl 1075.35052
[25] Lenells, J.: A variational approach to the stability of periodic peakons. J. nonlinear math. Phys. 11, 151-163 (2004) · Zbl 1067.35076
[26] Lenells J, Traveling wave solutions of the Camassa-Holm equation. J Diff Eq, submitted · Zbl 1082.35127
[27] Li, Y.; Olver, P.: Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system. I. compactons and peakons. Discrete cont. Dynam. syst. 3, 419-432 (1997) · Zbl 0949.35118
[28] Li, Y.; Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. differ. Eq. 162, 27-63 (2000) · Zbl 0958.35119
[29] Misiolek, G.: A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. geom. Phys. 24, 203-208 (1998) · Zbl 0901.58022
[30] Mckean, H.: Breakdown of a shallow water equation. Asian J. Math 2, 867-874 (1998) · Zbl 0959.35140