zbMATH — the first resource for mathematics

Stability of peakons. (English) Zbl 1049.35149
The authors prove the orbital stability of peakon solutions of the Camassa-Holm equation in the \(H^1\) norm under the assumption of their existence. The proof is based on estimates for two conservation laws of the equation. Some comments on known results concerning the well-posedness of the initial value problem and the existence of solutions conclude the article.

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35B35 Stability in context of PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI
[1] Arnold, Ann Inst Fourier 16 pp 319– (1966) · Zbl 0148.45301 · doi:10.5802/aif.233
[2] Benjamin, J Fluid Mech 125 pp 137– (1982)
[3] Camassa, Phys Rev Lett 71 pp 1661– (1993)
[4] Camassa, Adv Appl Mech 31 pp 1– (1994)
[5] Constantin, Ann Inst Fourier;
[6] Constantin, Ann Scuola Norm Sup Pisa Cl Sci (4) 26 pp 303– (1998)
[7] Constantin, Acta Math 181 pp 229– (1998)
[8] Constantin, Comm Pure Appl Math 52 pp 949– (1999)
[9] ; Orbital stability of the solitary waves for a shallow water equation. Preprint, 1998.
[10] Dai, Acta Mech 127 pp 193– (1998)
[11] Fuchssteiner, Phys D 4 pp 47– (1981)
[12] Kouranbaeva, J Math Phys 40 pp 857– (1999)
[13] Li, J Differential Equations
[14] McKean, Asian J Math
[15] Misiołek, J Geom Phys 24 pp 203– (1998)
[16] On the Cauchy problem for the Camassa-Holm equation. IMPA preprint, 155. Institute of Pure and Applied Mathematics, Rio de Janeiro, 1998.
[17] Nonlinear strain waves in elastic waveguides. Nonlinear waves in solids (Udine, 1993), 349-382, CISM Courses and Lectures, 341. Springer, Vienna, 1994. · Zbl 0806.73018 · doi:10.1007/978-3-7091-2444-4_6
[18] Toland, Proc Roy Soc London Ser A 363 pp 469– (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.