Stability in the energy space of the sum of \(N\) peakons for the Degasperis-Procesi equation. (English) Zbl 1331.35043
Summary: The Degasperis-Procesi equation possesses well-known peaked solitary waves that are called peakons. Their stability has been established by Z. Lin and Y. Liu [Commun. Pure Appl. Math. 62, No. 1, 125–146 (2009; Zbl 1165.35045)]. In this paper, we localize the proof (in some suitable sense detailed in Section 3) of the stability of a single peakon. Thanks to this, we extend the result of stability to the sum of \(N\) peakons traveling to the right with respective speeds \(c_1, \ldots, c_N\), such that the difference between consecutive locations of peakons is large enough.
MSC:
| 35B35 | Stability in context of PDEs |
| 35G25 | Initial value problems for nonlinear higher-order PDEs |
| 35C08 | Soliton solutions |
Citations:
Zbl 1165.35045References:
| [1] | Degasperis, A.; Kholm, D. D.; Khon, A. N.I., A new integrable equation with peakon solutions, Teoret. Mat. Fiz., 133, 2, 170-183 (2002) |
| [2] | El Dika, Khaled; Molinet, Luc, Stability of multi antipeakon-peakons profile, Discrete Contin. Dyn. Syst. Ser. B, 12, 3, 561-577 (2009) · Zbl 1180.35453 |
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| [5] | Lin, Zhiwu; Liu, Yue, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62, 1, 125-146 (2009) · Zbl 1165.35045 |
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| [7] | Martel, Yvan; Merle, Frank; Tsai, Tai-Peng, Stability and asymptotic stability in the energy space of the sum of \(N\) solitons for subcritical gKdV equations, Comm. Math. Phys., 231, 2, 347-373 (2002) · Zbl 1017.35098 |
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