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In this paper, we identify criteria that guarantees the nonlinear orbital stability of a given periodic traveling wave solution within the b-family Camassa-Holm equation. These periodic waves exist as 3-parameter families (up to spatial translations) of smooth traveling wave solu- tions, and their stability criteria are expressed in terms of Jacobians of the conserved quantities with respect to these parameters. The stability criteria utilizes a general Hamiltonian structure which exists for every b > 1, and hence applies outside of the completely integrable cases (b = 2 and b = 3).
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Orbital stability of smooth solitary waves for the Degasperis-Procesi equation
The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space.
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- Award ID(s):
- 1815079
- PAR ID:
- 10382651
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society
- Volume:
- 151
- Issue:
- 763
- ISSN:
- 0002-9939
- Page Range / eLocation ID:
- 151 to 160
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/proc/16087
