Orbital stability of smooth solitons for the modified Camassa-Holm equation.(English)Zbl 1546.35155
Summary: The modified Camassa-Holm equation with cubic nonlinearity is completely integrable and is considered a model for the unidirectional propagation of shallow-water waves. The localized smooth-wave solution exists uniquely, up to translation, within a certain range of the linear dispersive parameter. By constructing conserved \(H^1\) and \(L^1\) quantities in terms of the momentum variable \(m\), this study demonstrates that the smooth soliton, when regarded as a solution of the initial-value problem for the modified Camassa-Holm equation, is orbitally stable to perturbations in the Sobolev space \(H^3\). Furthermore, the global well-posedness of the solution is established for certain initial data in \(H^s\) with \(s \geq 3\).
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q51 | Soliton equations |
| 35C08 | Soliton solutions |
| 35B35 | Stability in context of PDEs |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
modified Camassa-Holm equation;smooth solitary waves;orbital stability;global well-posednessCite
Full Text:DOI
References:
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