On the Cauchy Problem for the New Shallow-water Models with Cubic Nonlinearity

This paper is devoted to the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa–Holm equation (also called Fokas–Olver–Rosenau–Qiao equation) and the Novikov equation as two special cases. On the...

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Bibliographic Details
Published in:Frontiers of Mathematics 2024-05, Vol.19 (3), p.435-455
Main Authors: Mi, Yongsheng, Guo, Boling
Format: Article
Language:English
Subjects:
Cauchy problems
Existence theorems
Fluid dynamics
Lower bounds
Mathematics
Mathematics and Statistics
Nonlinearity
Physical simulation
Research Article
Shallow water
Sobolev space
Solitary waves
Uniqueness theorems
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Summary:This paper is devoted to the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa–Holm equation (also called Fokas–Olver–Rosenau–Qiao equation) and the Novikov equation as two special cases. On the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey–Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map for the system. On the other hand, we prove the persistence properties in weighted spaces of the solution, provided that the initial potential satisfies a certain sign condition.
ISSN:2731-8648
1673-3452
2731-8656
1673-3576
DOI:10.1007/s11464-021-0319-9