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