On the Cauchy problem for the two-component Novikov system with peakons

ABSTRACT

Considered herein is the Cauchy problem of the two-component Novikov system with peakons. Based on the local well-posedness results for this problem, it is shown that the solution map $z_0\mapsto z(t)$ of this problem on the line is not uniformly continuous in Besov spaces $(B_{2,r}^s(\mathbb{R}){)}^2$ with $s>3/2,2\le r\le \mathrm{\infty }$ through the method of approximate solutions. Next, in the periodic case, the non-uniform continuity of the solution map in Besov spaces $(B_{p,r}^s(\mathbb{T}){)}^2$ and Hölder spaces $(C^q(\mathbb{T}){)}^2$ with $s>max\{3/2,1+1/p\},q>3/2,q\in {\mathbb{R}}^{+}\setminus \mathbb{N},1\le p,r\le \mathrm{\infty }$ is discussed. Finally, the Hölder continuity of this solution map in Sobolev spaces is investigated.

KEYWORDS:

• The two-component Novikov system with peakons
• non-uniformly continuous dependence
• Hölder continuity

MSC (2020):

• 35B30
• 35G25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Fundamental Research Program of Shanxi Province [Program No. 20210302124259].