Ill_posedness for a two_component Novikov system in Besov space

Abstract

In this paper, we consider the Cauchy problem for a two-component Novikov system on the line. By specially constructed initial data $(\ddot{I}{}_0,u_0)$ in $B_{p,\infty }^{s-1}(\mathbb{R})\times B_{p,\infty }^s(\mathbb{R})$ with $s>\max \{2+\frac{1}{p},\frac{5}{2}\}$ and $1\le p\le \infty$, we show that any energy bounded solution starting from $(\ddot{I}{}_0,u_0)$ does not converge back to $(\ddot{I}{}_0,u_0)$ in the metric of $B_{p,\infty }^{s-1}(\mathbb{R})\times B_{p,\infty }^s(\mathbb{R})$ as time goes to zero, thus results in discontinuity of the data-to-solution map and ill-posedness.