Well-posedness and ill-posedness of the Cauchy problem for the generalized Thirring model

Abstract

We consider the Cauchy problem for the generalized Thirring model $(\partial _t \pm \partial _x ) U_{\pm} = i |U_{\pm}|^k |U_{\mp}|^{m-k} U_{\pm}$ in one spatial dimension which was introduced in [4]. Several results on well-posedness and ill-posedness have been obtained. Since the nonlinearity is not smooth if $k$ or $m$ is odd, an upper bound of $s$ to be well-posed appears. We prove that the upper bound is essential. More precisely, we show ill-posedness in $H^s(\mathbb{R})$ for sufficiently large $s$ which is a novel feature of this paper.