Sharp well-posedness for a coupled system of mKdV-type equations

Abstract

We consider the initial value problem associated with a system consisting modified Korteweg–de Vries-type equations $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tv + \partial _x^3v + \partial _x(vw^2) =0,&{}\quad u(x,0)=\phi \partial _tw + \alpha \partial _x^3w + \partial _x(v^2w) =0,&{}\quad v(x,0)=\psi (x), \end{array}\right. } \end{aligned}$$and prove the local well-posedness results for given data in low regularity Sobolev spaces $$H^{s}({\mathbb {R}})\times H^{s}({\mathbb {R}})$$, $$s> -\,\frac{1}{2}$$, for $$0 1$$ as well.