We study the global in time existence of small solutions to the Cauchy problem for the subcritical modified Korteweg-de Vries (KdV) equation Formula: see text We suppose that Formula: see text . We remark that Formula: see text means that equation is subcritical in the sense of the large time asymptotic behavior of solutions. We assume that the initial data have an analytic extension on the sector and are small. Then we find the large time asymptotics of the solutions.
Hayashi et al. (Wed,) studied this question.