Enhanced existence time of solutions to the fractional Korteweg de Vries equation

Abstract

We consider the fractional Korteweg–de Vries equation ut+ uux−|D| αux = 0 in the range of −1 < α < 1, α 6= 0. Using basic Fourier techniques in combination with the modified energy method we extend the existence time of classical solutions with initial data of size ε from 1 ε to a time scale of 1 ε2 . This analysis, which is carried out in Sobolev space HN (R), N ≥ 3, answers positively a question posed by Linares, Pilod and Saut in [SIAM J. Math. Anal., 46 (2014), pp. 1505--1537].