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

Abstract

We consider the fractional Korteweg-de Vries equation $u_t+u{u}_x-|D{|}^{\alpha }{u}_x=0$ in the range of $-1<\alpha <1$ , $\alpha \ne 0$. Using basic Fourier techniques in combination with the modified energy method we extend the existence time of classical solutions with initial data of size $\epsilon$ from $\frac{1}{\epsilon }$ to a time scale of $\frac{1}{{\epsilon }^2}$. This analysis, which is carried out in Sobolev space $H^N(\mathbb{R})$, $N\ge 3$, answers positively a question posed by Linares, Pilod and Saut (SIAM J. Math. Anal. 46 (2014), no. 2, 1505-1537).