Enhanced Existence Time of Solutions to the Fractional Korteweg--de Vries Equation

Source abstract

We consider the fractional Korteweg--de Vries equation $u_t + u u_x - |D|^\alpha u_x = 0$ in the range of $-1<\alpha<1$, $\alpha\neq0$. Using basic Fourier techniques in combination with the modified energy method, we extend the existence time of classical solutions with initial data of size $\varepsilon$ from $\frac1ǎrepsilon$ to a time scale of $\frac1ǎrepsilon^2$. This analysis, which is carried out in Sobolev space \(H^N(\mathbbR)\), $N \geq 3$, answers positively a question posed by Linares, Pilod, and Saut in [ SIAM J. Math. Anal., 46 (2014), pp. 1505--1537].