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

TL;DR

This paper extends the existence time of solutions to the fractional Korteweg-de Vries equation using Fourier and energy methods, showing solutions persist longer than previously known for small initial data.

Contribution

It introduces a novel analysis that increases the known existence time from 1/ε to 1/ε^2 for solutions in Sobolev space, answering a prior open question.

Findings

Extended solution existence time from 1/ε to 1/ε^2

Applied Fourier and modified energy techniques

Confirmed positive answer to an open problem

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 $\frac{1}{\varepsilon}$ to a time scale of $\frac{1}{\varepsilon^2}$. This analysis, which is carried out in Sobolev space $H^N(\mathbb{R})$, $N \geq 3$, answers positively a question posed by Linares, Pilod and Saut (SIAM J. Math. Anal. 46 (2014), no. 2, 1505-1537).