Growth of Fourier--Lebesgue norms for mKdV

Abstract

We demonstrate inflation of Fourier--Lebesgue norms for solutions to the focusing modified Korteweg--de Vries equation posed on the real line. For $p\ne 2$ and all $s\in \mathbb{R}$, we construct a sequence of solutions $u_n$ whose initial data $u_n(0)$ converges to zero in the Fourier--Lebesgue spaces $\mathcal{F}{L}_s^p(\mathbb{R})$, but whose evolutions at later times $t_n$ diverge to infinity.