Growth of Fourier--Lebesgue norms for mKdV

TL;DR

This work shows that conservation laws for the focusing mKdV on $\mathbb{R}$ do not control Fourier--Lebesgue norms $\mathcal{F}L^p_s$ when $p\neq 2$, by constructing solutions that start arbitrarily small in $\mathcal{F}L^p_s$ but become unbounded in finite or infinite time. The authors build a coordinated family of rescaled multisolitons $u_{N,\lambda}$ derived from $u_N(0,x)=(-1)^N N\mathrm{sech}(x)$, and analyze their long-time behavior using explicit multisoliton formulas, long-time asymptotics, and a delicate matrix-factorization argument to demonstrate soliton resolution in $L^p$ and, by extension, in $\mathcal{F}L^p_s$. A key contribution is showing that spectral broadening and interval-wise decoupling drive norm inflation: for $p>2$, time-averaging plus Bourgain $L^p$ bounds yields small average norms over long times, while for $1\le p<2$ a duality argument reveals large norms at specific times. The resulting inflation is then propagated to the full solution via time-translation and time-reversal symmetry, establishing the main theorem that the norm remains unbounded despite vanishing initial data in $\mathcal{F}L^p_s$.

Abstract

We demonstrate inflation of Fourier--Lebesgue norms for solutions to the focusing modified Korteweg--de Vries equation posed on the real line. For $p\neq 2$ 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 $\mathcal F L^p_s(\mathbb{R})$, but whose evolutions at later times $t_n$ diverge to infinity.

Theorems & Definitions (22)

• Theorem 1.1
• Proposition 2.1: Multisolitons
• Lemma 2.2: Cauchy Matrices
• Corollary 2.3
• proof
• proof : Proof of Proposition \ref{['P:psi']}
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3