Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

TL;DR

This paper investigates the low regularity behavior of defocusing canonical equations, constructing special solutions to demonstrate ill-posedness and analyzing the endpoint regularity for KdV using a generalized Miura transform.

Contribution

It introduces classes of global solutions for defocusing equations that show ill-posedness and extends endpoint regularity results for KdV via a novel transform.

Findings

Demonstrates lack of uniform continuity in solution dependence for defocusing mKdV in L^2.

Constructs solutions at the KdV endpoint regularity H^{-3/4} with Lipschitz continuity.

Provides a generalized Miura transform linking mKdV and KdV endpoint theories.

Abstract

In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schr\"odinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities. In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces,in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the $L^2$ topology, despite the existence of global weak solutions at this regularity. Finally, we investigate the KdV equation at the endpoint regularity $H^{-3/4}$, and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval $[-T,T]$ and a locally Lipschitz continuous map taking the initial data in $H^{-3/4}$ to a distributional solution $u \in C^0 ([-T,T]; $H^{-3/4})$ which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.