Nonlinear PDEs with modulated dispersion II: Korteweg–de Vries equation

We continue the study of various nonlinear PDEs under the effect of a time–inhomogeneous and irregular modulation of the dispersive term. In this paper we con sider the modulated versions of the 1d periodic or non-periodic Korteweg–de Vries (KdV) equatio n and of the modified KdV equation. For that we use a deterministic notion of ”irregularity” fo r the modulation and obtain local and global results similar to those valid without modulation. In so me cases the irregularity of the modulation improves the well-posedness theory of the equat ions. Our approach is based on estimates for the regularising effect of the modulated dispersion on the non-linear term using the theory of controlled paths and estimates stemming from Young’s th eory of integration.no