Well-posedness of generalized KdV and one-dimensional fourth-order derivative nonlinear Schr\"odinger equations for data with an infinite $L^2$ norm

TL;DR

This paper proves the global well-posedness of certain nonlinear dispersive equations for initial data that may have infinite $L^2$ norm, expanding understanding of solution behavior in rough data regimes.

Contribution

It establishes global well-posedness results for generalized KdV and fourth-order derivative NLS with initial data in modulation spaces that include some with infinite $L^2$ norm.

Findings

Global solutions exist for small rough data in specific modulation spaces.

The results include data with infinite $L^2$ norm.

The study advances the theory of dispersive equations with non-standard initial data.

Abstract

We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schr\"odinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of modulation spaces is shown, which contain some data with infinite $L^{2}$ norm.