A higher dispersion KdV equation on the half-line

TL;DR

This paper extends the Fokas Unified Transform Method to analyze the m-th order dispersion KdV equation on the half-line with rough data, establishing well-posedness in Bourgain spaces, a significant generalization from previous smooth data results.

Contribution

It advances the Fokas method to handle dispersive equations with rough data in Bourgain spaces on the half-line, broadening the scope of well-posedness analysis.

Findings

Successfully established well-posedness for rough initial and boundary data.

Extended the Fokas method to Bourgain spaces framework.

Provided estimates for the linear and nonlinear IBVPs in this setting.

Abstract

The initial-boundary value problem (ibvp) for the $m$-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method (UTM). Thus, this work advances the implementation of the Fokas method, used earlier for the KdV on the half-line with smooth data and solution in the classical Hadamard space, consisting of function that are continuous in time and Sobolev in the spatial variable, to the more general Bourgain spaces framework of dispersive equations with rough data on the half-line. The spaces needed and the estimates required arise at the linear level and in particular in the estimation of the linear pure ibvp, which has forcing and initial data zero but non-zero boundary data. Using the iteration map defined by the Fokas solution formula of the forced linear ibvp in combination with the bilinear estimates in modified Bourgain spaces introduced by this map, well-posedness of the nonlinear ibvp is established for rough initial and boundary data belonging in Sobolev spaces of the same optimal regularity as in the case of the initial value problem for this equation on the whole line.