  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.
