  We elaborate on a new methodology, which starting with an integrable
evolution equation in one spatial dimension, constructs an integrable forced
version of this equation. The forcing consists of terms involving quadratic
products of certain eigenfunctions of the associated Lax pair. Remarkably, some
of these forced equations arise in the modelling of important physical
phenomena. The initial value problem of these equations can be formulated as a
Riemann-Hilbert problem, where the "jump matrix" has explicit x and t
dependence and can be computed in terms of the initial data. Thus, these
equations can be solved as efficiently as the nonlinear integrable equations
from which they are generated. Details are given for the forced versions of the
nonlinear Schrodinger.

Exactly Solvable and Integrable Systems