The nonlinear Schr\"odinger equation with forcing involving products of eigenfunctions

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.