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 forced version of this equation. The forcing consists terms involving quadratic products certain eigenfunctions the associated Lax pair. Remarkably, some these equations arise modelling important physical phenomena. initial value problem can be formulated as Riemann-Hilbert problem, where "jump matrix" has explicit x and t dependence computed data. Thus, solved efficiently nonlinear from they are generated. Details given for versions Schrodinger.