Nonlinear Schrödinger equations "with attraction" and "with repulsion" (NSE(+) and NSE(-)) and the Korteweg-de Vries equation perturbed by small dissipative terms are considered. Near the linear instability threshold, where nonlinear and spatially nonuniform dissipations dominate, explicit solutions expressing the local amplitude and occupation numbers of a nonsoliton (dispersive) wavetrain in terms of arbitrary initial data are obtained, at sufficiently large times, in the logarithmic approximation. For NSE(+) the particular case of the wavetrain near the soliton-birth threshold is also considered. The influence of the dissipation on the evolution of a "quasi-classical" (strongly nonlinear) wavetrain is studied. It is demonstrated that for NSE(-) the dissipation can be taken into account within the framework of the eikonal approximation, while for the NSE(+) a perturbation theory based on the inverse scattering transform is relevant.