Evolution of solitonless wave packets in the nonlinear Schrödinger equation and the Korteweg–de Vries equation with dissipative perturbations

The nonlinear Schrödinger equation “with attraction” and “with repulsion” (NSE($+$) and NSE($-$)) and the Korteweg–de Vries equation, and also the sine-Gordon equation are considered with small dissipative perturbations. For the case when the nonlinear and spatially inhomogeneous (diffusion) dissipations are dominant, explicit solutions are obtained that in the logarithmic approximation at sufficiently large times express the local amplitude and population number of the solitonless wave packet in terms of the initial data. For NSE(+), the singular case of a wave packet near the threshold of soliton creation is also considered.