Nontrivial Galilean-like invariance of the generalized higher-order nonlinear Schrödinger equation model with gravitation-like potential and the soliton analogies of the cosmic dark energy and antigravitation effects

We reveal remarkable, but obviously only formal analogies and parallels between the soliton behavior in external gravitational-like potentials and the cosmic dark energy and antigravitation effects. The introduced generalized and exactly integrable higher-order nonlinear Schrödinger equation model for the wave function of a soliton in the attractive gravitational-like potential implies the existence of many fundamental nonlinear phenomena in completely different systems, independent of the physical origin of the nonlinearity and the higher-order linear and nonlinear dispersion. We show that the proposed model is governed by nontrivial Galilean-like invariance and, under the exact integrability conditions given by the Lax pair, demonstrates important formal nonlinear-optical analogies with enigmatic cosmic dark energy and antigravitation effects. Guided by interesting, but only formal analogy between the linear Doppler red shift and the nonlinear soliton self-frequency red shift effects, we reveal nontrivial features of the soliton propagation scenario in the attractive gravitation-like potential. Namely, the soliton attraction in the gravitation-like potential is changed to strong repulsion similar to the Newtonian particle dynamics in the repulsive antigravitational field. The wave-function red-shift effect (which is accumulated linearly with the propagation distance) is conserved under the transformation from attractive to repulsive scenarios in the ordinary space-time domain.