  Topological soliton is a nonperturbative excitation in commensurate density
wave states and connects degenerate ground states. In incommensurate density
wave states, ground states are continuously degenerate and topological soliton
is reckoned to be smoothly connected to the perturbative phason excitation. We
study the ultrafast nonequilibrium dynamics due to photoexcited electron-hole
pair in a one-dimensional chain with an incommensurate charge density wave
ground state. Time-resolved evolution reveals both perturbative excitation of
collective modes and nonperturbative topological phase transition due to
creating novel topological solitons, where the continuous complex order
parameter with amplitude and phase is essential. We identify the nontrivial
phase-winding solitons in the complex plane unique to this nonequilibrium state
and capture it by a low-energy effective model. The perturbative temporal gap
oscillation and the solitonic in-gap states enter the optical conductivity
absorption edge and the spectral density related to spectroscopic measurement,
providing concrete connections to real experiments.
