Analytical expressions for dark soliton solution of a Gross–Pitaevskii equation

In this paper, by improving the classical variational approximation (VA) theory and applying the method of asymptotic analysis (AA), we obtain two different analytical expressions for dark soliton solutions to the Gross–Pitaevskii equation with an optical lattice potential. By direct simulations of the underlying Gross–Pitaevskii equation, we demonstrate that the analytical results established by the VA method agree entirely with the numerical simulation solutions, but the AA results can only qualitatively capture the dark solitons in the whole range of the potential parameters.