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Equations of motion for nematic liquid crystals under time-dependent shear are derived. Soliton solutions are investigated. When the soliton velocity is large and the shear varies slowly in time, approximate analytic solutions for single solitons of the A and B types are found with use of multiplescale analysis. These perturbed solitons move with time-dependent velocities but are constant in shape and carry no tails. The velocity is proportional to the shear rate. Numerical calculations of the director equation of motion are performed and are in agreement with the analytic results. A recent experiment in which dark lines (under white light) excited by a periodically moving plate at one end of a nematic cell are observed is analyzed and interpreted according to our theory. Cfood agreement between theory and experiment is obtained.