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Employing a generalized resistive-capacitive shunted junction model for Josephson junctions (JJs), the nonlinear wave propagation in the series-connected discrete Josephson transmission line (DJTL) is investigated. A DJTL consists of a finite number of unit cells, each including a segment of superconducting transmission line with a single array stack, or generally a block including an N identical lumped JJ element. As the governing nonlinear wave propagation is a system of nonlinear partial differential equations with mixed boundary conditions, the method of the finite difference time domain is used to solve the equations. By this numerical technique, the behavior of wave propagation along the DJTL can be monitored in time and space domains. Cutoff propagation, dispersive behavior, and shock-wave formation through the DJTL is addressed in this paper.