With the aid of a discrete nonlinear Schrödinger equation (NLSE), the nonlinear interaction among the periodically placed ultrashort pulses is analyzed. If the amplitudes of these pulses are chosen to be secant-hyperbolic, it is found that they propagate without exchanging energy and hence the envelope of the peak of the short pulses is termed the discrete soliton in analogy with its counterpart in the spatial domain. In addition, we develop the concept of discrete chirp transform (DChT) and its inverse, and show that the weights of the pulses can be extracted from the field envelope using the discrete chirp transform (DChT). The computational cost of evaluating the output of a linear dispersive fiber using DChT approach is nearly half of the conventional frequency domain approach based on fast Fourier transform (FFT). We found that an isolated pump sinc pulse is not stable and it generates temporally separated sinc pulses if the dispersion of the transmission fiber is anomalous. By choosing a proper time separation between signal pulse and pump pulse, it is possible to amplify the signal pulse. The nonlinear interaction between signal pulse and pump pulse generates an idler pulse that is a phase-conjugated copy of the signal pulse. Hence, this result could have potential applications for time domain optical amplification and phase-conjugation.