We study the propagations of optical self-similar solutions in a tapered graded-index nonlinear-fiber amplifier with an external source through asymmetric twin-core fiber amplifiers. Various types of exact self-similar solutions, including the W-shaped and U-shaped solutions, trigonometric function solutions, and periodic wave solutions are found. The results show that these different types of self-similar optical structures can be generated and effectively controlled by modulating the amplitude of the source. The influences of nonlinear tunneling effects on the propagation of optical pulses are investigated as well. The obtained results may have potential applications in a tapered graded-index nonlinear-fiber amplifier with an external source.