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Nonlinear phenomena have many important applications in several aspects of physics as well as other natural and applied sciences. Essentially all the fundamental equations of physics are nonlinear and, in general, such nonlinear equations are often very difficult to solve explicitly. In this report, we consider the mechanical calculation of explicit exact solutions for the nonlinear evolution equations (NLEEs). We mainly consider the calculation of such solutions as the solitary solutions, the soliton-like solutions, the multi-soliton solutions, the multi-soliton-like solutions, the rational form solutions and the formal periodic solutions, etc. We present constructive algorithms as well as their implementations.In Chapter 1, we briefly review some effective constructive algorithms for obtaining exact solitary solutions (or multi-soliton solutions) of NLEEs. We also present the MMP implementation of an ans(a)ts-based method developed by Yan [15].In Chapter 2, we give introduction to a further extended Tanh function method [16]and present its new applications. We show that this method can also be applied to linearize some NLEEs to obtain multi-soliton-like solutions. Furthermore, we obtain using this method an explicit B(a)cklund transformation for the Burgers equation.Based on the exact solutions and B(a)cklund transformation of Burgers equation obtained in Chapter 2, we present in Chapter 3 a new constructive algorithm for solving NLEEs. This algorithm enables us to generate the solutions of NLEEs by the solutions of Burgers equation. We also reveal by 3D-plots some interesting soliton-like wave motions possibly observable in the future experiments.