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Novel (G'/G)-expansion method is one of the powerful and competent methods for establishing exact traveling wave solutions to NLEEs. Exact solutions of NLEEs play a key role to reveal the internal structure of intricate physical phenomena. We apply the novel (G'/G)-expansion method to solve several systems of NLEEs and construct traveling wave solutions expressed in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. The obtained solutions might be imperative and significant for the explanation of practical problems. The solutions include soliton, singular soliton, cuspon, peakon, compacton, bell-shape solitary wave solution, kink, singular kink, periodic and singular periodic solution. The performance of this method is reliable, compatible, and gives us more new exact solutions than the existing methods. We highlight the power of the novel ( G'/G)-expansion method in providing generalized solitary wave solutions of different complex physical phenomena. The method is direct, concise and simple to implement compared with other existing methods. This method presents a wider applicability for handling nonlinear wave equations.