| Summary: | In this manuscript, we focus on the application of recently developed analytical scheme, namely, the rational sine-Gordon expansion method (SGEM). Some new exact solutions of Riemann wave system and the nonlinear Gross−Pitaevskii equation (GPE) by using this method are extracted. This method is based on the general properties of the SGEM which uses the fundamental properties of trigonometric functions. Many novel analytical solutions such as dark, bright, mixed dark–bright, hyperbolic, and periodic wave solutions are successfully extracted. Physical meanings of solutions are simulated by the various figures in 2D and 3D along with the contour graphs. Strain conditions of the existence are also reported in detail. Finally, modulation instability analysis of the nonlinear GPE is studied in detail.
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