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The cubic-quintic nonlinear Schr\"{o}dinger equation emerges in models of light propagation in diverse optical media, such as non-Kerr crystals, chalcogenide glasses, organic materials, colloids, dye solutions and ferroelectrics. The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. By using the extended first integral method, we construct exact solutions of a fourth-order dispersive cubic-quintic nonlinear Schr$\ddot{o}$dinger equation and the variant Boussinesq system. The stability analysis for these solutions are discussed.

Soliton solutions of cubic-quintic nonlinear Schrödinger and variant boussinesq equations by the first integral method