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The current work studies the bifurcation and the classification of single traveling wave solutions of the coupled version of Radhakrishnan-Kundu-Lakshmanan equation that usually describes the dynamics of optical pulses in fiber Bragg gratings, which is also described by a family of nonlinear Schrödinger equations with cubic nonlinear terms. The solutions of the hyperbolic functions, the rational functions, the trigonometric functions and the Jacobian functions are retrieved by using the complete discrimination system of polynomial. By selecting appropriate parameters, phase portraits, two-dimension graphics and three-dimension graphics of the obtained solutions are drawn.