The main purpose of this paper is to study the single traveling wave solutions of the fractional coupled nonlinear SchrÖdinger equation. By using the complete discriminant system method and computer algebra with symbolic computation, a series of new single traveling wave solutions are obtained, which include trigonometric function solutions, Jacobi elliptic function solutions, hyperbolic function solutions, solitary wave solutions and rational function solutions. In order to further explain the propagation of the fractional coupled nonlinear Schr\"{o}dinger equation in nonlinear optics, two-dimensional and three-dimensional graphs are drawn.