Abstract: The coupled nonlinear Schr\"{o}dinger equations have important applications in many areas of physics, such as quantum physics, nonlinear optics, crystal physics, Bose-Einstein condensates, water wave dynamics and so on. In this paper, a local discontinuous Petrov-Galerkin method is developed. The coupled nonlinear Schr\"{o}dinger equations are firstly rewritten as a first order differential system. The spacial discretization is accomplished by the discontinuous Petrov-Galerkin method. The third order total variation diminishing Runge-Kutta method is used to finish the temporal discretization. The numerical experiments shown that the algori-thm can reach its optimal convergence order for both linear and quadratic elements. The mass, momentum and energy conservation quantities are evaluated by some numerical examples. The algorithm can simulate the single soliton propagation, double soliton collision and three soliton collision phenomena very well. In addition, the algorithm can simulate the complex wave interaction or propagation in a long time interval. Furthermore, the scheme can simulate the soliton transmission and soliton creation phenomena.

Key words: local discontinuous Petrov-Galerkin method, coupled nonlinear Schr\"{o}dinger equations, soliton collision, conservation quantity