In this study, the complex modified Korteweg-de Vries (CMKdV) equation is solved numerically by three different split-step Fourier schemes. The main difference among the three schemes is in the order of the splitting approximation used to factorize the exponential operator. The space variable is discretized by means of a Fourier method for both linear and nonlinear subproblems. A fourth-order Runge-Kutta scheme is used for the time integration of the nonlinear subproblem. Classical problems concerning the motion of a single solitary wave with a constant polarization angle are used to compare the schemes in terms of the accuracy and the computational cost. Furthermore, the interaction of two solitary waves with orthogonal polarizations is investigated and particular attention is paid to the conserved quantities as an indicator of the accuracy. Numerical tests show that the split-step Fourier method provides highly accurate solutions for the CMKdV equation. (C) 2003 Elsevier Science Ltd. All rights reserved.