We present a double-mapping method (D-MM), a natural combination of a similarity with F-expansion methods, for obtaining general solvable nonlinear evolution equations. We focus on variable-coefficients complex Ginzburg-Landau equations (VCCGLE) with multi-body interactions. We show that it is easy by this method to find a large class of exact solutions of Gross-Pitaevskii and Gross-Pitaevskii-Ginzburg equations. We apply the D-MM to investigate the dynamics of Bose-Einstein condensation with two- and three-body interactions. As a surprising result, we obtained that it is very easy to use the built D-MM to obtain a large class of exact solutions of VCCGLE with two-body interactions via a generalized VCCGLE with two- and three-body interactions containing cubic-derivative terms. The results show that the proposed method is direct, concise, elementary, and effective, and can be a very effective and powerful mathematical tool for solving many other nonlinear evolution equations in physics.