Nonlinear Time-Dependent One-Dimensional Schrödinger Equation with Double-Well Potential

We consider time-dependent Schrödinger equations in one dimension with doublewell potential and an external nonlinear perturbation. If the initial state belongs to the eigenspace spanned by the eigenvectors associated to the two lowest eigenvalues, then, in the semiclassical limit, we show that the reduction of the time-dependent equation to a 2-mode equation gives the dominant term of the solution with a precise estimate of the error. By means of this stability result we are able to prove the absence of the beating motion for large enough nonlinearity.