Dynamics of Nonautonomous Matter Waves in a Waveguide With an External Source

We explore explicit exact Lorentzian-type matter wave solutions of nonautonoumous nonlinear Schrödinger equation (NLSE) with an external source and a harmonic trap. We explicate the mechanism to control the dynamical behaviour of these waves of Bose-Einstein condensate (BEC) for different sets of nonlinearity and trapping modulations. Our results raise the possibility of some experiments and potential applications related to BEC in the presence of an inhomogeneous external source.