Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose--Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger (GNLS) equations, which are two important models of nonlinear excitations of matter waves in Bose--Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schrodinger (NLS) equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two GNLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.