Parameters controlling matter waves in the one-dimensional generalized Gross-Pitaevskii equation with the varying potential and source

In this paper, the one-dimensional generalized Gross-Pitaevskii (GP) equation with the varying external potential and source which plays an important role in Bose-Einstein condensates is investigated. For the different types of external potentials, based on some powerful transformation methods with different types of expressions in Jacobi elliptic functions or polynomial functions, many types of matter-wave solutions with traveling-wave and non-trivial phases are found for the generalized GP equation with the external potential and source. These matter-wave solutions contain the doubly periodic wave solutions, solitary wave solutions, periodic wave solutions and the bounded rational solutions. Moreover, we analyze the properties of some obtained solutions and the corresponding potentials on the basis of the different parameters including the source amplitude. These results may be useful to explain some nonlinear wave phenomena in Bose-Einstein condensates, and nonlinear optics.