Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential

Properties of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation (ICQNLSE) with an external potential are studied. When it is associated with the homogeneous CQNLSE, a general condition exists linking the external potential and inhomogeneous cubic and quintic (ICQ) nonlinearities. Besides for the nonpresence of an external potential, two classes of Jacobian elliptic periodic potentials are discussed in detail, and the corresponding ICQ nonlinearities are found to be either periodic or localized. Exact analytical soliton solutions in these cases are presented, such as the bright, dark, kink, and periodic solitons, etc. An appealing aspect is that the ICQNLSE can support bound states with any number of solitons when the ICQ nonlinearities are localized and an external potential is either applied or not.