We add an on-site potential to the integrable lattice nonlinear Schrödinger equation and show how a number of interesting and novel features can be understood with the help of a simple soliton collective variable approximation. Results include: trapping of a soliton in a linear potential and on a maximum of a smooth potential; trapping of a soliton on a repulsive impurity and breaking into two solitons beyond a critical impurity strength; and a crossover from a soliton state to a local impurity mode upon increasing the strength of an attractive potential. In addition, we prove and illustrate the complete integrability of the system for a linear on-site potential. Results are compared with those for a nonintegrable discretization of the cubic Schrödinger equation.

• Received 8 November 1990

DOI:https://doi.org/10.1103/PhysRevA.43.6535