Properties of the nonlinear Schrödinger equation on a lattice

We add an on-site potential to the integrable lattice nonlinear Schr\"odinger 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 Schr\"odinger equation.

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