Stability of multiple pulses in discrete systems

The stability of multiple-pulse solutions to the discrete nonlinear Schrodinger equation is considered. A bound state of widely separated single pulses is rigorously shown to be unstable, unless the phase shift Delta phi between adjacent pulses satisfies Delta phi = pi. This instability is accounted for by positive real eigenvalues in the linearized system. The analysis leading to the instability result does not, however, determine the linear stability of those multiple pulses for which Delta phi = pi between adjacent pulses. A direct variational approach for a two-pulse predicts that it is linearly stable if Delta phi = pi, and if the separation between the individual pulses satisfies a certain condition. The variational approach can easily be generalized to study the stability of N pulses for any N greater than or equal to 3. The analysis is supplemented with a detailed numerical stability analysis.