Travelling solitary waves in the discrete Schrِdinger equation with saturable nonlinearity: Existence, stability and dynamics

The present work examines in detail the existence, stability and dynamics of travelling solitary waves in a Schrِdinger lattice with saturable nonlinearity. After analysing the linear spectrum of the problem in the travelling wave frame, a pseudo-spectral numerical method is used to identify weakly nonlocal solitary waves. By finding zeros of an appropriately crafted tail condition, we can obtain the genuinely localized pulse-like solutions. Subsequent use of continuation methods allows us to obtain the relevant branches of solutions as a function of the system parameters, such as the frequency and intersite coupling strength. We examine the stability of the solutions in two ways: both by imposing numerical perturbations and observing the solution dynamics, as well as by considering the solutions as fixed points of an appropriate map and computing the corresponding Floquet matrix and its eigenvalues. Both methods indicate that our solutions are robustly localized. Finally, the interactions of these solutions are examined in collision type phenomena, observing that relevant collisions are near-elastic, although they may, under appropriate conditions, lead to the generation of an additional pulse.