Exponential asymptotics of dark and bright solitons in the discrete nonlinear Schrödinger equation

Abstract: We investigate the existence and linear stability of solitons in the nonlinear Schr\"odinger lattices in the strong coupling regime. Focusing and defocusing nonlinearities are considered, giving rise to bright and dark solitons. In this regime, the effects of lattice discreteness become exponentially small, requiring a beyond-all-orders analysis. To this end, we employ exponential asymptotics to derive soliton solutions and examine their stability systematically. We show that only two symmetry-related soliton configurations are permissible: onsite solitons centered at lattice sites and intersite solitons positioned between adjacent sites. Although the instability of intersite solitons due to real eigenvalue pairs is known numerically, a rigorous analytical account, particularly for dark solitons, has been lacking. Our work fills this gap, yielding analytical predictions that match numerical computations with high accuracy. We also establish the linear stability of onsite bright solitons. While the method cannot directly resolve the quartet eigenvalue-induced instability of onsite dark solitons due to the continuous spectrum covering the entire imaginary axis, we conjecture an eigenvalue-counting argument that supports their instability. Overall, our application of the exponential asymptotics method shows the versatility of this approach for addressing multiscale problems in discrete nonlinear systems.