Solitary waves in the Ablowitz–Ladik equation with power-law nonlinearity

We introduce a generalized version of the Ablowitz–Ladik model with a power-law nonlinearity, as a discretization of the continuum nonlinear Schrödinger equation with the same type of nonlinearity. In this model, we study the interplay of discreteness and generic nonlinearity features. We identify stationary discrete-soliton states for different values of nonlinearity power σ, and address changes of their stability as the frequency ω of the standing wave varies for given σ. Along with numerical methods, a variational approximation is used to predict the form of the discrete solitons, their stability changes, and bistability features by means of the Vakhitov–Kolokolov criterion (developed from first principles). Development of instabilities and the resulting asymptotic dynamics are explored by means of direct simulations.