Higher-order lattice diffraction: Solitons in the discrete NLS equation with next-nearest-neighbor interactions

We introduce a general model of a one-dimensional dynamical lattice with the on-site cubic nonlinearity and both nearest-neighbor (NN) and next-nearest-neighbor (NNN) linear interactions between lattice sites (i.e., a generalized discrete nonlinear Schrödinger equation). Unlike some previously considered cases, our model includes a complex coefficient of the NNN interaction, which applies, e.g., to optical waveguide arrays, where fields in adjacent cores may be phase shifted. The application to optical arrays is especially important in the case in which the effective lattice diffraction coefficient generated by the NN coupling is close to zero, which may be achieved by means of the so-called diffraction management. Three types of fundamental solitons are considered: site-centered and intersite-centered ones, and twisted localized modes (TLMs). It is found that, with the increase of the imaginary part of the NNN coupling constant, site-centered solitons lose their stability, but then regain it. The instability region disappears if the real part of the NNN coupling constant is negative and sufficiently large. If the site-centered soliton is unstable, it rearranges itself into a quasi-periodic (in time) breathing soliton. Intersite-centered solitons cannot be fully stabilized by the NNN interactions. TLM solitons are stable in a limited parametric region, then they become unstable, and eventually disappear. Direct simulations of the evolution of the intersite-centered solitons and unstable TLMs show that the instability reshapes them into site-centered solitons with intrinsic vibrations (breathers).