Gap solitons and truncated nonlinear Bloch waves in combined linear and quintic nonlinear lattices

This paper presents a systematic investigation of the existence, stability, and dynamics of localized modes—specifically fundamental gap solitons and nonlinear truncated Bloch waves (TBWs)—in a one-dimensional system with a combined linear lattice and a purely quintic nonlinear lattice. The model is governed by the nonlinear Schrödinger equation with a periodic linear potential and a spatially modulated quintic nonlinearity. Using numerical methods, including the Newton-conjugate gradient technique for stationary solutions, linear stability analysis, and direct propagation simulations, we demonstrate that the spatially periodic modulation of the quintic nonlinear term plays a crucial role in stabilizing fundamental gap solitons. Stable families emerge when specific commensurability conditions are satisfied between the linear and nonlinear lattice periods. Furthermore, we report the first discovery of stable TBWs in a quintic nonlinear setting, demonstrating that their formation and spatial period are intrinsically governed by the coupling between the linear and nonlinear lattices, independent of initial conditions. These multi-peak localized states exhibit high robustness during propagation when phase-matching conditions are satisfied. Our results underscore the critical role of quintic nonlinear lattice modulation in stabilizing diverse localized states and offer insights for designing advanced photonic devices based on higher-order nonlinear periodic media.