Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential

Abstract

Localization and dynamics of the one-dimensional biharmonic nonlinear Schr\"odinger (NLS) equation in the presence of an external periodic potential is studied. The band-gap structure is determined using the Floquet-Bloch theory and the shape of its dispersion curves as a function of the fourth-order dispersion coupling constant $\ensuremath{\beta}$ is discussed. Contrary to the classical NLS equation ($\ensuremath{\beta}=0$) with an external periodic potential for which a gap in the spectrum opens for any nonzero potential, here it is found that for certain negative $\ensuremath{\beta},$ there exists a nonzero threshold value of potential strength below which there is no gap. For increasing values of potential amplitudes, the shape of the dispersion curves change drastically leading to the formation of localized nonlinear modes that have no counterpart in the classical NLS limit. A higher-order two-band tight-binding model is introduced that captures and intuitively explains most of the numerical results related to the spectral bands. Lattice solitons corresponding to spectral eigenvalues lying in the semi-infinite and first band gaps are constructed. In the anomalous dispersion case, i.e., $\ensuremath{\beta}<0$ (where for the self-focusing nonlinearity no localized nonradiating solitons exist in the absence of an external potential), nonlinear finite-energy stationary modes with eigenvalues residing in the first band gap are found and their properties are discussed. The stability of various localized lattice modes is studied via linear stability analysis and direct numerical simulation.