From nonlocal gap solitary waves to bound states in periodic media

TL;DR

This paper studies gap solitons in periodic media using the nonlinear Schrödinger equation, constructing bound states from nonlocal solitary waves and analyzing their bifurcations and power characteristics.

Contribution

It introduces a method to construct and analyze bound states of gap solitons by matching their exponentially-small tails, revealing new families of solutions and their bifurcation structure.

Findings

Bound states characterized by separation distance are identified.

Three solution branches bifurcate near Bloch-band edges.

Asymptotic power curves match numerical results.

Abstract

Solitary waves in one-dimensional periodic media are discussed employing the nonlinear Schr\"odinger equation with a spatially periodic potential as a model. This equation admits two families of gap solitons that bifurcate from the edges of Bloch bands in the linear wave spectrum. These fundamental solitons may be positioned only at specific locations relative to the potential; otherwise, they become nonlocal owing to the presence of growing tails of exponentially-small amplitude with respect to the wave peak amplitude. Here, by matching the tails of such nonlocal solitary waves, higher-order locally confined gap solitons, or bound states, are constructed. Details are worked out for bound states comprising two nonlocal solitary waves in the presence of a sinusoidal potential. A countable set of bound-state families, characterized by the separation distance of the two solitary waves, is found, and each family features three distinct solution branches that bifurcate near Bloch-band edges at small, but finite, amplitude. Power curves associated with these solution branches are computed asymptotically for large solitary-wave separation, and the theoretical predictions are consistent with numerical results.