Gap Solitons and Bloch Waves in Nonlinear Periodic Systems

TL;DR

This paper explores the relationship between gap solitons and Bloch waves in nonlinear periodic systems, revealing a composition relation that simplifies understanding their properties and stability without extensive computation.

Contribution

It introduces a composition relation between Bloch waves and gap solitons, linking them to nonlinear Wannier functions, and analyzes the stability and existence of various gap soliton families.

Findings

Bloch waves at zone edges are chains of fundamental gap solitons.

Number of FGS families varies with nonlinearity type and band gaps.

Some FGSs have cutoffs in propagation constant, explaining their limited existence.

Abstract

We comprehensively investigate gap solitons and Bloch waves in one-dimensional nonlinear periodic systems. Our results show that there exists a composition relation between them: Bloch waves at either the center or edge of the Brillouin zone are infinite chains composed of fundamental gap solitons(FGSs). We argue that such a relation is related to the exact relation between nonlinear Bloch waves and nonlinear Wannier functions. With this composition relation, many conclusions can be drawn for gap solitons without any computation. For example, for the defocusing nonlinearity, there are $n$ families of FGS in the $n$th linear Bloch band gap; for the focusing case, there are infinite number of families of FGSs in the semi-infinite gap and other gaps. In addition, the stability of gap solitons is analyzed. In literature there are numerical results showing that some FGSs have cutoffs on propagation constant (or chemical potential), i.e. these FGSs do not exist for all values of propagation constant (or chemical potential) in the linear band gap. We offer an explanation for this cutoff.