Composition Relation between Nonlinear Bloch Waves and Gap Solitons in Periodic Fractional Systems.

Liangwei Dong,Changming Huang

doi:10.3390/ma11071134

Abstract

Evolution of beams in nonlinear optical media with a fractional-order diffraction is currently attracting a growing interest. We address the existence of linear and nonlinear Bloch waves in fractional systems with a periodic potential. Under a defocusing nonlinearity, nonlinear Bloch waves at the centers or edges of the first Brillouin zone bifurcate from the corresponding linear Bloch modes at different band edges. They can be constructed by directly copying a fundamental gap soliton (in one lattice site) or alternatively copying it and its mirror image to infinite lattice channels. The localized truncated-Bloch-wave solitons bridging nonlinear Bloch waves and gap solitons are also revealed. We thus prove that fundamental gap solitons can be used as unit cells to build nonlinear Bloch waves or truncated-Bloch-wave solitons, even in fractional configurations. Our results provide helpful hints for understanding the dynamics of localized and delocalized nonlinear modes and the relation between them in periodic fractional systems with an optical nonlinearity.

Highlights

Periodic systems in optics always show a wide range of complex dynamics and intriguing properties [1,2]
Periodic nonlinear Bloch waves (NLBWs) with infinite spatial scale and finite amplitudes are supported by nonlinear systems with an optical lattice [4,5,6,7,8]
Recalling the fact that NLBWs in nonlinear Shrödinger equation (NLSE) or Gross-Pitaevskii equation (GPE) with an optical lattice can be regarded as infinite chains composed of fundamental gap solitons (FGSs) [7], one naturally asks: do NLBWs exist in a periodic system described by the nonlinear fractional Schrödinger equation (NLFSE)? If yes, another question arises: how they are influenced by the Lévy index of the system? are there any links between NLBWs and FGSs? Up to now, LBWs at the edges of the first and second bands of a fractional system were addressed only in [21]

Summary

Introduction

Periodic systems in optics always show a wide range of complex dynamics and intriguing properties [1,2]. Periodic nonlinear Bloch waves (NLBWs) with infinite spatial scale and finite amplitudes are supported by nonlinear systems with an optical lattice [4,5,6,7,8]. Such Bloch waves originate from the corresponding linear Bloch waves (LBWs) at the edges of bands in the first Brillouin zone (BZ). Recalling the fact that NLBWs in NLSE or GPE with an optical lattice can be regarded as infinite chains composed of fundamental gap solitons (FGSs) [7], one naturally asks: do NLBWs exist in a periodic system described by the nonlinear fractional Schrödinger equation (NLFSE)? The elucidation of NLBWs in NLFSE and their relation to FGSs is, a central goal of this paper

Theoretical Model

Band Gap Structures and Bloch Modes in Linear Fractional Systems

Conclusions