Propagation and interaction between special fractional soliton and soliton molecules in the inhomogeneous fiber

Abstract

Introduction: Fractional nonlinear models have been widely used in the research of nonlinear science. A fractional nonlinear Schrödinger equation with distributed coefficients is considered to describe the propagation of pi-second pulses in inhomogeneous fiber systems. However, soliton molecules based on the fractional nonlinear Schrödinger equation are hardly reported although many fractional soliton structures have been studied.

Objectives: This paper discusses the propagation and interaction between special fractional soliton and soliton molecules based on analytical solutions of a fractional nonlinear Schrödinger equation.

Methods: Two analytical methods, including the variable-coefficient fractional mapping method and Hirota method with the modified Riemann-Liouville fractional derivative rule, are used to obtain analytical non-travelling wave solutions and multi-soliton approximate solutions.

Results: Analytical non-travelling wave solutions and multi-soliton approximate solutions are derived. The form conditions of soliton molecules are given, and the dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed in the periodic inhomogeneous fiber and the exponential dispersion decreasing fiber.

Conclusion: Analytical chirp-free and chirped non-traveling wave solutions and multi-soliton approximate solutions including soliton molecules are obtained. Based on these solutions, dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed. These theoretical studies are of great help to understand the propagation of optical pulses in fibers.

Keywords: Distributed coefficients; Fractional nonlinear Schrödinger equation; Interaction; Soliton molecules; Special soliton.

Graphical abstract

Fig. 1

Chirp-free bright soliton (16) with the intensity${\left|Q\right|}^2$. Parameters are $C_1=0.8,C_2=C_3=1.5,$$C_4=C_{\text{5}}\text{=}0,$$\alpha (x)=exp(-0.05x)sin(x),\rho (x)=-0.025$with (a) $\lambda =1$and (b) $\lambda =0.5$

Fig. 2

Density map of chirp-free bright soliton ${\left|Q_{21}\right|}^2$. Parameters are same as Fig. 1.

Fig. 3

The dynamical interaction of the two-soliton solution: (a) density plot, (b) intensity plot and (c) numerical rerun with 5% white random noise. Parameters are$s_{11}=s_{21}=1,s_{12}=0.3,$$s_{22}=0.1,k_{11}=-5,$$k_{12}=2,k_{21}=3,k_{22}=-2,\lambda =0.5,$$P=2,\rho (x)=0,\alpha (x)=0.8exp(-0.01x).$

Fig. 4

A soliton molecule consisting of two solitons: (a) density plot and (b) intensity plot. Parameters are$s_{11}=1.5,s_{12}=-0.3,s_{21}=2,s_{22}=-0.3,$$k_{11}=-2,$$k_{12}=-2,k_{21}=2,k_{22}=-2,\lambda =0.5,$$P=2,\rho (x)=0,\alpha (x)=0.0034exp(-0.076x).$

Fig. 5

The numerical rerun of the soliton molecule in Fig. 4 with 5% white random noise: (a) intensity plot and (b) density plot. Parameters are $s_{11}=1.5,s_{12}=-0.3,s_{21}=2,s_{22}=-0.3,$$k_{11}=-2,$$k_{12}=-2,k_{21}=2,k_{22}=-2,\lambda =0.5,$$P=2,\rho (x)=0,\alpha (x)=0.0034exp(-0.076x)$

Fig. 6

A soliton molecule consisting of three solitons: (a) density plot and (b) intensity plot. Parameters are $s_{11}=1.5,s_{12}=-0.3,s_{21}=2,s_{22}=-0.3,s_{31}=2.5,s_{32}=\text{-}0.3,k_{11}=0.1,k_{12}=-2,$$k_{21}=2,k_{22}=2,k_{31}=4,k_{32}=-2,P=2,\rho (x)=0,\lambda =0.5,$$\alpha (x)=0.0034exp(-0.076x).$

Fig. 7

The interaction of soliton molecules and a bright solitons: (a) density plot and (b) intensity plot. Parameters are$s_{11}=2,s_{12}=-2,s_{21}=-3,s_{22}=2,s_{31}=3,s_{32}=-2,k_{11}=-1,k_{12}=-2,$$k_{21}=2,k_{22}=4,k_{31}=4,k_{32}=-2,P=2,\rho (x)=0,\lambda =0.5,\alpha (x)=0.6.$

Fig. 8

The numerical rerun of the three-soliton molecule in Fig. 6 with 5% white random noise: (a) intensity plot and (b) density plot. Parameters are $s_{11}=1.5,s_{12}=s_{22}=s_{32}=-0.3,s_{21}=2,s_{31}=2.5,$$k_{11}=0.1,k_{12}=k_{32}=-2,k_{21}=k_{22}=2,k_{31}=4,P=2,\rho (x)=0,\lambda =0.5,$$\alpha (x)=0.0034exp(-0.076x).$