Research progress on solitons in fractional nonlinear wave models

LI Pengfei; HE Xueqing; CAI Qiang

doi:10.12405/j.issn.2097-1486.2024.02.005

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Research progress on solitons in fractional nonlinear wave models

更新时间：2024-11-05

- Research progress on solitons in fractional nonlinear wave models
- Emerging Science and Technology(Nat.Sci.Ed.) Vol. 3, Issue 2, Pages: 167-179(2024)
- 作者机构：
  
  1.太原师范学院物理系，山西晋中 030619
  2.太原师范学院计算与应用物理研究所，山西晋中 030619
  3.太原师范学院智能优化计算与区块链技术山西省重点实验室，山西晋中 030619
- 作者简介：
- 基金信息：
- DOI：10.12405/j.issn.2097-1486.2024.02.005
  CLC： O437
- Published：15 June 2024，
  
  Received：19 March 2024，
  
  Revised：10 April 2024，
- 稿件说明：
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李鹏飞,贺雪晴,蔡强.分数阶非线性波模型中孤子研究进展[J].新兴科学和技术趋势,2024,3(2):167-179.

LI Pengfei,HE Xueqing,CAI Qiang.Research progress on solitons in fractional nonlinear wave models[J].Emerging Science and Technology,2024,3(2):167-179.
李鹏飞,贺雪晴,蔡强.分数阶非线性波模型中孤子研究进展[J].新兴科学和技术趋势,2024,3(2):167-179. DOI： 10.12405/j.issn.2097-1486.2024.02.005.

LI Pengfei,HE Xueqing,CAI Qiang.Research progress on solitons in fractional nonlinear wave models[J].Emerging Science and Technology,2024,3(2):167-179. DOI： 10.12405/j.issn.2097-1486.2024.02.005.

摘要

孤子理论和实验研究涉及的非线性波模型大多数是整数阶导数，在分数阶导数非线性波模型中研究孤子存在性、稳定性及传输特性，进一步将孤子的研究拓展到分数维度。此外，分数衍射和分数色散效应为孤子传输的调控提供了新的自由度。本文简要介绍了分数量子力学的理论研究进展，概述了光学领域中分数薛定谔方程的理论和实验方面的研究进展，回顾了分数阶非线性薛定谔方程中孤子的数值算法、孤子解的存在形式、自发对称破缺以及演化，具体介绍了宇称时间对称的分数阶非线性薛定谔方程中孤子的自发对称破缺和鬼态的产生机制，并对该领域未来的发展做了展望。

Abstract

Most of the nonlinear wave models involved in soliton theory and experimental research are integer derivatives. The study of of solitons has been further extended from the existence， the stability and the transmission characteristics of solitons in the fractional derivative nonlinear wave model to fractional dimension. In addition， fractional diffraction and fractional dispersion effects provide a new freedom to control the soliton evolutions. This article briefly introduces the theoretical research progress of fractional quantum mechanics， outlines the theoretical and experimental research progress of the fractional Schrödinger equation in the field of optics， and reviews the numerical algorithm of solitons in the fractional nonlinear Schrödinger equation， the existence form of soliton solutions， and spontaneous symmetry breaking and evolutions. It also introduces in details the spontaneous symmetry breaking of solitons and the generation mechanism of ghost states in the fractional nonlinear Schrödinger equation with parity-time symmetric potentials， and gives an outlook for the future development of this field.

关键词

分数阶非线性薛定谔方程孤子宇称时间对称自发对称破缺

Keywords

fractional nonlinear Schrödinger equationsolitonparity-time symmetryspontaneous symmetry breaking

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