EXACT SOLUTIONS AND DYNAMICS OF THE RAMAN SOLITON MODEL IN NANOSCALE OPTICAL WAVEGUIDES, WITH METAMATERIALS, HAVING PARABOLIC LAW NON-LINEARITY

Yan Zhou

doi:10.11948/2019.159

2019 Volume 9 Issue 1

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Yan Zhou. EXACT SOLUTIONS AND DYNAMICS OF THE RAMAN SOLITON MODEL IN NANOSCALE OPTICAL WAVEGUIDES, WITH METAMATERIALS, HAVING PARABOLIC LAW NON-LINEARITY[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 159-186. doi: 10.11948/2019.159

Citation:

Yan Zhou. EXACT SOLUTIONS AND DYNAMICS OF THE RAMAN SOLITON MODEL IN NANOSCALE OPTICAL WAVEGUIDES, WITH METAMATERIALS, HAVING PARABOLIC LAW NON-LINEARITY[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 159-186. doi: 10.11948/2019.159

EXACT SOLUTIONS AND DYNAMICS OF THE RAMAN SOLITON MODEL IN NANOSCALE OPTICAL WAVEGUIDES, WITH METAMATERIALS, HAVING PARABOLIC LAW NON-LINEARITY

Yan Zhou^1,2, ,

1.
Wu Wen-Tsun Key Laboratory, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
2.
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China

Corresponding author: Email address:zy4233@hqu.edu.cn(Y. Zhou)
Fund Project: The authors were supported by National Natural Science Foundation of China (11571318, 11162020, 11701197, 11771414) and the Project for Young and Middle-aged Teacher in Education and Science Research of Fujian Province of China (JAT170028)

Abstract

Abstract

This paper investigate the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity by using the method of dynamical systems. The functions $ q(x, t) = \phi(\xi)\exp(i(-kx+\omega t)) $ are solutions of the equation (1.1) that governs the propagation of Raman solitons through optical metamaterials, where $ \xi = x-vt $ and $ \phi(\xi) $ in the solutions satisfy a singular planar dynamical system (1.5) which has two singular straight lines. By using the bifurcation theory method of dynamical systems to the equation of $ \phi(\xi) $, bifurcations of phase portraits for this dynamical system are obtained under 28 different parameter conditions. Based on those phase portraits, 62 exact solutions of system (1.5) including periodic solutions, heteroclinic and homoclinic solutions, periodic peakons and peakons as well as compacton solutions are derived.
- Integrable system /
- exact solution /
- homoclinic and heteroclinic orbit /
- periodic solution /
- peakon /
- compacton
MSC: 34C60, 35Q51, 35C05, 35C07, 35C08

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References

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