Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential

Xiankun Yao; Xueming Liu

doi:doi:10.1364/PRJ.6.000875

Photonics Research, 2018, 6 (9): 09000875, Published Online: Aug. 15, 2018

Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential

Xiankun Yao Xueming Liu ^*

Author Affiliations

State Key Laboratory of Modern Optical Instrumentation, Department of Optical Engineering, Zhejiang University, Hangzhou 310027, China

Figures & Tables

Fig. 1. (a) Photonic band structure for FSE with PT-symmetric periodic potential, when Vi=0.4 (red line) and 0.55 (blue line). (b) Imaginary parts of the band structure for Vi=0.55.

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Fig. 2. (a) Soliton field profile (real part: blue line, imaginary part: red line) for β=0.8. (b) The spectral distribution |ϕ˜(k)| in the inverse space. (c) Transverse power flow (red line) of the soliton. (d) Spectrum of the linearization operator for the soliton solution in (a). (e) Stable propagation perturbed with weak noise. The light gray line in (a) and (d) represents the scaled real part of the potential. For all cases, Vi=0.4 and A=1.

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Fig. 3. (a) Soliton field profile (real part: blue line, imaginary part: red line). The light gray line denotes the real part of the potential. (b) The perturbation growth rate profile for the soliton solution in (a). (c) Unstable PT soliton propagation perturbed with weak noise. In all cases, Vi=0.55 and β=0.8.

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Fig. 4. (a) Domains of stability on the plane (β, Vi) for 1D solitons. Solitons are stable in the blue region. The red line denotes the lower edge of the semi-infinite forbidden gap. (b) The maximum real part of perturbation growth rate versus β for Vi=0.4 and A=1.

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Fig. 5. (a) Band structure of a 2D-PT potential for Eq. (4) in linear form. The inset displays the band structure in the reduced Brillouin zone. (b) The profile of a 2D soliton when β=5.55. (c) The spectral profile |ϕ˜(kx,ky)| in the inverse space. (d) Transverse power flow (indicated by arrows) for the soliton in (b) within one cell. “L” and “G” indicate the loss and gain regions, respectively. For all cases, A=4 and Vi=0.2.

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Fig. 6. (a) Domains of stability on the plane (β, Vi) for 2D solitons. Solitons are stable in the blue region. The red line denotes the lower edge of the semi-infinite forbidden gap. (b) Real part of perturbation growth rate versus β for A=4 and Vi=0.2.

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Xiankun Yao, Xueming Liu. Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential[J]. Photonics Research, 2018, 6(9): 09000875.

Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential

Fig. 1. (a) Photonic band structure for FSE with PT-symmetric periodic potential, when Vi=0.4 (red line) and 0.55 (blue line). (b) Imaginary parts of the band structure for Vi=0.55.

Fig. 4. (a) Domains of stability on the plane (β, Vi) for 1D solitons. Solitons are stable in the blue region. The red line denotes the lower edge of the semi-infinite forbidden gap. (b) The maximum real part of perturbation growth rate versus β for Vi=0.4 and A=1.

Fig. 6. (a) Domains of stability on the plane (β, Vi) for 2D solitons. Solitons are stable in the blue region. The red line denotes the lower edge of the semi-infinite forbidden gap. (b) Real part of perturbation growth rate versus β for A=4 and Vi=0.2.

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Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential

Fig. 1. (a) Photonic band structure for FSE with PT-symmetric periodic potential, when Vi=0.4 (red line) and 0.55 (blue line). (b) Imaginary parts of the band structure for Vi=0.55.

Fig. 4. (a) Domains of stability on the plane (β, Vi) for 1D solitons. Solitons are stable in the blue region. The red line denotes the lower edge of the semi-infinite forbidden gap. (b) The maximum real part of perturbation growth rate versus β for Vi=0.4 and A=1.

Fig. 6. (a) Domains of stability on the plane (β, Vi) for 2D solitons. Solitons are stable in the blue region. The red line denotes the lower edge of the semi-infinite forbidden gap. (b) Real part of perturbation growth rate versus β for A=4 and Vi=0.2.

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