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A generalized nonlinear Schrdinger equation is numerically studied using the split-step Fourier method. For a fixed external potential field and an initial pulse disturbance, the effects of the complex coefficients p and q in the nonlinear Schrdinger equation on the evolution of the wave field are investigated. From a large number of simulations, it is found that the evolution of the wave field remains similar for different signs of the real parts of p and q, and different values of the real part of p. The initial pulse consisting of the longest wavelength modes (in the smallest-|k| corner of the phase space) of the spectrum first suffers modulational instability. Collapse begins at t～0.1, followed by inverse cascade of the shortest wavelength modes to longer wavelength ones, so that the whole k space becomes turbulent. For p = 1+0.04i, and q = 1+0.6i, it is found that first modulational instability occurs in the longer wavelength regime and the wave energy is transferred to the larger |k| modes. Then the wave collapse appears with increasing wave energy. Next, the large-|k| modes condense into a smaller-|k| mode by inverse cascade before spreading to the center of the phase space, until a turbulent state occurs there. Finally, most of the wave energy is condensed to the neighborhoods of three modes.
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Keywords:
- nonlinear Schrdinger equation /
- collapse /
- turbulence /
- inverse cascade
[1] Zhang Y D 2008 Quantum Mechanics (2nd Ed.) (Beijing: Science Press) p239 (in Chinese) [张永德 2008 量子力学 (第二版) (北京: 科学出版社) 第239页]
[2] Zhao L, Sui Z, Zhu Q H, Zhang Y, Zuo Y L 2009 Acta Phys. Sin. 58 4731 (in Chinese) [赵磊, 隋展, 朱启华, 张颖, 左言磊 2009 物理学报 58 4731]
[3] Goldman M V 1984 Rev. Mod. Phys. 56 709
[4] Zhou C T, Yu M Y, He X T 2006 Phys. Rev. E 73 026209
[5] Zhao D, Yu M Y 2011 Phys. Rev. E 83 036405
[6] Zhao D, Tian L P, Cui S Y, Yu M Y 2012 Phys. Scr. 86 035501
[7] Cui S Y, Yu M Y, Zhao D 2013 Phys. Rev. E 87 053104
[8] Yu M Y, Cui S Y, Zhao D 2015 Europhys. Lett. 109 65001
[9] Itoh S I, Itoh K 2012 Chin. Phys. B 21 095201
[10] Feng C H, Wang W H, He Y X, Gao Z, Zeng L, Zhang G P, Xie L F 2004 Chin. Phys. 13 2091
[11] Nicolis G, Prigogine I 1977 Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations (New York: Wiley) p169
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[1] Zhang Y D 2008 Quantum Mechanics (2nd Ed.) (Beijing: Science Press) p239 (in Chinese) [张永德 2008 量子力学 (第二版) (北京: 科学出版社) 第239页]
[2] Zhao L, Sui Z, Zhu Q H, Zhang Y, Zuo Y L 2009 Acta Phys. Sin. 58 4731 (in Chinese) [赵磊, 隋展, 朱启华, 张颖, 左言磊 2009 物理学报 58 4731]
[3] Goldman M V 1984 Rev. Mod. Phys. 56 709
[4] Zhou C T, Yu M Y, He X T 2006 Phys. Rev. E 73 026209
[5] Zhao D, Yu M Y 2011 Phys. Rev. E 83 036405
[6] Zhao D, Tian L P, Cui S Y, Yu M Y 2012 Phys. Scr. 86 035501
[7] Cui S Y, Yu M Y, Zhao D 2013 Phys. Rev. E 87 053104
[8] Yu M Y, Cui S Y, Zhao D 2015 Europhys. Lett. 109 65001
[9] Itoh S I, Itoh K 2012 Chin. Phys. B 21 095201
[10] Feng C H, Wang W H, He Y X, Gao Z, Zeng L, Zhang G P, Xie L F 2004 Chin. Phys. 13 2091
[11] Nicolis G, Prigogine I 1977 Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations (New York: Wiley) p169
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