多组分耦合非线性薛定谔方程的3-孤子解及其相互作用
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摘要
研究了带有变系数的N阶耦合非线性薛定谔方程,获得了其3-孤子解,并通过渐近分析和图像分析研究了孤子的相互作用。结果表明,当本征值不同时, 3-孤子解分别为常规孤子、束缚态孤子以及常规孤子和束缚态孤子的组合;当满足特定条件时,常规亮孤子和束缚态亮孤子可实现弹性相互作用,也可实现非弹性相互作用,而暗孤子仅存在非弹性相互作用;对于常规孤子和束缚态孤子的组合,亮孤子分量的相互作用规律较为复杂,受参数取值影响较大,但暗孤子分量依然保持弹性相互作用。
The N-order coupled nonlinear Schr?dinger equation with variable coefficients is studied and its 3-soliton solutions are obtained. The soliton interaction is studied by the asymptotic analysis and graph analysis. The results show that when the eigenvalues are different, the 3-soliton solutions can be expressed as the regular solitons, the bounded solitons, and the combination of them, respectively. Under the certain conditions, both the regular bright solitons and the bounded bright solitons can realize the elastic and the inelastic interactions, respectively. However, the dark solitons only have the inelastic interaction. But for the combination of the regular and the bounded solitons, the interacting rules of the bright soliton components are very complex and significantly influenced by the value of the parameters, but the dark soliton components still maintain the elastic interaction.
The N-order coupled nonlinear Schr?dinger equation with variable coefficients is studied and its 3-soliton solutions are obtained. The soliton interaction is studied by the asymptotic analysis and graph analysis. The results show that when the eigenvalues are different, the 3-soliton solutions can be expressed as the regular solitons, the bounded solitons, and the combination of them, respectively. Under the certain conditions, both the regular bright solitons and the bounded bright solitons can realize the elastic and the inelastic interactions, respectively. However, the dark solitons only have the inelastic interaction. But for the combination of the regular and the bounded solitons, the interacting rules of the bright soliton components are very complex and significantly influenced by the value of the parameters, but the dark soliton components still maintain the elastic interaction.
引文
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[34] Jiang Y,Tian B,Liu W J,et al.Soliton interactions and complexes for coupled nonlinear Schr?dinger equations[J].Physical Review E,2012,85(3):036605.
[35] Hirota R.A new form of backlund transformations and its relation to the inverse scattering problem[J].Progress of Theoretical Physics,1974,52(5):1498-1512.
[2] Emplit P,Hamaide J P,Reynaud F,et al.Picosecond steps and dark pulses through nonlinear single mode fibers[J].Optics Communications,1987,62(6):374-379.
[3] Kivshar Y.Dark optical solitons:physics and applications[J].Physics Reports,1998,298(2/3):81-197.
[4] Tian B,and Gao Y T.Variable-coefficient higher-order nonlinear Schr?dinger model in optical fibers:new transformation with burstons,brightons and symbolic computation[J].Physics Letters A,2006,359(3):241-248.
[5] Agrawal G P.Nonlinear fiber optics[M].4th ed.New York:Academic Press,2006.
[6] Chakravarty S,Sauer J R,Jenkins R B,et al.Multisoliton interactions and wavelength-division multiplexing[J].Optics Letters,1995,20(2):136.
[7] Yeh C,Bergman L.Enhanced pulse compression in a nonlinear fiber by a wavelength division multiplexed optical pulse[J].Physical Review E,1998,57(2):2398-2404.
[8] Dutta M,Ghosh S,Chakrabarti N.Nonlinear coupling of Langmuir and electron acoustic waves in a viscous plasma[J].Physics of Plasmas,2018,25(1):012103.
[9] Bashkin E P,Vagov A V.Instability and stratification of a two-component Bose-Einstein condensate in a trapped ultracold gas[J].Physical Review B,1997,56(10):6207-6212.
[10] Wu C F,Grimshaw R H J,Chow K W,et al.A coupled “AB” system:rogue waves and modulation instabilities[J].Chaos:an Interdisciplinary Journal of Nonlinear Science,2015,25(10):103113.
[11] Sun J Q,Ma Z Q,Qin M Z.Simulation of envelope Rossby solitons in a pair of cubic Schr?dinger equations[J].Applied Mathematics and Computation,2006,183(2):946-952.
[12] Kumar H,Malik A,Chand F.Soliton solutions of some nonlinear evolution equations with time-dependent coefficients[J].Pramana,2013,80(2):361-367.
[13] Serkin V N,Hasegawa A.Novel soliton solutions of the nonlinear Schr?dinger equation model[J].Physical Review Letters,2000,85(21):4502.
[14] Wang P,Feng L,Shang T,et al.Analytical soliton solutions for the cubic-quintic nonlinear Schr?dinger equation with Raman effect in the nonuniform management systems[J].Nonlinear Dynamics,2015,79(1):387-395.
[15] Serkin V N,Hasegawa A,Belyaeva T L.Nonautonomous solitons in external potentials[J].Physical Review Letters,2007,98(7):074102.
[16] Serkin V N,Hasegawa A,Belyaeva T L.Nonautonomous matter-wave solitons near the Feshbach resonance[J].Physical Review A,2010,81(2):023610.
[17] Zhang J F,Zhao P,Hu W C,et al.Interaction propagation of optical vortex solitons in inhomogeneous nonlinear waveguides[J].Acta Optica Sinica,2013,33(4):0419001.张解放,赵辟,胡文成,等.非均匀非线性波导中涡旋光孤子的相互作用传播[J].光学学报,2013,33(4):0419001.
[18] Liu J L,Gu Z,Tian E G,et al.New results on Hafilter design for nonlinear systems with time-delay through a T-S fuzzy model approach[J].International Journal of Systems Science,2012,43(3):426-442.
[19] Liu W J,Yang C Y,Liu M L,et al.Effect of high-order dispersion on three-soliton interactions for the variable-coefficients Hirota equation[J].Physical Review E,2017,96(4):042201.
[20] Wang C,Chen Y L,Chen X F,et al.Propagation of the unbounded bright spatical soliton[J].Acta Optica Sinica,1999,19(4):513-518.王超,陈英礼,陈险锋,等.非束缚态空间亮孤子的传输稳定性[J].光学学报,1999,19(4):513-518.
[21] Milián C,Marest T,Kudlinski A,et al.Spectral wings of the fiber supercontinuum and the dark-bright soliton interaction[J].Optics Express,2017,25(9):10494.
[22] He X G,Zhao D,Li L,et al.Engineering integrable nonautonomous nonlinear Schr?dinger equations[J].Physical Review E,2009,79(5):056610.
[23] Yang Z Y,Zhao L C,Zhang T,et al.Dynamics of a nonautonomous soliton in a generalized nonlinear Schr?dinger equation[J].Physical Review E,2011,83(6):066602.
[24] Quintero N R,Mertens F G,Bishop A R.Soliton stability criterion for generalized nonlinear Schr?dinger equations[J].Physical Review E,2015,91(1):012905.
[25] Shi J,Wang J F,Zhang C,et al.Transmission characteristics of bright-dark soliton pair in fiber lasers[J].Acta Optica Sinica,2018,38(5):0519001.石佳,王娟芬,张聪,等.光纤激光器中亮暗孤子对的传输特性[J].光学学报,2018,38(5):0519001.
[26] Stegeman G I,Segev M.Optical spatial solitons and their interactions:universality and diversity[J].Science,1999,286(5444):1518-1523.
[27] Si L G,Yang W X,Lü X Y,et al.Slow vector optical solitons in a cold four-level inverted-Y atomic system[J].The European Physical Journal D,2009,55(1):161-166.
[28] Menyuk C R.Stability of solitons in birefringent optical fibers II Arbitrary amplitudes[J].Journal of the Optical Society of America B,1988,5(2):392-402.
[29] Radhakrishnan R,Dinda P T,Millot G.Efficient control of the energy exchange due to the Manakov vector-soliton collision[J].Physical Review E,2004,69(4):046607.
[30] Radhakrishnan R,Aravinthan K.A dark-bright optical soliton solution to the coupled nonlinear Schr?dinger equation[J].Journal of Physics A,2007,40(43):13023-13030.
[31] Sun Z Y,Gao Y T,Yu X,et al.Bound vector solitons and soliton complexes for the coupled nonlinear Schr?dinger equations[J].Physical Review E,2009,80(6):066608.
[32] Kanna T,Sakkaravarthi K.Multicomponent coherently coupled and incoherently coupled solitons and their collisions[J].Journal of Physics A,2011,44(28):285211.
[33] Chakraborty S,Nandy S,Barthakur A.Bilinearization of the generalized coupled nonlinear Schr?dinger equation with variable coefficients and gain and dark-bright pair soliton solutions[J].Physical Review E,2015,91(2):023210.
[34] Jiang Y,Tian B,Liu W J,et al.Soliton interactions and complexes for coupled nonlinear Schr?dinger equations[J].Physical Review E,2012,85(3):036605.
[35] Hirota R.A new form of backlund transformations and its relation to the inverse scattering problem[J].Progress of Theoretical Physics,1974,52(5):1498-1512.