非线性薛定谔方程的呼吸子解及其怪波极限
|
摘要
采用达布变换法得到了标准非线性薛定谔方程的一阶呼吸子解及其怪波极限,研究了一阶呼吸子解的动力学特性。借助达布变换的递推关系得到了非线性薛定谔方程的高阶呼吸子解,并分别研究了碰撞叠加、分离、简并和并行传输模式。当各呼吸子的频率趋于零时,得到非线性薛定谔方程怪波极限。研究结果表明,怪波幅值、凸起数以及怪波分裂后中心波峰的阶数和周围的波峰个数均与怪波阶数有关。
Based on the standard nonlinear Schr?dinger equation,the first-order breather solution and its rouge wave limit are obtained with Darboux transform method,and the dynamic characteristics of first-order breather solution are studied.High-order breather solutions of nonlinear Schr?dinger equation are obtained by means of recurrence relation of Darboux transformation.And their collision superposition,separation,degeneracy and parallel transmission modes are studied,respectively.Nonlinear Schr?dinger equation′s rouge wave limit can be obtained when each breather frequency tends to zero.Research results show that the rouge wave′s amplitude,number of bumps,order of center peaks and number of surrounding peaks after splitting are related to rouge wave′s order.
Based on the standard nonlinear Schr?dinger equation,the first-order breather solution and its rouge wave limit are obtained with Darboux transform method,and the dynamic characteristics of first-order breather solution are studied.High-order breather solutions of nonlinear Schr?dinger equation are obtained by means of recurrence relation of Darboux transformation.And their collision superposition,separation,degeneracy and parallel transmission modes are studied,respectively.Nonlinear Schr?dinger equation′s rouge wave limit can be obtained when each breather frequency tends to zero.Research results show that the rouge wave′s amplitude,number of bumps,order of center peaks and number of surrounding peaks after splitting are related to rouge wave′s order.
引文
[1]Zakharov V E,Shabat A B.Exact theory of twodimensional self-focussing and one-dimensional selfmodulating waves in nonlinear media[J].Journal of Mathematical Physics,2015,34(15):62-69.
[2]Akhmediev N,Ankiewicz A.Solitons:nonlinear pulses and beams[M].London:Chapman and Hall,1997.
[3]Kharif C,Pelinovsky E.Physical mechanisms of the rogue wave phenomenon[J].European Journal of Mechanics,2003,22(6):603-634.
[4]Janssen P A E M.Nonlinear four-wave interactions and freak waves[J].Journal of Physical Oceanography,2003,33(4):863-884.
[5]Onorato M,Osborne A R,Serio M,et al.Freak waves in random oceanic sea states[J].Physical Review Letters,2001,86(25):5831-5834.
[6]Dyachenko A I,Zakharov V E.Spatial equation for water waves[J].JETP Letters,2016,103(3):181-184.
[7]Shrira V I,Geogjaev V V.What makes the Peregrine soliton so special as a prototype of freak waves?[J].Journal of Engineering Mathematics,2010,67(1/2):11-22.
[8]Osborne A R,Onorato M,Serio M.The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains[J].Physics Letters A,2000,275(5/6):386-393.
[9]Osborne A R.The random and deterministic dynamics of‘rogue waves’in unidirectional,deepwater wave trains[J].Marine Structures,2001,14(3):275-293.
[10]Perrie W.Nonlinear ocean waves[M].Southampton:Computational Mechanics Publications,2006.
[11]Hasegawa A,Tappert F.Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers.I.Anomalous dispersion[J].Applied Physics Letters,1973,23(3):142-144.
[12]Agrawal G P.Nonlinear fiber optics[M].4th ed.Amsterdam:Elsevier,2006.
[13]Andreev P A.First principles derivation of NLSequation for BEC with cubic and quintic nonlinearities at nonzero temperature:dispersion of linear waves[J].International Journal of Modern Physics B,2013,27(6):1350017.
[14]Slunyaev A V.A high-order nonlinear envelope equation for gravity waves in finite-depth water[J].Journal of Experimental and Theoretical Physics,2005,101(5):926-941.
[15]Wu D,Wang J F,Shi J,et al.Generation and transmission of Peregrine solitons in doped fiber[J].Acta Optica Sinica,2017,37(4):0406002.武达,王娟芬,石佳,等.掺杂光纤中Peregrine孤子的产生和传输[J].光学学报,2017,37(4):0406002.
[16]Vishnu P N,Senthilvelan M,Lakshmanan M.Akhmediev breathers,Ma solitons,and general breathers from rogue waves:a case study in the Manakov system[J].Physical Review E,2013,88(2):022918.
[17]Frisquet B,Kibler B,Millot G.Collision of akhmediev breathers in nonlinear fiber optics[J].Physical Review X,2013,3(4):041032.
[18]Kedziora D J,Ankiewicz A,Akhmediev N.Secondorder nonlinear Schr9dinger equation breather solutions in the degenerate and rogue wave limits[J].Physical Review E,2012,85(6):066601.
[19]Yang G Y,Li L,Tian J P.Study on Transformation of Kuznetsov-Ma soliton to quasi-fundamental soliton based on spectral-filtering method[J].Acta Optica Sinica,2016,36(6):0619002.杨光晔,李禄,田晋平.基于谱过滤方法的Kuznetsov-Ma孤子向准基态孤子转化研究[J].光学学报,2016,36(6):0619002.
[20]Bergé L.Wave collapse in physics:principles and applications to light and plasma waves[J].Physics Reports,1998,303(5/6):259-370.
[21]Efimov V B,Ganshin A N,Kolmakov G V,et al.Rogue waves in superfluid helium[J].The European Physical Journal Special Topics,2010,185(1):181-193.
[22]Shats M,Punzmann H,Xia H.Capillary rogue waves[J].Physical Review Letters,2010,104(10):104503.
[23]Kibler B,Fatome J,Finot C,et al.The Peregrine soliton in nonlinear fibre optics[J].Nature Physics,2010,6(10):790-795.
[24]Kichenassamy S.Breather solutions of the nonlinear wave equation[J].Communications on Pure and Applied Mathematics,1991,44(7):789-818.
[25]Tajiri M,Watanabe Y.Breather solutions to the focusing nonlinear Schr9dinger equation[J].Physical Review E,1998,57(3):3510-3519.
[26]Kovalyov M.Modulating properties of harmonic breather solutions of KdV[J].Journal of Physics A:Mathematical and General,1998,31(22):5117-5128.
[27]Wang C J,Dai Z D,Lin S Q,et al.Breather-type soliton and two-soliton solutions for modified Korteweg-de Vries equation[J].Applied Mathematics and Computation,2010,216(1):341-343.
[28]Sarkar R,Dey B.Exact compact breather-like solutions of two-dimensional Fermi-Pasta-Ulam lattice[J].Journal of Physics A:Mathematical and General,2006,39(4):L99-L104.
[29]Zhu Y Q,Hu W.Propagation of breathers in the nematic liquid crystal cell without bias voltage[J].Acta Optica Sinica,2015,35(9):0919001.朱叶青,胡巍.无外置偏压的向列相液晶盒中的呼吸子传输[J].光学学报,2015,35(9):0919001.
[30]Benjamin T B,Feir J E.The disintegration of wave trains on deep water.Part 1.Theory[J].Journal of Fluid Mechanics,1967,27(3):417-430.
[31]Lu X,Wang D S.Modulation instability in dispersion managed soliton systems[J].Laser Technology,2012,36(4):557-561.卢洵,王东升.色散管理孤子系统的调制不稳定性[J].激光技术,2012,36(4):557-561.
[32]Akhmediev N N,Korneev V I,Mitskevich N V.Modulation instability in an optical fiber induced by cross-phase modulation[J].Radiophysics and Quantum Electronics,1991,34(1):73-77.
[33]Akhmediev N,Soto-Crespo J M,Ankiewicz A.Extreme waves that appear from nowhere:on the nature of rogue waves[J].Physics Letters A,2009,373(25):2137-2145.
[34]Akhmediev N,Ankiewicz A,Taki M.Waves that appear from nowhere and disappear without a trace[J].Physics Letters A,2009,373(6):675-678.
[35]Shukla P K,Kourakis I,Eliasson B,et al.Instability and evolution of nonlinearly interacting water waves[J].Physical Review Letters,2006,97(9):094501.
[36]Tao Y S,He J S.Multisolitons,breathers,and rogue waves for the Hirota equation generated by the Darboux transformation[J].Physical Review E,2012,85(2):026601.
[37]He J S,Xu S W,Porsezian K.Rogue waves of the Fokas-Lenells equation[J].Journal of the Physical Society of Japan,2012,81(12):124007.
[38]Li C,He J,Porsezian K,et al.Rogue waves of the Hirota and the Maxwell-Bloch equations[J].Physical Review E,2013,87(1):012913.
[39]Bandelow U,Akhmediev N.Sasa-Satsuma equation:soliton on a background and its limiting cases[J].Physical Review E,2012,86(2):026606.
[40]Soto-Crespo J M,Grelu P,Akhmediev N.Dissipative rogue waves:extreme pulses generated by passively mode-locked lasers[J].Physical Review E,2011,84:016604.
[41]Bespalov V I,Talanov V I.Filamentary structure of light beams in nonlinear liquids[J].Zhetf Pisma Redaktsiiu,1966,3(12):471.
[42]Voronovich V V,Shrira V I,Thomas G.Can bottom friction suppress'freak wave'formation?[J].Journal of Fluid Mechanics,2008,604:263-296.
[43]He J S,Zhang H R,Wang L H,et al.Generating mechanism for higher-order rogue waves[J].Physical Review E,2013,87(5):052914.
[44]Kedziora D J,Ankiewicz A,Akhmediev N.Circular rogue wave clusters[J].Physical Review E,2011,84(5):056611.
[45]Kedziora D J,Ankiewicz A,Akhmediev N.Classifying the hierarchy of nonlinear-Schr9dingerequation rogue-wave solutions[J].Physical Review E,2013,88:013207.
[46]Chowdury A,Kedziora D J,Ankiewicz A,et al.Breather solutions of the integrable quintic nonlinear Schr9dinger equation and their interactions[J].Physical Review E,2015,91(2):022919.
[47]Chowdury A,Krolikowski W.Breather-to-soliton transformation rules in the hierarchy of nonlinear Schr9dinger equations[J].Physical Review E,2017,95(6):062226.
[2]Akhmediev N,Ankiewicz A.Solitons:nonlinear pulses and beams[M].London:Chapman and Hall,1997.
[3]Kharif C,Pelinovsky E.Physical mechanisms of the rogue wave phenomenon[J].European Journal of Mechanics,2003,22(6):603-634.
[4]Janssen P A E M.Nonlinear four-wave interactions and freak waves[J].Journal of Physical Oceanography,2003,33(4):863-884.
[5]Onorato M,Osborne A R,Serio M,et al.Freak waves in random oceanic sea states[J].Physical Review Letters,2001,86(25):5831-5834.
[6]Dyachenko A I,Zakharov V E.Spatial equation for water waves[J].JETP Letters,2016,103(3):181-184.
[7]Shrira V I,Geogjaev V V.What makes the Peregrine soliton so special as a prototype of freak waves?[J].Journal of Engineering Mathematics,2010,67(1/2):11-22.
[8]Osborne A R,Onorato M,Serio M.The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains[J].Physics Letters A,2000,275(5/6):386-393.
[9]Osborne A R.The random and deterministic dynamics of‘rogue waves’in unidirectional,deepwater wave trains[J].Marine Structures,2001,14(3):275-293.
[10]Perrie W.Nonlinear ocean waves[M].Southampton:Computational Mechanics Publications,2006.
[11]Hasegawa A,Tappert F.Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers.I.Anomalous dispersion[J].Applied Physics Letters,1973,23(3):142-144.
[12]Agrawal G P.Nonlinear fiber optics[M].4th ed.Amsterdam:Elsevier,2006.
[13]Andreev P A.First principles derivation of NLSequation for BEC with cubic and quintic nonlinearities at nonzero temperature:dispersion of linear waves[J].International Journal of Modern Physics B,2013,27(6):1350017.
[14]Slunyaev A V.A high-order nonlinear envelope equation for gravity waves in finite-depth water[J].Journal of Experimental and Theoretical Physics,2005,101(5):926-941.
[15]Wu D,Wang J F,Shi J,et al.Generation and transmission of Peregrine solitons in doped fiber[J].Acta Optica Sinica,2017,37(4):0406002.武达,王娟芬,石佳,等.掺杂光纤中Peregrine孤子的产生和传输[J].光学学报,2017,37(4):0406002.
[16]Vishnu P N,Senthilvelan M,Lakshmanan M.Akhmediev breathers,Ma solitons,and general breathers from rogue waves:a case study in the Manakov system[J].Physical Review E,2013,88(2):022918.
[17]Frisquet B,Kibler B,Millot G.Collision of akhmediev breathers in nonlinear fiber optics[J].Physical Review X,2013,3(4):041032.
[18]Kedziora D J,Ankiewicz A,Akhmediev N.Secondorder nonlinear Schr9dinger equation breather solutions in the degenerate and rogue wave limits[J].Physical Review E,2012,85(6):066601.
[19]Yang G Y,Li L,Tian J P.Study on Transformation of Kuznetsov-Ma soliton to quasi-fundamental soliton based on spectral-filtering method[J].Acta Optica Sinica,2016,36(6):0619002.杨光晔,李禄,田晋平.基于谱过滤方法的Kuznetsov-Ma孤子向准基态孤子转化研究[J].光学学报,2016,36(6):0619002.
[20]Bergé L.Wave collapse in physics:principles and applications to light and plasma waves[J].Physics Reports,1998,303(5/6):259-370.
[21]Efimov V B,Ganshin A N,Kolmakov G V,et al.Rogue waves in superfluid helium[J].The European Physical Journal Special Topics,2010,185(1):181-193.
[22]Shats M,Punzmann H,Xia H.Capillary rogue waves[J].Physical Review Letters,2010,104(10):104503.
[23]Kibler B,Fatome J,Finot C,et al.The Peregrine soliton in nonlinear fibre optics[J].Nature Physics,2010,6(10):790-795.
[24]Kichenassamy S.Breather solutions of the nonlinear wave equation[J].Communications on Pure and Applied Mathematics,1991,44(7):789-818.
[25]Tajiri M,Watanabe Y.Breather solutions to the focusing nonlinear Schr9dinger equation[J].Physical Review E,1998,57(3):3510-3519.
[26]Kovalyov M.Modulating properties of harmonic breather solutions of KdV[J].Journal of Physics A:Mathematical and General,1998,31(22):5117-5128.
[27]Wang C J,Dai Z D,Lin S Q,et al.Breather-type soliton and two-soliton solutions for modified Korteweg-de Vries equation[J].Applied Mathematics and Computation,2010,216(1):341-343.
[28]Sarkar R,Dey B.Exact compact breather-like solutions of two-dimensional Fermi-Pasta-Ulam lattice[J].Journal of Physics A:Mathematical and General,2006,39(4):L99-L104.
[29]Zhu Y Q,Hu W.Propagation of breathers in the nematic liquid crystal cell without bias voltage[J].Acta Optica Sinica,2015,35(9):0919001.朱叶青,胡巍.无外置偏压的向列相液晶盒中的呼吸子传输[J].光学学报,2015,35(9):0919001.
[30]Benjamin T B,Feir J E.The disintegration of wave trains on deep water.Part 1.Theory[J].Journal of Fluid Mechanics,1967,27(3):417-430.
[31]Lu X,Wang D S.Modulation instability in dispersion managed soliton systems[J].Laser Technology,2012,36(4):557-561.卢洵,王东升.色散管理孤子系统的调制不稳定性[J].激光技术,2012,36(4):557-561.
[32]Akhmediev N N,Korneev V I,Mitskevich N V.Modulation instability in an optical fiber induced by cross-phase modulation[J].Radiophysics and Quantum Electronics,1991,34(1):73-77.
[33]Akhmediev N,Soto-Crespo J M,Ankiewicz A.Extreme waves that appear from nowhere:on the nature of rogue waves[J].Physics Letters A,2009,373(25):2137-2145.
[34]Akhmediev N,Ankiewicz A,Taki M.Waves that appear from nowhere and disappear without a trace[J].Physics Letters A,2009,373(6):675-678.
[35]Shukla P K,Kourakis I,Eliasson B,et al.Instability and evolution of nonlinearly interacting water waves[J].Physical Review Letters,2006,97(9):094501.
[36]Tao Y S,He J S.Multisolitons,breathers,and rogue waves for the Hirota equation generated by the Darboux transformation[J].Physical Review E,2012,85(2):026601.
[37]He J S,Xu S W,Porsezian K.Rogue waves of the Fokas-Lenells equation[J].Journal of the Physical Society of Japan,2012,81(12):124007.
[38]Li C,He J,Porsezian K,et al.Rogue waves of the Hirota and the Maxwell-Bloch equations[J].Physical Review E,2013,87(1):012913.
[39]Bandelow U,Akhmediev N.Sasa-Satsuma equation:soliton on a background and its limiting cases[J].Physical Review E,2012,86(2):026606.
[40]Soto-Crespo J M,Grelu P,Akhmediev N.Dissipative rogue waves:extreme pulses generated by passively mode-locked lasers[J].Physical Review E,2011,84:016604.
[41]Bespalov V I,Talanov V I.Filamentary structure of light beams in nonlinear liquids[J].Zhetf Pisma Redaktsiiu,1966,3(12):471.
[42]Voronovich V V,Shrira V I,Thomas G.Can bottom friction suppress'freak wave'formation?[J].Journal of Fluid Mechanics,2008,604:263-296.
[43]He J S,Zhang H R,Wang L H,et al.Generating mechanism for higher-order rogue waves[J].Physical Review E,2013,87(5):052914.
[44]Kedziora D J,Ankiewicz A,Akhmediev N.Circular rogue wave clusters[J].Physical Review E,2011,84(5):056611.
[45]Kedziora D J,Ankiewicz A,Akhmediev N.Classifying the hierarchy of nonlinear-Schr9dingerequation rogue-wave solutions[J].Physical Review E,2013,88:013207.
[46]Chowdury A,Kedziora D J,Ankiewicz A,et al.Breather solutions of the integrable quintic nonlinear Schr9dinger equation and their interactions[J].Physical Review E,2015,91(2):022919.
[47]Chowdury A,Krolikowski W.Breather-to-soliton transformation rules in the hierarchy of nonlinear Schr9dinger equations[J].Physical Review E,2017,95(6):062226.