Abstract

We find that diverse nonlinear waves, such as soliton, Akhmediev breather, and rogue waves (RWs), can emerge and interplay with each other in a two-mode coupled system. It provides a good platform to study interaction between different kinds of nonlinear waves. In particular, we obtain dark RWs analytically for the first time in the coupled system, and find that two RWs can appear in the temporal-spatial distribution. Possible ways to observe these nonlinear waves are discussed.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. K. H. Han and H. J. Shin, “Nonautonomous integrable nonlinear Schrödinger equations with generalized external potentials,” J. Phys. A 42, 335202 (2009).
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  5. L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
    [CrossRef]
  6. W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar-shaped Bose-Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
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  7. S. Burger, K. Bongs, S. Dettmer, W. Ertmer, and K. Sengstock, “Dark solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5198–5201 (1999).
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    [CrossRef]
  9. T. Busch, and J. R. Anglin, “Motion of dark solitons in trapped Bose-Einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. M. Vijayajayanthi, T. Kanna, and M. Lakshmanan, “Multisoliton solutions and energy sharing collisions in coupled nonlinear Schrödinger equations with focusing, defocusing and mixed type nonlinearities,” Eur. Phys. J., Spec. Top. 173, 57–80 (2009).
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    [CrossRef]
  23. C. Becker, S. Stellmer, P. S. Panahi, S. Dorscher, M. Baumert, E.-M. Richter, J. Kronjager, K. Bongs, and K. Sengstock, “Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,” Nat. Phys. 4, 496–501 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  28. M. Haelterman and A. Sheppard, “Bifurcation phenomena and multiple soliton-bound states in isotropic Kerr media,” Phys. Rev. E 49, 3376–3381 (1994).
    [CrossRef]
  29. Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Vector rogue waves in binary mixtures of Bose-Einstein condensates,” Eur. Phys. J. Spec. Top. 185, 169–180 (2010).
    [CrossRef]
  30. B. L. Guo, L. M. Ling, and Q. P. Liu, “Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions,” Phys. Rev. E 85, 026607 (2012).
    [CrossRef]
  31. Y. Ohta and J. K. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proc. R. Soc. Lond. Ser. A468, 1716–1740 (2012).
  32. Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
    [CrossRef]
  33. N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
    [CrossRef]
  34. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
    [CrossRef]
  35. J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16, 3644–3651 (2008).
    [CrossRef]
  36. T. Ueda and W. L. Lath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
    [CrossRef]
  37. J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
    [CrossRef]
  38. C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, Ph. Emplit, and M. Haelterman, “Symmetry-breaking instability of multimode vector solitons,” Phys. Rev. Lett. 89, 083901 (2002).
    [CrossRef]
  39. P. Das, T. S. Raju, U. Roy, and P. K. Panigrahi, “Sinusoidal excitations in two-component Bose-Einstein condensates in a trap,” Phys. Rev. A 79, 015601 (2009).
    [CrossRef]
  40. E. V. Doktorov, V. M. Rothos, and Y. S. Kivshar, “Full-time dynamics of modulational instability in spinor Bose-Einstein condensates,” Phys. Rev. A 76, 013626 (2007).
    [CrossRef]
  41. M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
    [CrossRef]
  42. F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
    [CrossRef]

2012

B. L. Guo, L. M. Ling, and Q. P. Liu, “Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions,” Phys. Rev. E 85, 026607 (2012).
[CrossRef]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[CrossRef]

2011

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

L. C. Zhao and S. L. He, “Matter wave solitons in coupled system with external potentials,” Phys. Lett. A 375, 3017–3020(2011).
[CrossRef]

B. L. Guo and L. M. Ling, “Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations,” Chin. Phys. Lett. 28, 110202 (2011).
[CrossRef]

2010

W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar-shaped Bose-Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
[CrossRef]

V. I. Shrira and V. V. Geogjaev, “What makes the Peregrine soliton so special as a prototype of freak waves?,” J. Eng. Math. 67, 11–22 (2010).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Vector rogue waves in binary mixtures of Bose-Einstein condensates,” Eur. Phys. J. Spec. Top. 185, 169–180 (2010).
[CrossRef]

2009

P. Das, T. S. Raju, U. Roy, and P. K. Panigrahi, “Sinusoidal excitations in two-component Bose-Einstein condensates in a trap,” Phys. Rev. A 79, 015601 (2009).
[CrossRef]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[CrossRef]

K. H. Han and H. J. Shin, “Nonautonomous integrable nonlinear Schrödinger equations with generalized external potentials,” J. Phys. A 42, 335202 (2009).
[CrossRef]

M. Vijayajayanthi, T. Kanna, and M. Lakshmanan, “Multisoliton solutions and energy sharing collisions in coupled nonlinear Schrödinger equations with focusing, defocusing and mixed type nonlinearities,” Eur. Phys. J., Spec. Top. 173, 57–80 (2009).

Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

2008

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16, 3644–3651 (2008).
[CrossRef]

M. Vijayajayanthi, T. Kanna, and M. Lakshmanan, “Bright-dark solitons and their collisions in mixed N-coupled nonlinear Schrödinger equations,” Phys. Rev. A 77, 013820 (2008).
[CrossRef]

C. Becker, S. Stellmer, P. S. Panahi, S. Dorscher, M. Baumert, E.-M. Richter, J. Kronjager, K. Bongs, and K. Sengstock, “Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,” Nat. Phys. 4, 496–501 (2008).
[CrossRef]

2007

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

E. V. Doktorov, V. M. Rothos, and Y. S. Kivshar, “Full-time dynamics of modulational instability in spinor Bose-Einstein condensates,” Phys. Rev. A 76, 013626 (2007).
[CrossRef]

2006

P. Kockaert, P. Tassin, G. V. Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006).
[CrossRef]

2005

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, “Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: I. Bright solitons,” Quantum Electron. 35, 778–786 (2005).
[CrossRef]

2003

P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and Boris A. Malomed, “Feshbach resonance management for Bose-Einstein condensates,” Phys. Rev. Lett. 90, 230401 (2003).
[CrossRef]

2002

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef]

B. Wu, J. Liu, and Q. Niu, “Controlled generation of dark solitons with phase imprinting,” Phys. Rev. Lett. 88, 034101 (2002).
[CrossRef]

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[CrossRef]

C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, Ph. Emplit, and M. Haelterman, “Symmetry-breaking instability of multimode vector solitons,” Phys. Rev. Lett. 89, 083901 (2002).
[CrossRef]

2001

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose-Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef]

T. Kanna, and M. Lakshmanan, “Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 86, 5043–5046(2001).
[CrossRef]

2000

Q. H. Park and H. J. Shin, “Systematic construction of multicomponent optical solitons,” Phys. Rev. E 61, 3093–3106 (2000).
[CrossRef]

J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schneider, and W. D. Phillips, “Generating solitons by phase engineering of a Bose-Einstein condensate,” Science 287, 97–101 (2000).
[CrossRef]

T. Busch, and J. R. Anglin, “Motion of dark solitons in trapped Bose-Einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[CrossRef]

C. K. Law, C. M. Chan, P. T. Leung, and M.-C. Chu, “Motional dressed states in a Bose-Einstein condensate: superfluidity at supersonic speed,” Phys. Rev. Lett. 85, 1598–1601 (2000).
[CrossRef]

M. G. Forest, S. P. Sheu, and P. C. Wright, “On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems,” Phys. Lett. A 266, 24–33 (2000).
[CrossRef]

1999

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, and K. Sengstock, “Dark solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5198–5201 (1999).
[CrossRef]

1994

M. Haelterman and A. Sheppard, “Bifurcation phenomena and multiple soliton-bound states in isotropic Kerr media,” Phys. Rev. E 49, 3376–3381 (1994).
[CrossRef]

1990

T. Ueda and W. L. Lath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef]

1989

1981

Afanasyev, V. V.

Aguero, G. M.

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, “Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: I. Bright solitons,” Quantum Electron. 35, 778–786 (2005).
[CrossRef]

Akhmediev, N.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Vector rogue waves in binary mixtures of Bose-Einstein condensates,” Eur. Phys. J. Spec. Top. 185, 169–180 (2010).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, 1997).

Anderson, B. P.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose-Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef]

Anglin, J. R.

T. Busch, and J. R. Anglin, “Motion of dark solitons in trapped Bose-Einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[CrossRef]

Ankiewicz, A.

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, 1997).

Avelar, A. T.

W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar-shaped Bose-Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
[CrossRef]

Baronio, F.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[CrossRef]

Baumert, M.

C. Becker, S. Stellmer, P. S. Panahi, S. Dorscher, M. Baumert, E.-M. Richter, J. Kronjager, K. Bongs, and K. Sengstock, “Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,” Nat. Phys. 4, 496–501 (2008).
[CrossRef]

Bazeia, D.

W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar-shaped Bose-Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
[CrossRef]

Becker, C.

C. Becker, S. Stellmer, P. S. Panahi, S. Dorscher, M. Baumert, E.-M. Richter, J. Kronjager, K. Bongs, and K. Sengstock, “Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,” Nat. Phys. 4, 496–501 (2008).
[CrossRef]

Belyaeva, T. L.

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, “Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: I. Bright solitons,” Quantum Electron. 35, 778–786 (2005).
[CrossRef]

Bludov, Y. V.

Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Vector rogue waves in binary mixtures of Bose-Einstein condensates,” Eur. Phys. J. Spec. Top. 185, 169–180 (2010).
[CrossRef]

Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
[CrossRef]

Bongs, K.

C. Becker, S. Stellmer, P. S. Panahi, S. Dorscher, M. Baumert, E.-M. Richter, J. Kronjager, K. Bongs, and K. Sengstock, “Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,” Nat. Phys. 4, 496–501 (2008).
[CrossRef]

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, and K. Sengstock, “Dark solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5198–5201 (1999).
[CrossRef]

Bourdel, T.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef]

Burger, S.

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, and K. Sengstock, “Dark solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5198–5201 (1999).
[CrossRef]

Busch, T.

T. Busch, and J. R. Anglin, “Motion of dark solitons in trapped Bose-Einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[CrossRef]

Cambournac, C.

C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, Ph. Emplit, and M. Haelterman, “Symmetry-breaking instability of multimode vector solitons,” Phys. Rev. Lett. 89, 083901 (2002).
[CrossRef]

Cardoso, W. B.

W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar-shaped Bose-Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
[CrossRef]

Carr, L. D.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef]

Castin, Y.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef]

Chan, C. M.

C. K. Law, C. M. Chan, P. T. Leung, and M.-C. Chu, “Motional dressed states in a Bose-Einstein condensate: superfluidity at supersonic speed,” Phys. Rev. Lett. 85, 1598–1601 (2000).
[CrossRef]

Chu, M.-C.

C. K. Law, C. M. Chan, P. T. Leung, and M.-C. Chu, “Motional dressed states in a Bose-Einstein condensate: superfluidity at supersonic speed,” Phys. Rev. Lett. 85, 1598–1601 (2000).
[CrossRef]

Clark, C. W.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose-Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef]

J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schneider, and W. D. Phillips, “Generating solitons by phase engineering of a Bose-Einstein condensate,” Science 287, 97–101 (2000).
[CrossRef]

Collins, L. A.

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S. Burger, K. Bongs, S. Dettmer, W. Ertmer, and K. Sengstock, “Dark solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5198–5201 (1999).
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Serkin, V. N.

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, “Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: I. Bright solitons,” Quantum Electron. 35, 778–786 (2005).
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M. Haelterman and A. Sheppard, “Bifurcation phenomena and multiple soliton-bound states in isotropic Kerr media,” Phys. Rev. E 49, 3376–3381 (1994).
[CrossRef]

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M. G. Forest, S. P. Sheu, and P. C. Wright, “On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems,” Phys. Lett. A 266, 24–33 (2000).
[CrossRef]

Shin, H. J.

K. H. Han and H. J. Shin, “Nonautonomous integrable nonlinear Schrödinger equations with generalized external potentials,” J. Phys. A 42, 335202 (2009).
[CrossRef]

Q. H. Park and H. J. Shin, “Systematic construction of multicomponent optical solitons,” Phys. Rev. E 61, 3093–3106 (2000).
[CrossRef]

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V. I. Shrira and V. V. Geogjaev, “What makes the Peregrine soliton so special as a prototype of freak waves?,” J. Eng. Math. 67, 11–22 (2010).
[CrossRef]

Simsarian, J. E.

J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schneider, and W. D. Phillips, “Generating solitons by phase engineering of a Bose-Einstein condensate,” Science 287, 97–101 (2000).
[CrossRef]

Solli, D. R.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Soto-Crespo, J. M.

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[CrossRef]

Stellmer, S.

C. Becker, S. Stellmer, P. S. Panahi, S. Dorscher, M. Baumert, E.-M. Richter, J. Kronjager, K. Bongs, and K. Sengstock, “Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,” Nat. Phys. 4, 496–501 (2008).
[CrossRef]

Strecker, K. E.

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[CrossRef]

Sylvestre, T.

C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, Ph. Emplit, and M. Haelterman, “Symmetry-breaking instability of multimode vector solitons,” Phys. Rev. Lett. 89, 083901 (2002).
[CrossRef]

Taki, M.

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

Tassin, P.

P. Kockaert, P. Tassin, G. V. Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006).
[CrossRef]

Theocharis, G.

P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and Boris A. Malomed, “Feshbach resonance management for Bose-Einstein condensates,” Phys. Rev. Lett. 90, 230401 (2003).
[CrossRef]

Tlidi, M.

P. Kockaert, P. Tassin, G. V. Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006).
[CrossRef]

Truscott, A. G.

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[CrossRef]

Ueda, T.

T. Ueda and W. L. Lath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef]

Vanderlinden, B.

C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, Ph. Emplit, and M. Haelterman, “Symmetry-breaking instability of multimode vector solitons,” Phys. Rev. Lett. 89, 083901 (2002).
[CrossRef]

Veretennicoff, I.

P. Kockaert, P. Tassin, G. V. Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006).
[CrossRef]

Vijayajayanthi, M.

M. Vijayajayanthi, T. Kanna, and M. Lakshmanan, “Multisoliton solutions and energy sharing collisions in coupled nonlinear Schrödinger equations with focusing, defocusing and mixed type nonlinearities,” Eur. Phys. J., Spec. Top. 173, 57–80 (2009).

M. Vijayajayanthi, T. Kanna, and M. Lakshmanan, “Bright-dark solitons and their collisions in mixed N-coupled nonlinear Schrödinger equations,” Phys. Rev. A 77, 013820 (2008).
[CrossRef]

Villargan, V. E.

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, “Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: I. Bright solitons,” Quantum Electron. 35, 778–786 (2005).
[CrossRef]

Wabnitz, S.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[CrossRef]

Wright, P. C.

M. G. Forest, S. P. Sheu, and P. C. Wright, “On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems,” Phys. Lett. A 266, 24–33 (2000).
[CrossRef]

Wu, B.

B. Wu, J. Liu, and Q. Niu, “Controlled generation of dark solitons with phase imprinting,” Phys. Rev. Lett. 88, 034101 (2002).
[CrossRef]

Yang, J. K.

Y. Ohta and J. K. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proc. R. Soc. Lond. Ser. A468, 1716–1740 (2012).

Zhao, L. C.

L. C. Zhao and S. L. He, “Matter wave solitons in coupled system with external potentials,” Phys. Lett. A 375, 3017–3020(2011).
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Chin. Phys. Lett.

B. L. Guo and L. M. Ling, “Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations,” Chin. Phys. Lett. 28, 110202 (2011).
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Eur. Phys. J. Spec. Top.

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[CrossRef]

Eur. Phys. J., Spec. Top.

M. Vijayajayanthi, T. Kanna, and M. Lakshmanan, “Multisoliton solutions and energy sharing collisions in coupled nonlinear Schrödinger equations with focusing, defocusing and mixed type nonlinearities,” Eur. Phys. J., Spec. Top. 173, 57–80 (2009).

J. Eng. Math.

V. I. Shrira and V. V. Geogjaev, “What makes the Peregrine soliton so special as a prototype of freak waves?,” J. Eng. Math. 67, 11–22 (2010).
[CrossRef]

J. Phys. A

K. H. Han and H. J. Shin, “Nonautonomous integrable nonlinear Schrödinger equations with generalized external potentials,” J. Phys. A 42, 335202 (2009).
[CrossRef]

Nat. Phys.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

C. Becker, S. Stellmer, P. S. Panahi, S. Dorscher, M. Baumert, E.-M. Richter, J. Kronjager, K. Bongs, and K. Sengstock, “Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,” Nat. Phys. 4, 496–501 (2008).
[CrossRef]

Nature

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

M. G. Forest, S. P. Sheu, and P. C. Wright, “On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems,” Phys. Lett. A 266, 24–33 (2000).
[CrossRef]

L. C. Zhao and S. L. He, “Matter wave solitons in coupled system with external potentials,” Phys. Lett. A 375, 3017–3020(2011).
[CrossRef]

W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar-shaped Bose-Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
[CrossRef]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

Phys. Rev. A

P. Das, T. S. Raju, U. Roy, and P. K. Panigrahi, “Sinusoidal excitations in two-component Bose-Einstein condensates in a trap,” Phys. Rev. A 79, 015601 (2009).
[CrossRef]

E. V. Doktorov, V. M. Rothos, and Y. S. Kivshar, “Full-time dynamics of modulational instability in spinor Bose-Einstein condensates,” Phys. Rev. A 76, 013626 (2007).
[CrossRef]

P. Kockaert, P. Tassin, G. V. Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006).
[CrossRef]

T. Ueda and W. L. Lath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef]

M. Vijayajayanthi, T. Kanna, and M. Lakshmanan, “Bright-dark solitons and their collisions in mixed N-coupled nonlinear Schrödinger equations,” Phys. Rev. A 77, 013820 (2008).
[CrossRef]

Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
[CrossRef]

Phys. Rev. E

B. L. Guo, L. M. Ling, and Q. P. Liu, “Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions,” Phys. Rev. E 85, 026607 (2012).
[CrossRef]

M. Haelterman and A. Sheppard, “Bifurcation phenomena and multiple soliton-bound states in isotropic Kerr media,” Phys. Rev. E 49, 3376–3381 (1994).
[CrossRef]

Q. H. Park and H. J. Shin, “Systematic construction of multicomponent optical solitons,” Phys. Rev. E 61, 3093–3106 (2000).
[CrossRef]

Phys. Rev. Lett.

T. Kanna, and M. Lakshmanan, “Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 86, 5043–5046(2001).
[CrossRef]

P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and Boris A. Malomed, “Feshbach resonance management for Bose-Einstein condensates,” Phys. Rev. Lett. 90, 230401 (2003).
[CrossRef]

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, and K. Sengstock, “Dark solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5198–5201 (1999).
[CrossRef]

T. Busch, and J. R. Anglin, “Motion of dark solitons in trapped Bose-Einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[CrossRef]

C. K. Law, C. M. Chan, P. T. Leung, and M.-C. Chu, “Motional dressed states in a Bose-Einstein condensate: superfluidity at supersonic speed,” Phys. Rev. Lett. 85, 1598–1601 (2000).
[CrossRef]

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose-Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef]

B. Wu, J. Liu, and Q. Niu, “Controlled generation of dark solitons with phase imprinting,” Phys. Rev. Lett. 88, 034101 (2002).
[CrossRef]

C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, Ph. Emplit, and M. Haelterman, “Symmetry-breaking instability of multimode vector solitons,” Phys. Rev. Lett. 89, 083901 (2002).
[CrossRef]

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901(2011).
[CrossRef]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[CrossRef]

Quantum Electron.

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, “Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: I. Bright solitons,” Quantum Electron. 35, 778–786 (2005).
[CrossRef]

Science

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef]

J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schneider, and W. D. Phillips, “Generating solitons by phase engineering of a Bose-Einstein condensate,” Science 287, 97–101 (2000).
[CrossRef]

Other

N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, 1997).

Y. Ohta and J. K. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proc. R. Soc. Lond. Ser. A468, 1716–1740 (2012).

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Figures (8)

Fig. 1.
Fig. 1.

Evolution plot of B-DAB in the coupled system, (a) for bright soliton in ψ 1 component and (b) for dark with an AB in ψ 2 component. It is seen that the bright soliton in one field is reflected like hard wall by the AB in the other one. The dark soliton is reflected well by the AB too. The coefficients are a 0 = 0 , b 0 = 1.2 , g 1 = g 2 = 0.25 , s 1 = 0.001 , s 2 = 2 , A 1 = 1 , A 2 = 4 , A 3 = 3 , k 1 = 0.5 , and k 2 = 0.5 .

Fig. 2.
Fig. 2.

Evolution plot of ABs-ABs in two components coupled system (a) for ABs soliton in ψ 1 component and (b) for ABs in ψ 2 component. It is shown that one AB collides with a AB inelastically. The coefficients are a 0 = 0 , b 0 = 1.2 , g 1 = g 2 = 0.25 , s 1 = 1 , s 2 = 1 , A 1 = 1 , A 2 = 4 , A 3 = 3 , k 1 = 0.5 , and k 2 = 0.5 .

Fig. 3.
Fig. 3.

Evolution of B-DRW in the coupled system, (a) for one bright soliton with RW in ψ 1 component and (b) for one dark with a RW in ψ 2 component. It is shown that one dark soliton collides with RW elastically. Bright soliton are attracted by RW when being next to it. The coefficients are a 0 = 0.1 , b 0 = 2 , g 1 = g 2 = 0.25 , s 1 = 0.01 , s 2 = 2 , A 1 = 1 , A 2 = 4 , A 3 = 3 , k 1 = 1 , and k 2 = 0.1 .

Fig. 4.
Fig. 4.

Plot of AB-ABRW in the coupled system, (a) for AB with a RW in ψ 1 component and (b) for AB with a dark RW in ψ 2 component. For (a) and (b), A 1 = 1 , A 2 = 4 . (c) The plot of one RW in ψ 1 component and (d) the plot of one RW in ψ 2 component. It is seen that the RW in ψ 2 component is a dark one (c) and (d) with A 1 = A 2 = 0 . The other coefficients are a 0 = 0.53 , b 0 = 2.26 , g 1 = g 2 = 0.5 , s 1 = 1 , s 2 = 2 , A 3 = 3 , k 1 = 1 , and k 2 = 0.1 .

Fig. 5.
Fig. 5.

Evolution plot of only one RW in coupled system, (a) for one RW in ψ 1 component and (b) for one RW in ψ 2 component. The coefficients are g 1 = g 2 = 0.25 , s 1 = s 2 = 1 , A 1 = 1 , A 2 = 1 , A 3 = 0 , k 1 = 0.55 and k 2 = 0.05 .

Fig. 6.
Fig. 6.

Evolution plot of two RWs in coupled system, (a) for two similar RWs in ψ 1 component and (b) for two similar RWs in ψ 2 component. The coefficients are g 1 = g 2 = 0.25 , s 1 = s 2 = 4 , A 1 = 1 , A 2 = 1 , A 3 = 0.2 , k 1 = 2.05 and k 2 = 0.05 .

Fig. 7.
Fig. 7.

Evolution plot of two vector RWs in the coupled system, (a) for two different RWs in ψ 1 component and (b) for two different RWs in ψ 2 component. It is seen that dark RW could be observed in one component of the coupled system. The coefficients are g 1 = g 2 = 0.5 , s 1 = s 2 = 2 , A 1 = 0 , A 2 = 0 , A 3 = 158 k 1 = 0.807 , and k 2 = 0.05 .

Fig. 8.
Fig. 8.

Evolution of whole ABs-ABs density in the coupled system. The initial condition approaches the ideal condition for RWs, and the corresponding required backgrounds are set. It is seen that the evolution of them is close to RWs in Fig. 6. The explicit coefficients are a 0 = 1.05 , b 0 = 5.196 , g 1 = g 2 = 0.25 , s 1 = s 2 = 4 , A 1 = 1 , A 2 = 1 , A 3 = 6 , k 1 = 2.05 and k 2 = 0.05 .

Equations (34)

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i ψ 1 z + σ 1 2 ψ 1 t 2 + [ 2 g 1 | ψ 1 | 2 + 2 g 2 | ψ 2 | 2 ] ψ 1 = 0 ,
i ψ 2 z + σ 2 2 ψ 2 t 2 + [ 2 g 1 | ψ 1 | 2 + 2 g 2 | ψ 2 | 2 ] ψ 2 = 0 ,
ψ 10 = s 1 exp [ i θ 1 [ t , z ] ] = s 1 exp [ i k 1 t + i ( 2 g 1 s 1 2 + 2 g 2 s 2 2 i k 1 2 ) z ] ,
ψ 20 = s 2 exp [ i θ 2 [ t , z ] ] = s 2 exp [ i k 2 t + i ( 2 g 1 s 1 2 + 2 g 2 s 2 2 i k 2 2 ) z ] ,
t ( Φ 1 Φ 2 Φ 3 ) = U ( Φ 1 Φ 2 Φ 3 ) ,
z ( Φ 1 Φ 2 Φ 3 ) = V ( Φ 1 Φ 2 Φ 3 ) ,
U = ( i 2 3 λ g 1 ψ 1 g 2 ψ 2 g 1 ψ ¯ 1 i 3 λ 0 g 2 ψ ¯ 2 0 i 3 λ ) ,
V = U λ + ( J 1 i g 1 ψ 1 t i g 2 ψ 2 t i g 1 ψ ¯ 1 t J 2 i g 1 g 2 ψ 2 ψ ¯ 1 i g 2 ψ ¯ 2 t i g 1 g 2 ψ ¯ 2 ψ 1 J 3 ) ,
J 1 = i g 1 | ψ 1 | 2 + i g 2 | ψ 2 | 2 J 2 = i g 1 | ψ 1 | 2 , J 3 = i g 2 | ψ 2 | 2 .
τ 3 + a 1 τ + b 1 = 0 ,
a 1 = b c a b a c + g 1 s 1 2 + g 2 s 2 2 , b 1 = a b c g 1 s 1 2 c g 2 s 2 2 b , a = i 2 λ 3 + i 3 ( k 1 + k 2 ) , b = i λ 3 + i 3 ( 2 k 1 k 2 ) , c = i λ 3 + i 3 ( 2 k 2 k 1 ) .
Φ 1 [ t , z ] = ( Φ 01 + Φ 02 + Φ 03 ) × exp [ I 3 ( θ 1 [ t , z ] + θ 2 [ t , z ] ) ] ,
Φ 2 [ t , z ] = ( g 1 s 1 τ 1 b Φ 01 g 1 s 1 τ 2 b Φ 02 g 1 s 1 τ 3 b Φ 03 ) × exp [ I 3 ( θ 2 [ t , z ] 2 θ 1 [ t , z ] ) ] ,
Φ 3 [ t , z ] = ( g 2 s 2 τ 1 c Φ 01 g 2 s 2 τ 2 c Φ 02 g 2 s 2 τ 3 c Φ 03 ) × exp [ I 3 ( θ 1 [ t , z ] 2 θ 2 [ t , z ] ) ] ,
Φ 01 [ t , z ] = A 1 exp [ τ 1 t + I τ 1 2 z + 2 ( λ k 1 k 2 ) τ 1 z / 3 ] × exp [ F [ z ] ] , Φ 02 [ t , z ] = A 2 exp [ τ 2 t + I τ 2 2 z + 2 ( λ k 1 k 2 ) τ 2 z / 3 ] × exp [ F [ z ] ] , Φ 03 [ t , z ] = A 3 exp [ τ 3 t + I τ 3 2 z + 2 ( λ k 1 k 2 ) τ 3 z / 3 ] × exp [ F [ z ] ] ,
ψ 1 = ψ 10 1 g 1 i ( λ λ ¯ ) u ¯ 1 + | u | 2 + | v | 2 ,
ψ 2 = ψ 20 1 g 2 i ( λ λ ¯ ) v ¯ 1 + | u | 2 + | v | 2 ,
[ 3 ( k 1 + k 2 ) 2 + 4 ( C A ) ] λ 4 + [ 4 ( k 1 + k 2 ) 3 + ( k 1 + k 2 ) ( 8 C 6 A ) 4 B ] λ 3 + [ 4 ( k 1 + k 2 ) 2 C + 4 C 2 / 3 A 2 6 ( k 1 + k 2 ) B ] λ 2 + [ 4 ( k 1 + k 2 ) C 2 / 3 2 A B ] λ + 4 C 3 27 B 2 = 0 ,
A = 2 ( 2 k 1 k 2 ) ( 2 k 2 k 1 ) + ( k 1 + k 2 ) 2 + 9 g 1 s 1 2 + 9 g 2 s 2 2 , B = 9 g 1 s 1 2 ( 2 k 2 k 1 ) + ( k 1 + k 2 ) ( 2 k 1 k 2 ) ( 2 k 2 k 1 ) + 9 g 2 s 2 2 ( 2 k 1 k 2 ) , C = ( k 1 + k 2 ) 2 ( 2 k 1 k 2 ) ( 2 k 2 k 1 ) + 9 g 1 s 1 2 + 9 g 2 s 2 2 .
τ 1 = 2 τ 2 ,
τ 2 = τ 3 = H 1 ( λ ) H 2 ( λ ) ,
H 1 ( λ ) = I [ 2 λ 3 + 3 ( k 1 + k 2 ) λ 2 + ( 2 ( 2 k 1 k 2 ) ( 2 k 2 k 1 ) + ( k 1 + k 2 ) 2 + 9 g 1 s 1 2 + 9 g 2 s 2 2 ) λ + 9 g 2 s 2 2 ( 2 k 1 k 2 ) + 9 g 1 s 1 2 ( 2 k 2 k 1 ) + ( k 1 + k 2 ) ( 2 k 1 k 2 ) ( 2 k 2 k 1 ) ] , H 2 ( λ ) = 6 λ 2 + 6 ( k 1 + k 2 ) λ + 2 ( k 1 + k 2 ) 2 2 ( 2 k 1 k 2 ) ( 2 k 2 k 1 ) + 18 g 1 s 1 2 + 18 g 2 s 2 2 .
g 1 s 1 2 = g 2 s 2 2 ,
| k 1 k 2 | = g 1 s 1 ,
λ = k 1 + k 2 2 + I 3 3 2 g 1 s 1 .
ψ 1 = [ 1 + 3 3 g 1 s 1 W 1 ( t , z ) 1 + | u | 2 + | v | 2 ] s 1 e i θ 1 ( t , z ) ,
ψ 2 = [ 1 + 3 3 g 1 s 1 W 2 ( t , z ) 1 + | u | 2 + | v | 2 ] s 2 e i θ 2 ( t , z ) ,
ψ 1 = ψ 10 1 g 1 i ( λ λ ¯ ) Φ 1 Φ ¯ 2 | Φ 1 | 2 + | Φ 2 | 2 + | Φ 3 | 2 ,
ψ 2 = ψ 20 1 g 2 i ( λ λ ¯ ) Φ 1 Φ ¯ 3 | Φ 1 | 2 + | Φ 2 | 2 + | Φ 3 | 2 ,
Φ 1 [ t , z ] = ( Φ 01 + Φ 02 + Φ 03 ) exp [ I 3 ( θ 1 [ t , z ] + θ 2 [ t , z ] ) ] , Φ 2 [ t , z ] = ( g 1 s 1 τ 1 I λ / 3 I ( 2 k 1 k 2 ) / 3 Φ 01 [ t , z ] + g 1 s 1 τ 2 I λ / 3 I ( 2 k 1 k 2 ) / 3 Φ 02 [ t , z ] + g 1 s 1 g 1 s 1 τ 2 I λ / 3 I ( 2 k 1 k 2 ) / 3 τ 2 I λ / 3 I ( 2 k 1 k 2 ) / 3 Φ 03 [ t , z ] ) × exp [ I 3 ( θ 2 [ t , z ] 2 θ 1 [ t , z ] ) ] , Φ 3 [ t , z ] = ( g 2 s 2 τ 1 I λ / 3 I ( 2 k 2 k 1 ) / 3 Φ 01 [ t , z ] + g 2 s 2 τ 2 I λ / 3 I ( 2 k 2 k 1 ) / 3 Φ 02 [ t , z ] + g 2 s 2 g 2 s 2 τ 2 I λ / 3 I ( 2 k 2 k 1 ) / 3 τ 2 I λ / 3 I ( 2 k 2 k 1 ) / 3 Φ 03 [ t , z ] ) × exp [ I 3 ( θ 1 [ t , z ] 2 θ 2 [ t , z ] ) ] ,
Φ 01 = A 1 exp [ τ 1 t + I τ 1 2 z + 2 ( λ k 1 k 2 ) τ 1 z / 3 ] , Φ 02 = ( A 3 t + 2 A 3 I τ 2 z + 2 / 3 A 3 ( λ k 1 k 2 ) z + A 2 ) × exp [ τ 2 t + I τ 2 2 z + 2 ( λ k 1 k 2 ) τ 2 z / 3 ] , Φ 02 = A 3 exp [ τ 2 t + I τ 2 2 z + 2 ( λ k 1 k 2 ) τ 2 z / 3 ] .
W 1 , 2 ( t , z ) = 1 K 1 , 2 + 1 K 1 , 2 2 + P 1 , 2 ( t , z ) Q ( t , z )
K 1 = i ( k 1 k 2 ) 2 3 g 1 s 1 2 , K 2 = i ( k 2 k 1 ) 2 3 g 1 s 1 2 ,
Q ( t , z ) = 1 2 A 3 t 2 + 2 9 ( λ k 1 k 2 ) 2 A 3 z 2 + ( A 2 + A 3 ) t + 2 3 ( λ k 1 k 2 ) ( A 2 + A 3 ) z + i A 3 z + 2 3 ( λ k 1 k 2 ) A 3 t z + A 1 + A 2 + A 3 , P 1 , 2 ( t , z ) = A 3 K 1 , 2 3 1 K 1 , 2 2 M ( t , z ) , M ( t , z ) = 1 2 A 3 t 2 + 2 9 ( λ k 1 k 2 ) 2 A 3 z 2 + A 2 t + 2 3 ( λ k 1 k 2 ) A 2 z + i A 3 z + 2 3 ( λ k 1 k 2 ) A 3 t z + A 1 .

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