Abstract

Nonlinear Schrödinger equation with simple quadratic potential modulated by a spatially-varying diffraction coefficient is investigated theoretically. Second-order rogue wave breather solutions of the model are constructed by using the similarity transformation. A modal quantum number is introduced, useful for classifying and controlling the solutions. From the solutions obtained, the behavior of second order Kuznetsov-Ma breathers (KMBs), Akhmediev breathers (ABs), and Peregrine solitons is analyzed in particular, by selecting different modulation frequencies and quantum modal parameter. We show how to generate interesting second order breathers and related hybrid rogue waves. The emergence of true rogue waves – single giant waves that are generated in the interaction of KMBs, ABs, and Peregrine solitons – is explicitly displayed in our analytical solutions.

© 2015 Optical Society of America

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References

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2014 (6)

Y. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, J. P. Song, and Y. P. Zhang, “Nonlinear Talbot effect of rogue waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(3), 032902 (2014).
[Crossref] [PubMed]

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

S. Loomba, R. Gupta, H. Kaur, and M. S. Mani Rajan, “Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation,” Phys. Lett. A 378(30), 2137–2141 (2014).
[Crossref]

W. P. Zhong, L. Chen, M. Belić, and N. Petrović, “Controllable parabolic-cylinder optical rogue wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 043201 (2014).
[Crossref] [PubMed]

C. Q. Dai and W. H. Huang, “Controllable mechanism of breathers in the (2+1)-dimensional nonlinear Schrödinger equation with different forms of distributed transverse diffraction,” Phys. Lett. A 378(16), 1113–1118 (2014).
[Crossref]

K. Manikandan, P. Muruganandam, M. Senthilvelan, and M. Lakshmanan, “Manipulating matter rogue waves and breathers in Bose-Einstein condensates,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(6), 062905 (2014).
[Crossref] [PubMed]

2013 (5)

J. S. He, Y. S. Tao, K. Porsezian, and A. S. Fokas, “Rogue wave management in an inhomogeneous Nonlinear Fibre with higher order effects,” J. Nonlin. Math. Phys. 20(3), 407–419 (2013).
[Crossref]

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 060201 (2013).
[Crossref]

W. P. Zhong, M. R. Belić, and T. Huang, “Rogue wave solutions to the generalized nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(6), 065201 (2013).
[Crossref] [PubMed]

Y. Zhang, M. R. Belić, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. P. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38(22), 4585–4588 (2013).
[PubMed]

2012 (4)

W. P. Zhong, “Rogue wave solutions of the generalized one-dimensional Gross-Pitaevskii equation,” J. Nonlinear Opt. Phys. Mater. 21(2), 1250026 (2012).
[Crossref]

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86(2), 025802 (2012).
[Crossref]

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1), 016603 (2012).
[Crossref] [PubMed]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(6), 066601 (2012).
[Crossref] [PubMed]

2010 (1)

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak Waves in the Linear Regime: A Microwave Study,” Phys. Rev. Lett. 104(9), 093901 (2010).
[Crossref] [PubMed]

2009 (4)

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373(6), 675–678 (2009).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “How to excite a rogue wave,” Phys. Rev. A 80(4), 043818 (2009).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103(17), 173901 (2009).
[Crossref] [PubMed]

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80(3), 033610 (2009).
[Crossref]

2008 (2)

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an Inverse Energy Cascade in Developed Acoustic Turbulence in Superfluid Helium,” Phys. Rev. Lett. 101(6), 065303 (2008).
[Crossref] [PubMed]

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16(6), 3644–3651 (2008).
[Crossref] [PubMed]

1988 (1)

N. Akhmediev and J. M. Dudley, “N-modulation signals in a single mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP 67(1), 89–95 (1988).

1986 (1)

N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69(2), 1089–1093 (1986).
[Crossref]

1983 (1)

D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. Ser. B 25, 16–43 (1983).

1979 (1)

Y. C. Ma, “The perturbed plane-wave solution of the cubic Schrödinger equation,” Stud. Appl. Math. 60, 43–58 (1979).

1977 (1)

E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).

Akhmediev, N.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 060201 (2013).
[Crossref]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(6), 066601 (2012).
[Crossref] [PubMed]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373(6), 675–678 (2009).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “How to excite a rogue wave,” Phys. Rev. A 80(4), 043818 (2009).
[Crossref]

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80(3), 033610 (2009).
[Crossref]

N. Akhmediev and J. M. Dudley, “N-modulation signals in a single mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP 67(1), 89–95 (1988).

N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69(2), 1089–1093 (1986).
[Crossref]

Ankiewicz, A.

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(6), 066601 (2012).
[Crossref] [PubMed]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “How to excite a rogue wave,” Phys. Rev. A 80(4), 043818 (2009).
[Crossref]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373(6), 675–678 (2009).
[Crossref]

Arecchi, F. T.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103(17), 173901 (2009).
[Crossref] [PubMed]

Belic, M.

W. P. Zhong, L. Chen, M. Belić, and N. Petrović, “Controllable parabolic-cylinder optical rogue wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 043201 (2014).
[Crossref] [PubMed]

Belic, M. R.

Y. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, J. P. Song, and Y. P. Zhang, “Nonlinear Talbot effect of rogue waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(3), 032902 (2014).
[Crossref] [PubMed]

W. P. Zhong, M. R. Belić, and T. Huang, “Rogue wave solutions to the generalized nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(6), 065201 (2013).
[Crossref] [PubMed]

Y. Zhang, M. R. Belić, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. P. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38(22), 4585–4588 (2013).
[PubMed]

Bludov, Yu. V.

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80(3), 033610 (2009).
[Crossref]

Bortolozzo, U.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103(17), 173901 (2009).
[Crossref] [PubMed]

Chen, H. X.

Y. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, J. P. Song, and Y. P. Zhang, “Nonlinear Talbot effect of rogue waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(3), 032902 (2014).
[Crossref] [PubMed]

Chen, L.

W. P. Zhong, L. Chen, M. Belić, and N. Petrović, “Controllable parabolic-cylinder optical rogue wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 043201 (2014).
[Crossref] [PubMed]

Dai, C. Q.

C. Q. Dai and W. H. Huang, “Controllable mechanism of breathers in the (2+1)-dimensional nonlinear Schrödinger equation with different forms of distributed transverse diffraction,” Phys. Lett. A 378(16), 1113–1118 (2014).
[Crossref]

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1), 016603 (2012).
[Crossref] [PubMed]

Dias, F.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

Dudley, J. M.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 060201 (2013).
[Crossref]

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16(6), 3644–3651 (2008).
[Crossref] [PubMed]

N. Akhmediev and J. M. Dudley, “N-modulation signals in a single mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP 67(1), 89–95 (1988).

Efimov, V. B.

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an Inverse Energy Cascade in Developed Acoustic Turbulence in Superfluid Helium,” Phys. Rev. Lett. 101(6), 065303 (2008).
[Crossref] [PubMed]

Eggleton, B. J.

Erkintalo, M.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

Fokas, A. S.

J. S. He, Y. S. Tao, K. Porsezian, and A. S. Fokas, “Rogue wave management in an inhomogeneous Nonlinear Fibre with higher order effects,” J. Nonlin. Math. Phys. 20(3), 407–419 (2013).
[Crossref]

Ganshin, A. N.

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an Inverse Energy Cascade in Developed Acoustic Turbulence in Superfluid Helium,” Phys. Rev. Lett. 101(6), 065303 (2008).
[Crossref] [PubMed]

Genty, G.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16(6), 3644–3651 (2008).
[Crossref] [PubMed]

Goyal, A.

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86(2), 025802 (2012).
[Crossref]

Gupta, R.

S. Loomba, R. Gupta, H. Kaur, and M. S. Mani Rajan, “Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation,” Phys. Lett. A 378(30), 2137–2141 (2014).
[Crossref]

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86(2), 025802 (2012).
[Crossref]

He, J. S.

J. S. He, Y. S. Tao, K. Porsezian, and A. S. Fokas, “Rogue wave management in an inhomogeneous Nonlinear Fibre with higher order effects,” J. Nonlin. Math. Phys. 20(3), 407–419 (2013).
[Crossref]

Heller, E. J.

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak Waves in the Linear Regime: A Microwave Study,” Phys. Rev. Lett. 104(9), 093901 (2010).
[Crossref] [PubMed]

Höhmann, R.

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak Waves in the Linear Regime: A Microwave Study,” Phys. Rev. Lett. 104(9), 093901 (2010).
[Crossref] [PubMed]

Huang, T.

W. P. Zhong, M. R. Belić, and T. Huang, “Rogue wave solutions to the generalized nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(6), 065201 (2013).
[Crossref] [PubMed]

Huang, W. H.

C. Q. Dai and W. H. Huang, “Controllable mechanism of breathers in the (2+1)-dimensional nonlinear Schrödinger equation with different forms of distributed transverse diffraction,” Phys. Lett. A 378(16), 1113–1118 (2014).
[Crossref]

Kaplan, L.

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak Waves in the Linear Regime: A Microwave Study,” Phys. Rev. Lett. 104(9), 093901 (2010).
[Crossref] [PubMed]

Kaur, H.

S. Loomba, R. Gupta, H. Kaur, and M. S. Mani Rajan, “Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation,” Phys. Lett. A 378(30), 2137–2141 (2014).
[Crossref]

Kedziora, D. J.

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(6), 066601 (2012).
[Crossref] [PubMed]

Kolmakov, G. V.

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an Inverse Energy Cascade in Developed Acoustic Turbulence in Superfluid Helium,” Phys. Rev. Lett. 101(6), 065303 (2008).
[Crossref] [PubMed]

Konotop, V. V.

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80(3), 033610 (2009).
[Crossref]

Korneev, V. I.

N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69(2), 1089–1093 (1986).
[Crossref]

Kuhl, U.

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak Waves in the Linear Regime: A Microwave Study,” Phys. Rev. Lett. 104(9), 093901 (2010).
[Crossref] [PubMed]

Kumar, C. N.

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86(2), 025802 (2012).
[Crossref]

Kuznetsov, E. A.

E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).

Lakshmanan, M.

K. Manikandan, P. Muruganandam, M. Senthilvelan, and M. Lakshmanan, “Manipulating matter rogue waves and breathers in Bose-Einstein condensates,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(6), 062905 (2014).
[Crossref] [PubMed]

Li, C. B.

Y. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, J. P. Song, and Y. P. Zhang, “Nonlinear Talbot effect of rogue waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(3), 032902 (2014).
[Crossref] [PubMed]

Li, Y.

Loomba, S.

S. Loomba, R. Gupta, H. Kaur, and M. S. Mani Rajan, “Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation,” Phys. Lett. A 378(30), 2137–2141 (2014).
[Crossref]

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86(2), 025802 (2012).
[Crossref]

Lu, K.

Ma, Y. C.

Y. C. Ma, “The perturbed plane-wave solution of the cubic Schrödinger equation,” Stud. Appl. Math. 60, 43–58 (1979).

Mani Rajan, M. S.

S. Loomba, R. Gupta, H. Kaur, and M. S. Mani Rajan, “Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation,” Phys. Lett. A 378(30), 2137–2141 (2014).
[Crossref]

Manikandan, K.

K. Manikandan, P. Muruganandam, M. Senthilvelan, and M. Lakshmanan, “Manipulating matter rogue waves and breathers in Bose-Einstein condensates,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(6), 062905 (2014).
[Crossref] [PubMed]

McClintock, P. V. E.

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an Inverse Energy Cascade in Developed Acoustic Turbulence in Superfluid Helium,” Phys. Rev. Lett. 101(6), 065303 (2008).
[Crossref] [PubMed]

Mezhov-Deglin, L. P.

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an Inverse Energy Cascade in Developed Acoustic Turbulence in Superfluid Helium,” Phys. Rev. Lett. 101(6), 065303 (2008).
[Crossref] [PubMed]

Montina, A.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103(17), 173901 (2009).
[Crossref] [PubMed]

Muruganandam, P.

K. Manikandan, P. Muruganandam, M. Senthilvelan, and M. Lakshmanan, “Manipulating matter rogue waves and breathers in Bose-Einstein condensates,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(6), 062905 (2014).
[Crossref] [PubMed]

Onorato, M.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

Panigrahi, P. K.

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86(2), 025802 (2012).
[Crossref]

Peregrine, D. H.

D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. Ser. B 25, 16–43 (1983).

Petrovic, N.

W. P. Zhong, L. Chen, M. Belić, and N. Petrović, “Controllable parabolic-cylinder optical rogue wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 043201 (2014).
[Crossref] [PubMed]

Porsezian, K.

J. S. He, Y. S. Tao, K. Porsezian, and A. S. Fokas, “Rogue wave management in an inhomogeneous Nonlinear Fibre with higher order effects,” J. Nonlin. Math. Phys. 20(3), 407–419 (2013).
[Crossref]

Raju, T. S.

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86(2), 025802 (2012).
[Crossref]

Residori, S.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103(17), 173901 (2009).
[Crossref] [PubMed]

Senthilvelan, M.

K. Manikandan, P. Muruganandam, M. Senthilvelan, and M. Lakshmanan, “Manipulating matter rogue waves and breathers in Bose-Einstein condensates,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(6), 062905 (2014).
[Crossref] [PubMed]

Solli, D. R.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 060201 (2013).
[Crossref]

Song, J. P.

Y. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, J. P. Song, and Y. P. Zhang, “Nonlinear Talbot effect of rogue waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(3), 032902 (2014).
[Crossref] [PubMed]

Soto-Crespo, J. M.

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “How to excite a rogue wave,” Phys. Rev. A 80(4), 043818 (2009).
[Crossref]

Stöckmann, H. J.

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak Waves in the Linear Regime: A Microwave Study,” Phys. Rev. Lett. 104(9), 093901 (2010).
[Crossref] [PubMed]

Taki, M.

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373(6), 675–678 (2009).
[Crossref]

Tao, Y. S.

J. S. He, Y. S. Tao, K. Porsezian, and A. S. Fokas, “Rogue wave management in an inhomogeneous Nonlinear Fibre with higher order effects,” J. Nonlin. Math. Phys. 20(3), 407–419 (2013).
[Crossref]

Turitsyn, S. K.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 060201 (2013).
[Crossref]

Wu, Z.

Zhang, J. F.

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1), 016603 (2012).
[Crossref] [PubMed]

Zhang, Y.

Y. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, J. P. Song, and Y. P. Zhang, “Nonlinear Talbot effect of rogue waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(3), 032902 (2014).
[Crossref] [PubMed]

Y. Zhang, M. R. Belić, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. P. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38(22), 4585–4588 (2013).
[PubMed]

Zhang, Y. P.

Y. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, J. P. Song, and Y. P. Zhang, “Nonlinear Talbot effect of rogue waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(3), 032902 (2014).
[Crossref] [PubMed]

Y. Zhang, M. R. Belić, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. P. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38(22), 4585–4588 (2013).
[PubMed]

Zheng, H.

Zheng, H. B.

Y. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, J. P. Song, and Y. P. Zhang, “Nonlinear Talbot effect of rogue waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(3), 032902 (2014).
[Crossref] [PubMed]

Zhong, W. P.

W. P. Zhong, L. Chen, M. Belić, and N. Petrović, “Controllable parabolic-cylinder optical rogue wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 043201 (2014).
[Crossref] [PubMed]

W. P. Zhong, M. R. Belić, and T. Huang, “Rogue wave solutions to the generalized nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(6), 065201 (2013).
[Crossref] [PubMed]

W. P. Zhong, “Rogue wave solutions of the generalized one-dimensional Gross-Pitaevskii equation,” J. Nonlinear Opt. Phys. Mater. 21(2), 1250026 (2012).
[Crossref]

Zhou, G. Q.

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1), 016603 (2012).
[Crossref] [PubMed]

Dokl. Akad. Nauk SSSR (1)

E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).

J. Aust. Math. Soc. Ser. B (1)

D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. Ser. B 25, 16–43 (1983).

J. Nonlin. Math. Phys. (1)

J. S. He, Y. S. Tao, K. Porsezian, and A. S. Fokas, “Rogue wave management in an inhomogeneous Nonlinear Fibre with higher order effects,” J. Nonlin. Math. Phys. 20(3), 407–419 (2013).
[Crossref]

J. Nonlinear Opt. Phys. Mater. (1)

W. P. Zhong, “Rogue wave solutions of the generalized one-dimensional Gross-Pitaevskii equation,” J. Nonlinear Opt. Phys. Mater. 21(2), 1250026 (2012).
[Crossref]

J. Opt. (1)

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15(6), 060201 (2013).
[Crossref]

Nat. Photonics (1)

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (3)

C. Q. Dai and W. H. Huang, “Controllable mechanism of breathers in the (2+1)-dimensional nonlinear Schrödinger equation with different forms of distributed transverse diffraction,” Phys. Lett. A 378(16), 1113–1118 (2014).
[Crossref]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373(6), 675–678 (2009).
[Crossref]

S. Loomba, R. Gupta, H. Kaur, and M. S. Mani Rajan, “Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation,” Phys. Lett. A 378(30), 2137–2141 (2014).
[Crossref]

Phys. Rep. (1)

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013).
[Crossref]

Phys. Rev. A (3)

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80(3), 033610 (2009).
[Crossref]

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86(2), 025802 (2012).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “How to excite a rogue wave,” Phys. Rev. A 80(4), 043818 (2009).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (6)

K. Manikandan, P. Muruganandam, M. Senthilvelan, and M. Lakshmanan, “Manipulating matter rogue waves and breathers in Bose-Einstein condensates,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(6), 062905 (2014).
[Crossref] [PubMed]

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1), 016603 (2012).
[Crossref] [PubMed]

W. P. Zhong, M. R. Belić, and T. Huang, “Rogue wave solutions to the generalized nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(6), 065201 (2013).
[Crossref] [PubMed]

Y. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, J. P. Song, and Y. P. Zhang, “Nonlinear Talbot effect of rogue waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(3), 032902 (2014).
[Crossref] [PubMed]

W. P. Zhong, L. Chen, M. Belić, and N. Petrović, “Controllable parabolic-cylinder optical rogue wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 043201 (2014).
[Crossref] [PubMed]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(6), 066601 (2012).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an Inverse Energy Cascade in Developed Acoustic Turbulence in Superfluid Helium,” Phys. Rev. Lett. 101(6), 065303 (2008).
[Crossref] [PubMed]

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak Waves in the Linear Regime: A Microwave Study,” Phys. Rev. Lett. 104(9), 093901 (2010).
[Crossref] [PubMed]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103(17), 173901 (2009).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

N. Akhmediev and J. M. Dudley, “N-modulation signals in a single mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP 67(1), 89–95 (1988).

Stud. Appl. Math. (1)

Y. C. Ma, “The perturbed plane-wave solution of the cubic Schrödinger equation,” Stud. Appl. Math. 60, 43–58 (1979).

Theor. Math. Phys. (1)

N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69(2), 1089–1093 (1986).
[Crossref]

Other (6)

B. A. Balomed, Nonlinear Schrödinger equations, in: Encyclopedia of Nonlinear Science, ed. by A. Scott (Routledge, 2005).

Y. S. Kivshar and G. Agrawal, Optical solitons: from fibers to photonic crystals (Academic, 2003).

V. B. Matveev and M. Salle, Darboux Transformations and Solitons (Springer-Verlag, 1991).

C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue waves in the ocean (Springer, 2009).

A. R. Osborne, Nonlinear Ocean Waves (Academic Press, 2009).

C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University Press, 2002)

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Figures (4)

Fig. 1
Fig. 1 Profiles of the second-order breathers, viewed as a nonlinear superposition of KMBs and Peregrine solitons. Here k1 = 1.7i and k2 = 0.3i for m = 0,1,2 from left to right. (a) Without transverse shifts. (b) With beam shifts x1 = 1 and x2 = 1.
Fig. 2
Fig. 2 Second-order breather solutions based on Peregrine solitons and KMBs. The setup and parameters are as in Fig. 1, except that k1 = 1.2i and k2 = 0.8i. (a) Without transverse shifts. (b) With shifts x1 = 1 and x2 = 1.
Fig. 3
Fig. 3 Patterns of the second-order breathers with k1 = 1.3i and k2 = 1.4i for m = 0,1 from left to right. (a) Without shifts. (b) With shifts x1 = 1, x2 = 1, z1 = 5 and z2 = 5, respectively.
Fig. 4
Fig. 4 KM breathers crossing an AB with k1 = 1.7i and k2 = 0.5 + 1.7i for m = 0,2 from left to right (without shifts).

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

i u z + 1 2 β( x ) 2 u x 2 +χ( x ) | u | 2 u+ 1 2 β( x )( 1 4 x 2 +m+ 1 2 )u=0,
u( z,x )= 1 2π m! D m ( x )V( z,Y ),
i V z + 1 2 2 V Y 2 +|V | 2 V=0.
V( z,Y )=[ 1+ G 2 ( z,Y )+i H 2 ( z,Y ) D 2 ( z,Y ) ] e iz ,
G 2 =( k 1 2 k 2 2 ) [ k 1 2 δ 2 k 2 cosh( δ 1 z s1 )cos( k 2 Y s2 )( k 1 2 k 2 2 )cosh( δ 1 z s1 )cosh( δ 2 z s2 ) k 2 2 δ 1 k 1 cosh( δ 2 z s2 )cos( k 1 Y s1 ) ], H 2 =2( k 1 2 k 2 2 ) δ 1 δ 2 [ 1 k 2 sinh( δ 1 z s1 )cos( k 2 Y s2 ) 1 k 1 sinh( δ 2 z s2 )cos( k 1 Y s1 ) 1 δ 2 sinh( δ 1 z s1 )cosh( δ 2 z s2 )+ 1 δ 1 sinh( δ 2 z s2 )cosh( δ 1 z s1 ) ], D 2 =2( k 1 2 + k 2 2 ) δ 1 δ 2 k 1 k 2 cos( k 1 Y s1 )cos( k 2 Y s2 )+ 4 δ 1 δ 2 [ sin( k 1 Y s1 )sin( k 2 Y s2 )+sinh( δ 1 z s1 )sinh( δ 2 z s2 ) ] ( 2 k 1 2 k 1 2 k 2 2 +2 k 2 2 )cosh( δ 1 z s1 )cosh( δ 2 z s2 ) 2( k 1 2 k 2 2 )[ δ 1 k 1 cos( k 1 Y s1 )cosh( δ 2 z s2 ) δ 2 k 2 cos( k 2 Y s2 )cosh( δ 1 z s1 ) ].

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