General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 468, Issue 2142, 2012, Pages 1716-1740

  Ohta, Yasuhiro ^a   Yang, Jianke ^b  

^a KOBE UNIVERSITY   (Japan)

^b University of Vermont ^*  (United States)

Author keywords

Bilinear method; Nonlinear Schr dinger equation; Rogue waves

Indexed keywords

DYNAMICS;

ALGEBRAIC EXPRESSION; BILINEAR METHOD; COMPLEX PARAMETER; DINGER EQUATION; FREE PARAMETERS; HIGH-ORDER; MATRIX ELEMENTS; PEAK AMPLITUDE; ROGUE WAVES; SOLUTION DYNAMICS; SPATIAL POSITIONS;

NONLINEAR EQUATIONS;

EID: 84862159587     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2011.0640     Document Type: Conference Paper

References (25)

1. Ablowitz, M. J. & Herbst, B. M. 1990 On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation. SIAM J. Appl. Math. 50, 339-351. (doi:10.1137/0150021) (Pubitemid 20670087)
2. Akhmediev, N., Eleonskii, V. M. & Kulagin, N. E. 1985 Generation of a periodic sequence of picosecond pulses in an optical fiber: exact solutions. Sov. Phys. JETP 89, 1542-1551. [In Russian.]
3. Akhmediev, N., Eleonskii, V. M. & Kulagin, N. E. 1988 Exact first-order solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 72, 809-818. (doi:10.1007/BF01017105)
4. Akhmediev, N., Ankiewicz, A. & Soto-Crespo, J. M. 2009a Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E 80, 026 601. (doi:10.1103/PhysRevE. 80.026601)
5. Akhmediev, N., Ankiewicz, A. & Taki, M. 2009b Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675-678. (doi:10.1016/j.physleta.2008.12.036)
6. Akhmediev, N., Soto-Crespo, J. M. & Ankiewicz, A. 2009c Extreme waves that appear from nowhere: on the nature of rogue waves. Phys. Lett. A 373, 2137-2145. (doi:10.1016/j.physleta. 2009.04.023)
7. Ankiewicz, A., Kedziora, D. J. & Akhmediev, N. 2011 Rogue wave triplets. Phys. Lett. A 375, 2782-2785. (doi:10.1016/j.physleta.2011.05.047)
8. Benney, D. J. & Newell, A. C. 1967 Nonlinear wave envelopes. J. Math. Phys. 46, 133-139.
9. Bertola, M. & Tovbis, A. 2010 Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: rational breathers and poles of the tritronquee solution to Painleve I. See http://arxiv.org/abs/ 1004.1828.
10. Dubard, P. & Matveev, V. B. 2011 Multi-rogue waves solutions to the focusing NLS equation and the KP-I equation. Nat. Hazards Earth. Syst. Sci. 11, 667-672. (doi:10.5194/nhess-11-667-2011)
11. Dubard, P., Gaillard, P., Klein, C. & Matveev, V. B. 2010 On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation. Eur. Phys. J. Spec. Top. 185, 247-258. (doi:10.1140/epjst/e2010-01252- 9)
12. Gaillard, P. 2011 Families of quasi-rational solutions of the NLS equation and multi-rogue waves. J. Phys. A Math. Theor. 44, 435204. (doi:10.1088/1751-8113/44/43/435204)
13. Hasegawa, A. & Tappert, F. 1973 Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. Appl. Phys. Lett. 23, 142. (doi:10.1063/1.1654836)
14. Hirota R. 2004 The direct method in soliton theory. Cambridge, UK: Cambridge University Press. doi:10.1017/CBO9780511543043)
15. Ikeda, T. & Yamada, H.-F. 2002 Polynomial t-functions of the NLS-Toda hierarchy and the Virasoro singular vectors. Lett. Math. Phys. 60, 147-156. (doi:10.1023/A:1016167008456)
16. Its, A. R., Rybin, A. V. & Salle, M. A. 1988 Exact integration of nonlinear Schrödinger equation. Theor. Math. Phys. 74, 29-45. (doi:10.1007/BF01018207)
17. Jimbo, M. & Miwa, T. 1983 Solitons and infinite dimensional Lie algebras. Publ. RIMS Kyoto Univ. 19, 943-1001. (doi:10.2977/prims/1195182017)
18. Kedziora, D. J., Ankiewicz, A. & Akhmediev, N. 2011 Circular rogue wave clusters. Phys. Rev. E 84, 056611. (doi:10.1103/PhysRevE.84.056611)
19. Kibler, B., Fatome, J., Finot, C., Millot, G., Dias, F., Genty, G., Akhmediev, N. & Dudley, J. M. 2010 The Peregrine soliton in nonlinear fibre optics. Nat. Phys. 6, 790-795. (doi:10.1038/ nphys1740)
20. Ohta, Y., Wang, D. & Yang, J. 2011 General N-dark-dark solitons in the coupled nonlinear Schrödinger equations. Stud. Appl. Math. 127, 345-371. (doi:10.1111/j.1467-9590.2011.00525.x)
21. Peregrine, D. H. 1983 Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. B 25, 16-43. (doi:10.1017/ S0334270000003891)
22. Sato, M. 1981 Soliton equations as dynamical systems on a infinite dimensional Grassmann manifolds. RIMS Kokyuroku 439, 30-46.
23. Solli, D. R., Ropers, C., Koonath, P. & Jalali, B. 2007 Optical rogue waves. Nature 450, 1054-1057. (doi:10.1038/nature06402) (Pubitemid 350273615)
24. Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190-194. (doi:10.1007/BF00913182)
25. Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62-69.