This paper is a continuation of a recent paper on the solutions of the focusing NLS equation. The representation in terms of a quotient of two determinants gives a very efficient method of determination of famous Peregrine breathers and their deformations. Here we construct Peregrine breathers of order and multi-rogue waves associated by deformation of parameters. The analytical expression corresponding to Peregrine breather is completely given.
A. R. Its and V. P. Kotlyarov, “Explicit expressions for the solutions of nonlinear Schrödinger equation,” Doklady Akademii Nauk SSSR A, vol. 965, no. 11, 1976.
View at: Google ScholarN. Akhmediev, V. Eleonsky, and N. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Soviet Physics, JETP, vol. 62, pp. 894–899, 1985.
View at: Google ScholarN. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Rogue waves and rational solutions of the nonlinear Schrödinger equation,” Physical Review E, vol. 80, no. 2, Article ID 026601, 2009.
View at: Publisher Site | Google ScholarN. N. Akhmediev, V. M. Eleonskiĭ, and N. E. Kulagin, “First-order exact solutions of the nonlinear Schrödinger equation,” Akademiya Nauk SSSR, vol. 72, no. 2, pp. 183–196, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNetP. Dubard, P. Gaillard, C. Klein, and V. B. Matveev, “On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation,” European Physical Journal, vol. 185, no. 1, pp. 247–258, 2010.
View at: Publisher Site | Google ScholarP. Gaillard, “Families of quasi-rational solutions of the NLS equation and multi-rogue waves,” Journal of Physics A, vol. 44, no. 43, Article ID 435204, pp. 1–15, 2011.
View at: Publisher Site | Google ScholarD. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” Australian Mathematical Society B, vol. 25, no. 1, pp. 16–43, 1983.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetA. Ankiewicz, P. A. Clarkson, and N. Akhmediev, “Rogue waves, rational solutions, the patterns of their zeros and integral relations,” Journal of Physics A, vol. 43, no. 12, Article ID 122002, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetB. Guo, L. Ling, and Q. P. Liu, “Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions,” Physical Review E, vol. 85, no. 2, Article ID 026607, 2012.
View at: Google ScholarY. Ohta and J. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proceedings of The Royal Society of London A, vol. 468, no. 2142, pp. 1716–1740, 2012.
View at: Publisher Site | Google Scholar | MathSciNetA. R. Its, A. V. Rybin, and M. A. Salle, “On the exact integration of the nonlinear Schrödinger equation,” Akademiya Nauk SSSR, vol. 74, no. 1, pp. 29–45, 1988.
View at: Publisher Site | Google Scholar | MathSciNetD. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Triangular rogue waves,” Physical Review E, vol. 86, no. 5, Article ID 056602, pp. 1–9, 2012.
View at: Publisher Site | Google Scholar