VII-th International Conference "SOLITONS, COLLAPSES AND TURBULENCE: Achievements, Developments and Perspectives" (SCT-14) in honor of Vladimir Zakharov's 75th birthday
August, 04-08, 2014
Chernogolovka, Russia

Nonlinear stage of modulation instability in the scalar and vector nonlinear Schrodinger equations.
Date/Time: 12:20 04-Aug-2014
Superregular solitonic solutions recently discovered by Vladimir E. Zakharov and Andrey A. Gelash [1] is the important scenario of modulation instability in the frame of the one-dimensional focusing nonlinear Schrodinger equation (NLSE). This 2N-solitonic solutions describe the evolution of a broad class of an initial plane wave (a condensate) localized perturbations. In this scenario a perturbation develops into 2N quasi-Akhmediev breathers. Reverse is also true ? at a certain moment of time 2N quasi-Akhmediev breathers are almost annihilate into small localized solution. Recently we have generalized the theory to the degenerate case [2]. Superregular solitonic solutions form the full solitonic part of localized perturbation spectrum with rational solutions (Peregrine soliton [3] and its multisolitonic analogs [4-5]) and Kuznetsov solitons with spectral parameter close to branch point [6]. In this work we discuss the theory of superregular solitonic solutions and the possibility of their experimental observation. We present the most appropriate solution parameters for hydrodynamics and optics.
The second part of this work is devoted to vector generalizations of superregular solutions. By using the dressing method we construct the N-solitonic solution on the condensate background for two-component coupled NLSE (Manakov system) and three-component coupled NLSE. In the scalar NLSE the mechanism of almost full quasi-Akhmediev breathers annihilation mathematically explained by the nulling of two-solitonic Akhmediev solution with the same spectral parameter [1,2]. We find the similar conditions in the vector case. Then we construct and discuss the vector superregular solitonic solutions.

1. V.E. Zakharov and A.A. Gelash. Nonlinear Stage of Modulation Instability// Phys. Rev. Lett. 111, 054101 (2013)
2. A.A. Gelash and V.E. Zakharov. Superregular solitonic solutions: a novel scenario for the nonlinear stage of Modulation Instability// Nonlinearity, 27, R1-R39 (2014)
3. D. H. Peregrine. Water waves, Nonlinear Schrodinger Equations and their solutions//J. Aust. Math. Soc. Ser. B. 25, 16--43 (1983)
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5. P. Dubard, P. Gaillard, C. Klein C and V.B. Matveev. Multi-rogue waves solutions of the focusing NLS equation and the KP-I equation// Eur. Phys. J. Special Topics. 185, 247--58 (2010)
6. V. I. Shrira and V. V. Geogjaev 2010. What makes the Peregrine soliton so special as a prototype of freak waves?// J. Eng. Math. 67, 11--22

Zakharov Vladimir E.
(no additional information)

Gelash Andrey A. (Presenter)
(no additional information)

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