

Nonlinear stage of modulation instability in the scalar and vector nonlinear Schrodinger equations.
Date/Time: 12:20 04Aug2014
Abstract:
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 onedimensional focusing nonlinear Schrodinger equation (NLSE). This 2Nsolitonic 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 quasiAkhmediev breathers. Reverse is also true ? at a certain moment of time 2N quasiAkhmediev 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 [45]) 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 Nsolitonic solution on the condensate background for twocomponent coupled NLSE (Manakov system) and threecomponent coupled NLSE. In the scalar NLSE the mechanism of almost full quasiAkhmediev breathers annihilation mathematically explained by the nulling of twosolitonic 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, R1R39 (2014) 3. D. H. Peregrine. Water waves, Nonlinear Schrodinger Equations and their solutions//J. Aust. Math. Soc. Ser. B. 25, 1643 (1983) 4. N. Akhmediev, A. Ankiewicz and J.M. SotoCrespo. Rogue waves and rational solutions of the nonlinear Schrodinger Equation//Phys. Rev. E. 80, 026601 (2009) 5. P. Dubard, P. Gaillard, C. Klein C and V.B. Matveev. Multirogue waves solutions of the focusing NLS equation and the KPI equation// Eur. Phys. J. Special Topics. 185, 24758 (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, 1122 Attachments:
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