Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability

We describe a general N-solitonic solution of the focusing NLSE in the presence of a condensate by using the dressing method. We give the explicit form of one- and two- solitonic solutions and study them in detail. We distinguish a special class of solutions that we call regular solitonic solutions. Regular solitonic solutions do not disturb phases of the condensate at infinity by coordinate. All of them can be treated as localized perturbations of the condensate. We find a broad class of superregular solitonic solutions which are small perturbations at certain a moment of time. Superregular solitonic solutions are generated by pairs of poles located on opposite sides of the cut. They describe the nonlinear stage of the modulation instability of the condensate and play an important role in the theory of freak waves.

[1]  V. Zakharov,et al.  On the formation of freak waves on the surface of deep water , 2008 .

[2]  A. Its,et al.  Exact integration of nonlinear Schrödinger equation , 1988 .

[3]  N. Hoffmann,et al.  Rogue wave observation in a water wave tank. , 2011, Physical review letters.

[4]  Yan‐Chow Ma,et al.  The Perturbed Plane‐Wave Solutions of the Cubic Schrödinger Equation , 1979 .

[5]  V. Zakharov,et al.  Soliton on Unstable Condensate , 2011, 1109.0620.

[6]  A. Balakin,et al.  Structural features of the self-action dynamics of ultrashort electromagnetic pulses , 2007 .

[7]  Vladimir E. Zakharov,et al.  Stability of periodic waves of finite amplitude on the surface of a deep fluid , 1968 .

[8]  E. Pelinovsky,et al.  Extreme ocean waves , 2008 .

[9]  J. Soto-Crespo,et al.  How to excite a rogue wave , 2009 .

[10]  Adrian Ankiewicz,et al.  Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  V. Matveev,et al.  Multi-rogue waves solutions to the focusing NLS equation and the KP-I equation , 2011 .

[12]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[13]  E. Pelinovsky,et al.  Nonlinear wave focusing on water of finite depth , 2002 .

[14]  Frédéric Dias,et al.  The Peregrine soliton in nonlinear fibre optics , 2010 .

[15]  V. Zakharov,et al.  New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface , 2002 .

[16]  Alexey Slunyaev.  Nonlinear analysis and simulations of measured freak wave time series , 2006 .

[17]  Hiroshi Inoue,et al.  Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions , 1978 .

[18]  Charles H. Townes,et al.  Self-trapping of optical beams , 1964 .

[19]  Yosuke Watanabe,et al.  Breather solutions to the focusing nonlinear Schrödinger equation , 1998 .

[20]  J. Soto-Crespo,et al.  Rogue waves and rational solutions of the nonlinear Schrödinger equation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Pierre Gaillard,et al.  On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation , 2010 .

[22]  Vladimir E. Zakharov,et al.  Optical solitons and quasisolitons , 1998 .

[23]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[24]  V. Zakharov,et al.  Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II , 1979 .

[25]  Dmitry Chalikov,et al.  Modeling extreme waves based on equations of potential flow with a free surface , 2005 .

[26]  J. Soto-Crespo,et al.  Extreme waves that appear from nowhere: On the nature of rogue waves , 2009 .

[27]  Yuri S. Kivshar,et al.  Optical Solitons: From Fibers to Photonic Crystals , 2003 .

[28]  V. Zakharov,et al.  Freak waves as nonlinear stage of Stokes wave modulation instability , 2006 .

[29]  V. Shrira,et al.  What makes the Peregrine soliton so special as a prototype of freak waves? , 2010 .

[30]  D. H. Peregrine,et al.  Water waves, nonlinear Schrödinger equations and their solutions , 1983, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[31]  Karsten Trulsen,et al.  NOTE ON BREATHER TYPE SOLUTIONS OF THE NLS AS MODELS FOR FREAK-WAVES , 1999 .

[32]  About shape of giant breather , 2010 .

[33]  V E Zakharov,et al.  Nonlinear stage of modulation instability. , 2012, Physical review letters.

[34]  J. Fatome,et al.  Observation of Kuznetsov-Ma soliton dynamics in optical fibre , 2012, Scientific Reports.

[35]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .