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PHYSICAL REVIEW E 83, 046603 (2011) Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift,
 

Summary: PHYSICAL REVIEW E 83, 046603 (2011)
Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift,
and exact solutions of the Ablowitz-Ladik equation
Nail Akhmediev and Adrian Ankiewicz
Optical Sciences Group, Research School of Physics and Engineering, Institute of Advanced Studies, The Australian National University,
Canberra ACT 0200, Australia
(Received 16 August 2010; revised manuscript received 18 February 2011; published 20 April 2011)
We study modulation instability (MI) of the discrete constant-background wave of the Ablowitz-Ladik (A-L)
equation. We derive exact solutions of the A-L equation which are nonlinear continuations of MI at longer
times. These periodic solutions comprise a family of two-parameter solutions with an arbitrary background
field and a frequency of initial perturbation. The solutions are recurrent, since they return the field state to the
original constant background solution after the process of nonlinear evolution has passed. These solutions can
be considered as a complete resolution of the Fermi-Pasta-Ulam paradox for the A-L system. One remarkable
consequence of the recurrent evolution is the nonlinear phase shift gained by the constant background wave after
the process. A particular case of this family is the rational solution of the first-order or fundamental rogue wave.
DOI: 10.1103/PhysRevE.83.046603 PACS number(s): 05.45.Yv, 45.05.+x, 02.30.Ik
I. INTRODUCTION
Modulation instability (MI) is a universal phenomenon that
occurs in many physical systems. It has been studied in optics
[1], hydrodynamics [2], and even in biology [3]. In the ocean,

  

Source: Akhmediev, Nail - Research School of Physical Sciences and Engineering, Australian National University
Australian National University, Research School of Physical Sciences and Engineering, Optical Sciences Group

 

Collections: Engineering; Physics