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Discrete rogue waves of the Ablowitz-Ladik and Hirota equations Adrian Ankiewicz,1
 

Summary: Discrete rogue waves of the Ablowitz-Ladik and Hirota equations
Adrian Ankiewicz,1
Nail Akhmediev,1
and J. M. Soto-Crespo2
1
Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra,
Australian Capital Territory 0200, Australia
2
Instituto de Óptica, C.S.I.C., Serrano 121, 28006 Madrid, Spain
Received 8 June 2010; published 11 August 2010
We show that the Ablowitz-Ladik equation, which is an integrable form of the discretized nonlinear
Schrödinger equation, has rogue wave solutions in the form of the rational solutions. We show that there is a
hierarchy of rational solutions and we derive the two lowest-order ones using the Hirota technique. More
generally, we present rational solutions for the discrete Hirota equation which includes, as particular cases,
both the discrete Ablowitz-Ladik equation and the discrete modified Korteweg-de Vries mKdV equation.
DOI: 10.1103/PhysRevE.82.026602 PACS number s : 05.45.Yv, 42.65.Tg, 42.65.Wi
I. INTRODUCTION
The Ablowitz-Ladik A-L equation 1,2 is an integrable
form of the discretized nonlinear Schrödinger equation
NLSE . As such, it has multiplicity of solutions which are

  

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