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Externally forced triads of resonantly interacting waves : boundedness and integrability properties
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Harris, J. (Jamie), Bustamante, Miguel D. and Connaughton, Colm. (2012) Externally forced triads of resonantly interacting waves : boundedness and integrability properties. Communications in Nonlinear Science and Numerical Simulation, Vol.17 (No.12). pp. 49885006. ISSN 10075704

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WRAP_Connaughton_Externally_Forced_triads1201.2867v1.pdf  Submitted Version Download (2421Kb) 
Official URL: http://dx.doi.org/10.1016/j.cnsns.2012.04.002
Abstract
We revisit the problem of a triad of resonantly interacting nonlinear waves driven by an external force applied to the unstable mode of the triad. The equations are Hamiltonian, and can be reduced to a dynamical system for 5 real variables with 2 conservation laws. If the Hamiltonian, H, is zero we reduce this dynamical system to the motion of a particle in a onedimensional timeindependent potential and prove that the system is integrable. Explicit solutions are obtained for some particular initial conditions. When explicit solution is not possible we present a novel numerical/analytical method for approximating the dynamics. Furthermore we show analytically that when H=0 the motion is generically bounded. That is to say the waves in the forced triad are bounded in amplitude for all times for any initial condition with the single exception of one special choice of initial condition for which the forcing is in phase with the nonlinear oscillation of the triad. This means that the energy in the forced triad generically remains finite for all time despite the fact that there is no dissipation in the system. We provide a detailed characterisation of the dependence of the period and maximum energy of the system on the conserved quantities and forcing intensity. When H ≠ 0 we reduce the problem to the motion of a particle in a onedimensional timeperiodic potential. Poincaré sections of this system provide strong evidence that the motion remains bounded when H ≠ 0 and is typically quasiperiodic although periodic orbits can certainly be found. Throughout our analyses, the phases of the modes in the triad play a crucial role in understanding the dynamics.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics  
Divisions:  Faculty of Science > Centre for Complexity Science Faculty of Science > Mathematics 

Library of Congress Subject Headings (LCSH):  Waves  Mathematical models, Hamiltonian systems, Dynamics of a particle  
Journal or Publication Title:  Communications in Nonlinear Science and Numerical Simulation  
Publisher:  Elsevier BV  
ISSN:  10075704  
Official Date:  December 2012  
Dates: 


Volume:  Vol.17  
Number:  No.12  
Page Range:  pp. 49885006  
Identifier:  10.1016/j.cnsns.2012.04.002  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access  
Funder:  University College Dublin (UCD), Engineering and Physical Sciences Research Council (EPSRC)  
Grant number:  SF304 (UCD), SF564 (UCD), EP/H051295/1  
Related URLs:  
References:  [1] A. I. Dyachenko, A. O. Korotkevich, and V. E. Zakharov, 

URI:  http://wrap.warwick.ac.uk/id/eprint/44644 
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