Externally forced triads of resonantly interacting waves: Boundedness and integrability properties

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Title: Externally forced triads of resonantly interacting waves: Boundedness and integrability properties
Authors: Harris, Jamie
Bustamante, Miguel
Connaughton, Colm
Permanent link: http://hdl.handle.net/10197/7506
Date: Dec-2012
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 one-dimensional time-independent 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 View the MathML source we reduce the problem to the motion of a particle in a one-dimensional time-periodic potential. Poincaré sections of this system provide strong evidence that the motion remains bounded when View the MathML source and is typically quasi-periodic 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.
Funding Details: University College Dublin
Type of material: Journal Article
Publisher: Elsevier
Copyright (published version): 2012 Elsevier
Keywords: Nonlinear dynamical systems;Integrable systems;Rossby waves
DOI: 10.1016/j.cnsns.2012.04.002
Language: en
Status of Item: Peer reviewed
Appears in Collections:Mathematics and Statistics Research Collection

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