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The Influence of Modulational Instability on Energy Exchange in Coupled Sine-Gordon Equations

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Abstract

We consider a two-component system of coupled sine-Gordon equations, particular solutions of which represent a continuum generalization of periodic energy exchange in a system of coupled pendulums. Weakly nonlinear solutions describing periodic energy exchange between waves traveling in the two components are governed, depending on the length scale of the amplitude variation, either by two nonlocally coupled nonlinear Schrödinger equations, with different transport terms due to the group velocity, or by a model that is nondispersive to the leading order. Using both asymptotic analysis and numerical simulations, we show that the effects of dispersion significantly influence the structure of these solutions, causing modulational instability and the formation of localized structures but preserving the pattern of energy exchange between the components.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK

    S. D. Griffiths, R. H. J. Grimshaw & K. R. Khusnutdinova

  2. Ufa Branch, RAS, Institute of Mechanics, Karl Marx Str. 6, Ufa, 450000, Russia

    K. R. Khusnutdinova

Authors
  1. S. D. Griffiths
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  2. R. H. J. Grimshaw
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  3. K. R. Khusnutdinova
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Griffiths, S.D., Grimshaw, R.H.J. & Khusnutdinova, K.R. The Influence of Modulational Instability on Energy Exchange in Coupled Sine-Gordon Equations. Theoretical and Mathematical Physics 137, 1448–1458 (2003). https://doi.org/10.1023/A:1026008907682

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