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Modulational Stability of Travelling Waves in 2D Anisotropic Systems

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Abstract

The modulational stability of travelling waves in 2D anisotropic systems is investigated. We consider normal travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for two envelopes of left- and right-travelling waves, and oblique travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for four envelopes associated with two counterpropagating pairs of travelling waves in two oblique directions. The Eckhaus stability boundary for these waves in the plane of wave numbers is computed from the linearized Ginzburg–Landau systems. We identify longitudinal long and finite wavelength instabilities as well as transverse long wavelength instabilities. The results of the stability calculations are confirmed through numerical simulations. In these simulations we observe a rich variety of behaviors, including defect chaos, elongated localized structures superimposed to travelling waves, and moving grain boundaries separating travelling waves in different oblique directions. The stability classification is applied to a reaction–diffusion system and to the weak electrolyte model for electroconvection in nematic liquid crystals.

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  1. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523-1874, USA

    G. Dangelmayr & I. Oprea

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  1. G. Dangelmayr
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  2. I. Oprea
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Correspondence to G. Dangelmayr.

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Communicated by G. Iooss.

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Dangelmayr, G., Oprea, I. Modulational Stability of Travelling Waves in 2D Anisotropic Systems. J Nonlinear Sci 18, 1–56 (2008). https://doi.org/10.1007/s00332-007-9009-3

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