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Diffusive mixing of periodic wave trains in reaction-diffusion systems. (English) Zbl 1298.35108

The paper addresses the evolution of periodic wavetrains with imposed phase perturbations, as solutions to one-dimensional reaction-diffusion systems. The unperturbed wavetrains are assumed to be \(2\pi\)-periodic functions of \(kx-\omega t\), with the wavenumber and frequency linked by a dispersion relation, \(\omega = \omega(k)\). Three different types of the initial perturbations are considered. The first is a localized perturbation that does not introduce a phase difference between the wavetrains at \(x=\pm\infty\). For this case, a rigorous proof is developed of the fact that the local perturbation of the wavenumber, corresponding to the phase perturbation, asymptotically converges to the \(x\)-derivative of the fundamental (Gaussian-shaped) solution to the diffusion equation. The second is the case when the asymptotic wavenumbers at \(x=\pm \infty\) are equal, but there is a nonzero phase shift (as a part of the perturbation) between \(x=+\infty\) and \(x=-\infty\). In that case, it is proven that the evolving field of the local wavenumber converges to a symmetric Gaussian profile if the dispersion relation has \(\omega''(k)=0\) at the carrier wavenumber. In the opposite case, with \(\omega''(k)\neq 0\), the local wavenumber-perturbation field converges to a nonsymmetric non-Gaussian profile, which is a solution of the Burgers equation. The latter result is the main result of this work. The rigorous consideration of the third, most difficult case, with different asymptotic values of the local wavenumber at \(x=\pm \infty\), is postponed to another work. In that case, formation of a dissipative shock wave may be naturally expected.

MSC:

35K57 Reaction-diffusion equations
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