Toward nonlinear stability of sources via a modified Burgers equation

Details Toward nonlinear stability of sources via a modified Burgers equation Toward nonlinear stability of sources via a modified Burgers equation

2011-03-16

arxiv.org/abs/1103.3197v2

Mathematics

Analysis of PDEs

revised version with some typos fixed

Scientific paper

Coherent structures are solutions to reaction-diffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at $x=\pm\infty$ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the linear level due to the outward transport. Using a modified Burgers equation as a model problem that captures some of the essential features of coherent structures, we show how this phenomenon can be analysed and nonlinear stability be established in this simpler context.

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