Target patterns in a 2D array of oscillators with nonlocal coupling. Jaramillo, G. & Venkataramani, S. C *Nonlinearity*, 31(9):4162-4201, IOP Publishing, 2018. Arxiv Journal doi abstract bibtex We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction–diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when , the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in . The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be `correct' to all orders in . We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution which was obtained using matched asymptotics.

@article{jaramillo2018target,
abstract = {We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction--diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when , the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in . The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be `correct' to all orders in . We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution which was obtained using matched asymptotics.},
author = {Jaramillo, Gabriela and Venkataramani, Shankar C},
da = {2018/07/27},
date-added = {2019-03-20 14:49:42 -0700},
date-modified = {2019-03-20 15:07:42 -0700},
doi = {10.1088/1361-6544/aac9a6},
isbn = {0951-7715; 1361-6544},
journal = {Nonlinearity},
keywords = {pubs},
number = {9},
pages = {4162-4201},
publisher = {IOP Publishing},
title = {Target patterns in a 2D array of oscillators with nonlocal coupling},
ty = {JOUR},
url_arxiv = {https://arxiv.org/abs/1706.00524},
url_journal = {https://iopscience.iop.org/article/10.1088/1361-6544/aac9a6/pdf},
volume = {31},
year = {2018},
Bdsk-Url-1 = {http://dx.doi.org/10.1088/1361-6544/aac9a6}}

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