Target patterns in a 2D array of oscillators with nonlocal coupling

Gabriela Jaramillo, Shankar C Venkataramani

Research output: Contribution to journalArticle

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 e, 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 e. 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 e 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 e. We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refned ansatz for the approximate solution which was obtained using matched asymptotics.

Original languageEnglish (US)
Pages (from-to)4162-4201
Number of pages40
JournalNonlinearity
Volume31
Issue number9
DOIs
StatePublished - Jul 26 2018

Fingerprint

Implicit Function Theorem
linearization
Linearization
Inhomogeneity
inhomogeneity
theorems
oscillators
Sobolev space
eikonal equation
Matched Asymptotics
Fredholm Property
Eikonal Equation
Sobolev spaces
Target
Phase Modulation
Perturbation Expansion
Convolution Operator
Phase modulation
Reaction-diffusion System
Convolution

Keywords

  • asymptotics beyond all orders
  • Fredholm operators
  • Kondratiev spaces
  • Target patterns

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Cite this

Target patterns in a 2D array of oscillators with nonlocal coupling. / Jaramillo, Gabriela; Venkataramani, Shankar C.

In: Nonlinearity, Vol. 31, No. 9, 26.07.2018, p. 4162-4201.

Research output: Contribution to journalArticle

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