### 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 language | English (US) |
---|---|

Pages (from-to) | 4162-4201 |

Number of pages | 40 |

Journal | Nonlinearity |

Volume | 31 |

Issue number | 9 |

DOIs | |

State | Published - Jul 26 2018 |

### Fingerprint

### 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

*Nonlinearity*,

*31*(9), 4162-4201. https://doi.org/10.1088/1361-6544/aac9a6

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

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 31, no. 9, pp. 4162-4201. https://doi.org/10.1088/1361-6544/aac9a6

}

TY - JOUR

T1 - Target patterns in a 2D array of oscillators with nonlocal coupling

AU - Jaramillo, Gabriela

AU - Venkataramani, Shankar C

PY - 2018/7/26

Y1 - 2018/7/26

N2 - 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.

AB - 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.

KW - asymptotics beyond all orders

KW - Fredholm operators

KW - Kondratiev spaces

KW - Target patterns

UR - http://www.scopus.com/inward/record.url?scp=85051721043&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051721043&partnerID=8YFLogxK

U2 - 10.1088/1361-6544/aac9a6

DO - 10.1088/1361-6544/aac9a6

M3 - Article

AN - SCOPUS:85051721043

VL - 31

SP - 4162

EP - 4201

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 9

ER -