Target patterns in a 2D array of oscillators with nonlocal coupling
| Autor: | Gabriela Jaramillo, Shankar C. Venkataramani |
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| Rok vydání: | 2018 |
| Předmět: |
Generalization
FOS: Physical sciences General Physics and Astronomy Pattern Formation and Solitons (nlin.PS) 01 natural sciences 010305 fluids & plasmas law.invention Mathematics - Analysis of PDEs Linearization law 0103 physical sciences FOS: Mathematics 0101 mathematics Mathematical Physics Mathematics Ansatz Eikonal equation Applied Mathematics Mathematical analysis Statistical and Nonlinear Physics State (functional analysis) Nonlinear Sciences - Pattern Formation and Solitons Implicit function theorem Functional Analysis (math.FA) Mathematics - Functional Analysis 010101 applied mathematics Sobolev space Invertible matrix Analysis of PDEs (math.AP) |
| Zdroj: | Nonlinearity. 31:4162-4201 |
| ISSN: | 1361-6544 0951-7715 |
| DOI: | 10.1088/1361-6544/aac9a6 |
| Popis: | 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 $\varepsilon$, 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 $\varepsilon$. 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 $\varepsilon$ 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 $\varepsilon$. 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 obtained using matched asymptotics. 39 pages, 1 figure |
| Databáze: | OpenAIRE |
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