Spiral waves : linear and nonlinear theory / Björn Sandstede, Arnd Scheel.

Format:

Book

Author/Creator:

Sandstede, Björn, author.

Scheel, Arnd, 1966- author.

Series:

Memoirs of the American Mathematical Society ; no. 1413.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1413

Language:

English

Subjects (All):

Reaction-diffusion equations.

Waves--Mathematics.

Waves.

Spirals--Mathematics.

Spirals.

Dynamics.

Solitons.

Physical Description:

v, 126 pages : illustrations (black & white) ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"Spiral waves are striking self-organized coherent structures that organize spatiotemporal dynamics in dissipative, spatially extended systems. In this paper, we provide a conceptual approach to various properties of spiral waves. Rather than studying existence in a specific equation, we study properties of spiral waves in general reaction-diffusion systems. We show that many features of spiral waves are robust and to some extent independent of the specific model analyzed. To accomplish this, we present a suitable analytic framework, spatial radial dynamics, that allows us to rigorously characterize features such as the shape of spiral waves and their eigenfunctions, properties of the linearization, and finite-size effects. We believe that our framework can also be used to study spiral waves further and help analyze bifurcations, as well as provide guidance and predictions for experiments and numerical simulations. From a technical point of view, we introduce non-standard function spaces for the well-posedness of the existence problem which allow us to understand properties of spiral waves using dynamical systems techniques, in particular exponential dichotomies. Using these pointwise methods, we are able to bring tools from the analysis of one-dimensional coherent structures such as fronts and pulses to bear on these inherently two-dimensional defects"-- Provided by publisher.

Contents:

Chapter 1. Introduction

Chapter 2. Background material on wave trains

Chapter 3. Main results

Chapter 4. Wave trains

Chapter 5. Exponential dichotomies

Chapter 6. Fredholm properties

Chapter 7. Robustness and asymptotics of spiral waves

Chapter 8. Shape of eigenfunctions, and transverse instabilities

Chapter 9. Spiral waves on large finite disks

Chapter 10. Spectra of spiral waves restricted to large finite disks

Chapter 11. Spectra of truncated spiral waves

Chapter 12. Applications to spiral-wave dynamics and discussion

Appendix A. Numerical computation of spiral waves in model systems

Bibliography

Index.

Notes:

Includes bibliographical references (pages 119-123) and index.

ISBN:

9781470463090

1470463091

OCLC:

1381796325