Cover Page
The handle
https://hdl.handle.net/1887/3147163
holds various files of this Leiden
University dissertation.
Author: Schouten-Straatman, W.M.
Title: Patterns on spatially structured domains
Issue Date: 2021-03-02
Patterns on Spatially Structured Domains
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van Rector Magnificus prof. dr. ir. H. Bijl,
volgens besluit van het College voor Promoties
te verdedigen op dinsdag 2 maart 2021
klokke 11.15 uur
door
Willem Migchel Schouten-Straatman
geboren te Baarn
in 1992
Promotor:
Prof. dr. Arjen Doelman
Copromotor:
Dr. Hermen Jan Hupkes
Promotiecommissie:
Prof. dr. Peter Bates Michigan State University
Dr. Gr´egory Faye Universit´e Toulouse III – Paul Sabatier Prof. dr. Erik Van Vleck University of Kansas
Prof. dr. Frank van der Duijn Schouten Prof. dr. Roeland Merks
c
Willem Schouten-Straatman, 2021 Print: Haveka — www.haveka.nl
Front Cover:
Whitehoune / stock.adobe.com
This work was supported by the Netherlands Organisation for Scientific Research (NWO), grant 639.032.612.
Contents
1 Introduction 1
1.1 Scalar LDEs and MFDEs . . . 2
1.1.1 The FPUT lattice . . . 3
1.1.2 The Nagumo equation . . . 4
1.2 The FitzHugh-Nagumo system . . . 9
1.3 Techniques . . . 13
1.3.1 Linear Fredholm theory . . . 13
1.3.2 The spectral convergence method . . . 16
1.3.3 Exponential dichotomies . . . 24
2 Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions 29 2.1 Introduction . . . 30
2.2 Main results . . . 38
2.3 The singular perturbation . . . 42
2.3.1 Strategy . . . 45
2.3.2 Preliminaries . . . 47
2.3.3 Proof of Proposition 2.3.4 . . . 48
2.4 Existence of pulse solutions . . . 58
2.5 The point and essential spectrum . . . 65
2.6 The resolvent set . . . 78
2.7 Green’s functions . . . 85
2.7.1 Construction of the Green’s function . . . 87
2.7.2 Meromorphic expansion of Gλ . . . 96
2.7.3 Decomposition into stable and center modes . . . 105
2.8 Nonlinear stability . . . 111
3 Travelling waves for spatially discrete systems of FitzHugh-Nagumo type with periodic coefficients 133 3.1 Introduction . . . 134
3.2 Main results . . . 140
3.3 The limiting system . . . 145
3.3.1 Properties of Lo . . . 147
3.4 Transfer of Fredholm properties . . . 149 3
4 CONTENTS
3.4.1 Strategy . . . 153
3.4.2 Proof of Proposition 3.4.7 . . . 156
3.5 Existence of travelling waves . . . 164
3.6 Stability of travelling waves . . . 171
3.6.1 The operator Lε . . . 173
3.6.2 Spectral stability . . . 175
4 Travelling wave solutions for fully discrete FitzHugh-Nagumo type equations with infinite-range interactions 179 4.1 Introduction . . . 180
4.2 Main result . . . 186
4.2.1 The spatially discrete system . . . 187
4.2.2 Spatially discrete travelling waves . . . 188
4.2.3 The fully discrete system . . . 190
4.2.4 Nonuniqueness and numerical examples . . . 193
4.3 Setup . . . 195
4.4 The limiting system . . . 199
4.5 Linear theory for ∆t → 0 . . . 205
4.5.1 Strategy . . . 205
4.5.2 Spectral convergence . . . 208
4.5.3 Exponential decay . . . 213
4.6 Proof of main result . . . 218
4.6.1 Existence of solutions . . . 219
4.6.2 Local uniqueness of solutions . . . 227
4.A Auxiliary results . . . 234
5 Exponential dichotomies for nonlocal differential operators with infinite-range interactions 239 5.1 Introduction . . . 240
5.2 Main results . . . 247
5.2.1 State spaces . . . 250
5.2.2 Exponential dichotomies on R . . . 252
5.2.3 Exponential dichotomies on half-lines . . . 255
5.3 The existence of exponential dichotomies . . . 256
5.3.1 Preliminaries . . . 258
5.3.2 Exponential decay . . . 259
5.3.3 The restriction operators π+ and π− . . . 265
5.3.4 Fundamental properties of the Hale inner product . . . 267
5.3.5 Exponential splitting of the state space X . . . 271
5.4 Fredholm properties of the projections Π b P and ΠQb . . . 274
5.5 Exponential dichotomies on half-lines . . . 277
5.5.1 Strategy . . . 278
5.5.2 Construction of Y(τ ) . . . 280
5.5.3 Exponential decay . . . 282
5.5.4 Projection operators . . . 286
5.6.1 Structural conditions . . . 290
5.6.2 Examples . . . 295
5.6.3 (Co)-dimension counting . . . 300
5.6.4 Cyclic coefficients . . . 302
5.6.5 Nondegeneracy of the Hale inner product . . . 304
5.6.6 The nontriviality condition (HKer) . . . 309
6 Parameter-dependent exponential dichotomies for nonlocal differen-tial operators 311 6.1 Introduction and main result . . . 311
6.2 One-sided exponential weights . . . 313
6.3 Construction of exponential splittings . . . 318
Bibliography 329
Samenvatting 343
Dankwoord 349
