Nonlinear PDEs. A Dynamical Systems Approach

Cover......Page 1
Title page......Page 4
Contents......Page 8
Preface......Page 12
1.1. The three classical linear PDEs......Page 16
1.2. Nonlinear PDEs......Page 19
1.3. Our choice of equations and the idea of modulation equations......Page 21
1.4. Overview......Page 26
Part I Nonlinear dynamics in \R^{��}......Page 29
Chapter 2. Basic ODE dynamics......Page 30
2.1. Linear systems......Page 32
2.2. Local existence and uniqueness for nonlinear systems......Page 49
2.3. Special solutions......Page 53
2.4. \om-limit sets and attractors......Page 64
2.5. Chaotic dynamics......Page 73
2.6. Examples......Page 79
Chapter 3. Dissipative dynamics......Page 90
3.1. Bifurcations......Page 91
3.2. Center manifold theory......Page 100
3.3. The Hopf bifurcation......Page 106
3.4. Routes to chaos......Page 113
4.1. Basic properties......Page 124
4.2. Some celestial mechanics......Page 131
4.3. Completely integrable systems......Page 136
4.4. Perturbations of completely integrable systems......Page 138
4.5. Homoclinic chaos......Page 143
Part II Nonlinear dynamics in countably many dimensions......Page 146
Chapter 5. PDEs on an interval......Page 148
5.1. From finitely to infinitely many dimensions......Page 149
5.2. Basic function spaces and Fourier series......Page 166
5.3. The Chafee-Infante problem......Page 182
6.1. Introduction......Page 194
6.2. The equations on a torus......Page 201
6.3. Other boundary conditions and more general domains......Page 212
Part III PDEs on the infinite line......Page 219
Chapter 7. Some dissipative PDE models......Page 220
7.1. The KPP equation......Page 221
7.2. The Allen-Cahn equation......Page 237
7.3. Intermezzo: Fourier transform......Page 240
7.4. The Burgers equation......Page 252
Chapter 8. Three canonical modulation equations......Page 264
8.1. The NLS equation......Page 265
8.2. The KdV equation......Page 274
8.3. The GL equation......Page 290
Chapter 9. Reaction-Diffusion systems......Page 310
9.1. Modeling, and existence and uniqueness......Page 312
9.2. Two classical examples......Page 317
9.3. The Turing instability......Page 322
Part IV Modulation theory and applications......Page 329
Chapter 10. Dynamics of pattern and the GL equation......Page 330
10.1. Introduction......Page 331
10.2. The Swift-Hohenberg equation......Page 334
10.3. The universality of the GL equation......Page 347
10.4. An abstract approximation result......Page 352
10.5. Reaction-Diffusion systems......Page 362
10.6. Convection problems......Page 369
10.7. The Couette-Taylor problem......Page 385
10.8. Attractors for pattern forming systems......Page 393
10.9. Further remarks......Page 410
Chapter 11. Wave packets and the NLS equation......Page 416
11.1. Introduction......Page 417
11.2. Justification in case of cubic nonlinearities......Page 419
11.3. The universality of the NLS equation......Page 426
11.4. Quadratic nonlinearities......Page 431
11.5. Extension of the theory......Page 436
11.6. Pulse dynamics in photonic crystals......Page 444
11.7. Nonlinear optics......Page 455
Chapter 12. Long waves and their modulation equations......Page 466
12.1. An approximation result......Page 467
12.2. The universality of the KdV equation......Page 471
12.3. Whitham, Boussinesq, BBM, etc.......Page 480
12.4. The long wave limit......Page 483
13.1. The center manifold theorem......Page 488
13.2. Local bifurcation theory on bounded domains......Page 493
13.3. Spatial dynamics for elliptic problems in a strip......Page 497
13.4. Applications......Page 499
Chapter 14. Diffusive stability......Page 512
14.1. Linear and nonlinear diffusive behavior......Page 513
14.2. Diffusive stability of spatially periodic equilibria......Page 522
14.3. The critical case......Page 538
14.4. Phase diffusion equations......Page 544
14.5. Dispersive dynamics......Page 550
Bibliography......Page 556
List of symbols......Page 582
Index......Page 584
Back Cover......Page 593