Nonlinear Pdes: A Dynamical Systems Approach (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 182)

This is an introductory textbook about nonlinear dynamics of PDEs, with a focus on problems over unbounded domains and modulation equations. The presentation is example-oriented, and new mathematical tools are developed step by step, giving insight into some important classes of nonlinear PDEs and nonlinear dynamics phenomena which may occur in PDEs. The book consists of four parts. Parts I and II are introductions to finite- and infinite-dimensional dynamics defined by ODEs and by PDEs over bounded domains, respectively, including the basics of bifurcation and attractor theory. Part III introduces PDEs on the real line, including the Korteweg-de Vries equation, the Nonlinear Schrödinger equation and the Ginzburg-Landau equation. These examples often occur as simplest possible models, namely as amplitude or modulation equations, for some real world phenomena such as nonlinear waves and pattern formation. Part IV explores in more detail the connections between such complicated physical systems and the reduced models. For many models, a mathematically rigorous justification by approximation results is given. The parts of the book are kept as self-contained as possible. The book is suitable for self-study, and there are various possibilities to build one- or two-semester courses from the book. 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 Cover 1 Title page 4 Contents 8 Preface 12 Chapter 1. Introduction 16 1.1. The three classical linear PDEs 16 1.2. Nonlinear PDEs 19 1.3. Our choice of equations and the idea of modulation equations 21 1.4. Overview 26 Part I Nonlinear dynamics in \R^{d} 29 Chapter 2. Basic ODE dynamics 30 2.1. Linear systems 32 2.2. Local existence and uniqueness for nonlinear systems 49 2.3. Special solutions 53 2.4. \om-limit sets and attractors 64 2.5. Chaotic dynamics 73 2.6. Examples 79 Chapter 3. Dissipative dynamics 90 3.1. Bifurcations 91 3.2. Center manifold theory 100 3.3. The Hopf bifurcation 106 3.4. Routes to chaos 113 Chapter 4. Hamiltonian dynamics 124 4.1. Basic properties 124 4.2. Some celestial mechanics 131 4.3. Completely integrable systems 136 4.4. Perturbations of completely integrable systems 138 4.5. Homoclinic chaos 143 Part II Nonlinear dynamics in countably many dimensions 146 Chapter 5. PDEs on an interval 148 5.1. From finitely to infinitely many dimensions 149 5.2. Basic function spaces and Fourier series 166 5.3. The Chafee-Infante problem 182 Chapter 6. The Navier-Stokes equations 194 6.1. Introduction 194 6.2. The equations on a torus 201 6.3. Other boundary conditions and more general domains 212 Part III PDEs on the infinite line 219 Chapter 7. Some dissipative PDE models 220 7.1. The KPP equation 221 7.2. The Allen-Cahn equation 237 7.3. Intermezzo: Fourier transform 240 7.4. The Burgers equation 252 Chapter 8. Three canonical modulation equations 264 8.1. The NLS equation 265 8.2. The KdV equation 274 8.3. The GL equation 290 Chapter 9. Reaction-Diffusion systems 310 9.1. Modeling, and existence and uniqueness 312 9.2. Two classical examples 317 9.3. The Turing instability 322 Part IV Modulation theory and applications 329 Chapter 10. Dynamics of pattern and the GL equation 330 10.1. Introduction 331 10.2. The Swift-Hohenberg equation 334 10.3. The universality of the GL equation 347 10.4. An abstract approximation result 352 10.5. Reaction-Diffusion systems 362 10.6. Convection problems 369 10.7. The Couette-Taylor problem 385 10.8. Attractors for pattern forming systems 393 10.9. Further remarks 410 Chapter 11. Wave packets and the NLS equation 416 11.1. Introduction 417 11.2. Justification in case of cubic nonlinearities 419 11.3. The universality of the NLS equation 426 11.4. Quadratic nonlinearities 431 11.5. Extension of the theory 436 11.6. Pulse dynamics in photonic crystals 444 11.7. Nonlinear optics 455 Chapter 12. Long waves and their modulation equations 466 12.1. An approximation result 467 12.2. The universality of the KdV equation 471 12.3. Whitham, Boussinesq, BBM, etc. 480 12.4. The long wave limit 483 Chapter 13. Center manifold reduction and spatial dynamics 488 13.1. The center manifold theorem 488 13.2. Local bifurcation theory on bounded domains 493 13.3. Spatial dynamics for elliptic problems in a strip 497 13.4. Applications 499 Chapter 14. Diffusive stability 512 14.1. Linear and nonlinear diffusive behavior 513 14.2. Diffusive stability of spatially periodic equilibria 522 14.3. The critical case 538 14.4. Phase diffusion equations 544 14.5. Dispersive dynamics 550 Bibliography 556 List of symbols 582 Index 584 Back Cover 593 This is an introductory textbook about nonlinear dynamics of PDEs, with a focus on problems over unbounded domains and modulation equations. The presentation is example-oriented, and new mathematical tools are developed step by step, giving insight into some important classes of nonlinear PDEs and nonlinear dynamics phenomena which may occur in PDEs.The book consists of four parts. Parts I and II are introductions to finite- and infinite-dimensional dynamics defined by ODEs and by PDEs over bounded domains, respectively, including the basics of bifurcation and attractor theory. Part III introduces PDEs on the real line, including the Korteweg-de Vries equation, the Nonlinear Schrodinger equation and the Ginzburg-Landau equation. These examples often occur as simplest possible models, namely as amplitude or modulation equations, for some real world phenomena such as nonlinear waves and pattern formation. Part IV explores in more detail the connections between such complicated physical systems and the reduced models. For many models, a mathematically rigorous justification by approximation results is given.The parts of the book are kept as self-contained as possible. The book is suitable for self-study, and there are various possibilities to build one- or two-semester courses from the book.