Analysis of Hamiltonian PDEs (Oxford Lecture Series in Mathematics and Its Applications, 19)

The book was written to present a proof of the following KAM theorem: most space-periodic finite-gap solutions of a Lax-integrable Hamiltonian partial differential equation (PDE) persist under a small Hamiltonian perturbation of the equation as time-quasiperiodic solutions of the perturbed equation. In order to prove the theorem we develop a theory of Hamiltonian PDEs (Chapter 1) and give short presentations of abstract Lax-integrable equations (Chapter 2) as well as of classical Lax-integrable PDEs (Chapters 3 and 4). Next, in Chapters 5-7 we develop normal forms for Lax-integrable PDEs in the vicinity of manifolds, formed by the finite-gap solutions. Finally, we prove the main theorem applying an abstract KAM theorem (Chapters 8 and 10 of Part II) to equations, written in the normal form. Our presentation is rather complete; the only nontrivial result which is given without a proof is the celebrated Its-Matveev theta formula for finite-gap solutions of a Lax-integrable PDE. The above-mentioned normal form results, and the abstract KAM theorem, are important effective tools to study non-linear PDEs, apart from the persistence of finite-gap solutions (e.g. see Kuksin 1993, Bobenko and Kuksin 1995a, and Kuksin and Poschel 1996 for some other KAM results). Kuksin,S.B. Analysis of Hamiltonian PDEs OLS Mathematics and Its Applications vol.19 ......Page 4 Copyright ......Page 5 Preface ......Page 6 Contents ......Page 8 Notation xi ......Page 12 I UNPERTURBED EQUATIONS ......Page 14 1.1 Differentiable and analytic maps 3 ......Page 16 1.2 Scales of Hilbert spaces and interpolation 5 ......Page 18 1.3 Differential forms 10 ......Page 23 1.4 Symplectic structures and Hamiltonian equations 14 ......Page 27 1.5 Symplectic transformations 19 ......Page 32 1.6 A Darboux lemma 26 ......Page 39 Appendix 1. Time-quasiperiodic solutions 27 ......Page 40 Appendix 2. Hilbert matrices and the Schur criterion 28 ......Page 41 2 Integrable subsystems of Hamiltonian equations and Lax-integrable equations 30 ......Page 43 2.1 Three examples 31 ......Page 44 2.2 Integrable subsystems 34 ......Page 47 2.3 Lax-integrable equations 37 ......Page 50 3.1 Finite-gap manifolds 40 ......Page 53 3.2 The Its-Matveev theta formulas 47 ......Page 60 3.3 Small-gap solutions 52 ......Page 65 3.4 Higher equations from the KdV hierarchy 58 ......Page 71 Appendix 3. On the Its-Matveev formulas 59 ......Page 72 Appendix 4. On the vectors V and W 61 ......Page 74 Appendix 5. A small-gap limit for theta functions 63 ......Page 76 Appendix 6. A Non-degeneracy Lemma 65 ......Page 78 4.1 The L, A pair 70 ......Page 83 4.2 Theta formulas 74 ......Page 87 4.3 Even periodic and odd periodic solutions 77 ......Page 90 4.4 Local structure of finite-gap manifolds 80 ......Page 93 4.5 Proof of Lemma 4.4 82 ......Page 95 Appendix 7. On the algebraic functions of infinite-dimensional arguments 86 ......Page 99 5.1 The linearized equation 87 ......Page 100 5.2 Floquet solutions 88 ......Page 101 5.3 Complete systems of Floquet solutions 92 ......Page 105 5.4 Lower-dimensional invariant tori in finite-dimensional systems and Floquet’s theorem 102 ......Page 115 6.1 Abstract setting 104 ......Page 117 6.2 Linearized KdV equation 105 ......Page 118 6.3 Higher KdV equations 112 ......Page 125 6.4 Linearized Sine-Gordon equation 113 ......Page 126 7.1 A normal form theorem 119 ......Page 132 7.2 Proof of Lemma 7.3 125 ......Page 138 7.3 Examples 128 ......Page 141 II PERTURBED EQUATIONS ......Page 144 8.1 The Main Theorem and related results 133 ......Page 146 8.2 Reduction to a parameter-depending case 136 ......Page 149 8.3 A KAM theorem for parameter-depending equations 138 ......Page 151 8.4 Completion of the proof of the Main Theorem 139 ......Page 152 8.5 Around the Main Theorem 141 ......Page 154 Appendix 8. Lipschitz analysis and Hausdorff measure 143 ......Page 156 9.1 Perturbed KdV equation 145 ......Page 158 9.2 Higher KdV equations 147 ......Page 160 9.3 Time-quasiperiodic perturbations of Lax-integrable equations 148 ......Page 161 9.4 Perturbed SG equation 151 ......Page 164 9.5 KAM persistence of lower-dimensional invariant tori of non-linear finite-dimensional systems 153 ......Page 166 10.1 Preliminary reductions 154 ......Page 167 10.2 Proof of the theorem 155 ......Page 168 10.3 Proof of Lemma 10.3 (estimation of the small divisors) 171 ......Page 184 Appendix 9. Some inequalities for Fourier series 174 ......Page 187 Appendix 10. On the Craig-Wayne-Bourgain KAM scheme 176 ......Page 189 11 Linearized equations 179 ......Page 192 12 First-order linear differential equations on the ft-torus 184 ......Page 197 A.2 Theorems A and B 192 ......Page 205 A.3 Sketch of the proof 195 ......Page 208 A.5 Proof of theorem B 196 ......Page 209 References 206 ......Page 219 Index 211 ......Page 224 cover......Page 1

For the last 20-30 years, interest among mathematicians and physicists in infinite-dimensional Hamiltonian systems and Hamiltonian partial differential equations has been growing strongly, and many papers and a number of books have been written on integrable Hamiltonian PDEs. During the last decade though, the interest has shifted steadily towards non-integrable Hamiltonian PDEs. Here, not algebra but analysis and symplectic geometry are the appropriate analysing tools. The present book is the first one to use this approach to Hamiltonian PDEs and present a complete proof of the KAM for PDEs theorem. It will be an invaluable source of information for postgraduate mathematics and physics students and researchers.

Booknews

Kuksin (mathematics, Heriot-Watt U., Edinburgh, Scotland and Steklov Mathematical Institute, Moscow, Russia) focuses on the use of analysis and symplectic geometry to analyze Hamiltonian PDEs. He develops a theory of Hamiltonian PDEs, offers a short presentation of abstract Lax-integrable equations and classical Lax-integrable PDEs, and develops normal forms for Lax-integrable PDEs in the vicinity of manifolds, formed by the finite-gap solutions. Finally, he proves the main KAM theorem applying an abstract KAM theorem to equations, written in the normal form. Of likely interest to postgraduate mathematics and physics students and researchers with some knowledge of basic symplectic geometry, non-linear PDEs, Sobolev spaces, and interpolation. Annotation c. Book News, Inc., Portland, OR (booknews.com)

"For the last 20-30 years, interest among mathematicians and physicists in infinite-dimensional Hamiltonian systems and Hamiltonian partial differential equations has been growing strongly, and many papers and a number of books have been written on integrable Hamiltonian PDEs. During the last decade though, the interest has shifted steadily towards non-integrable Hamiltonian PDEs. Here, not algebra but analysis and symplectic geometry are the appropriate analysing tools. The present book is the first one to use this approach to Hamiltonian PDEs and will be an invaluable source of information for postgraduate mathematics and physics students and researchers."--Jacket