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 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.
