Kuksin S.B. Analysis of Hamiltonian PDEs

Oxford University Press, 2000. — 226 p. — (OLS Mathematics and Its Applications vol.19). — ISBN: 0198503954The 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).