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Analysis of Hamiltonian PDE’s. (English) Zbl 0960.35001
Oxford Lecture Series in Mathematics and its Applications. 19. Oxford: Oxford University Press. xii, 212 p. (2000).
The book was written to present a complete proof of the following infinite-dimensional KAM theorem: most space-periodic finite-gap solutions of a Lax-integrable partial differential equation persist under a small Hamiltonian perturbation of the equation as time-periodic solutions of the perturbed equation. At the same time, the book is the first one which provides a rather systematic introduction into the following tools: symplectic geometry, Hamiltonian partial differential equations, Lax integrability, invariant tori, and finite-gap manifolds. A thorough discussion of KdV and sine-Gordon equations together with a simplified proof of the KAM theorem for the finite-dimensional case are included.
To draw the main result, let $$\{Z_s\}$$ $$(s\in\mathbb{R})$$ be a scale of Hilbert spaces equipped with the symplectic form $$\alpha= J= dz\wedge dz$$ and a quasilinear Hamiltonian $${\mathcal H}= {1\over 2}\langle Az, z\rangle+ H(z)$$, where $$A$$ is a selfadjoint isomorphism of the scale. The corresponding Hamilton equation reads $$\dot u=- J^{-1}(Au+\nabla H(u))$$. Assuming the invariant manifold $${\mathcal T}^{2n}= \Phi(\mathbb{R}\times \mathbb{T}^n)$$ is filled with quasiperiodic solutions, the behaviour of solutions near $${\mathcal T}^{2n}$$ of the perturbed equation $$\dot u= -J^{-1}(Au+\nabla H(u)+ \varepsilon\nabla H_1(u))$$ is described. In particular, under certain assumptions on $$\Phi$$, non-resonance and spectral asymptotics of Floquet solutions of the linearized equation, most of the invariant tori $$\mathbb{T}^n(r)$$ of the original equation persist when $$\varepsilon\to 0$$: a Borel subset $$\mathbb{R}_\varepsilon\subset\mathbb{R}$$ exists with $$\text{mes}(\mathbb{R}_\varepsilon-\mathbb{R})\to 0$$ such that each torus $$\mathbb{T}^n_\varepsilon(r)$$ $$(r\in\mathbb{R}_\varepsilon)$$ is invariant for the perturbed equation and is filled with time-quasiperiodic solutions. (We apologise for this very rough exposition.)
The book involves the following topics: differentiable and analytic maps, differential forms and symplectic structures in Hilbert scales, integrable subsystems of Hamiltonian equations, finite-gap manifolds for the KdV equation including the Its-Matveev formula, analogous result for the sine-Gordon equation, Floquet solutions and invariant tori, linearization of Lux-integrable equations, the normal form in the vinicity of the invariant manifold $${\mathcal T}^{2n}$$, the KAM theorem for parameter depending equations.
The only result given without proof is the Its-Matveev formula. The author deals with the “finite volume” case where the space variable belongs to a bounded domain supplemented by boundary conditions. No preliminary knowledge of KAM technique is assumed. The book provides a very useful source of information both for integrable and non-integrable differential equations.

##### MSC:
 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35Q53 KdV equations (Korteweg-de Vries equations) 37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems