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.

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.

Reviewer: Jan Chrastina (Brno)

##### 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 |