Nearly Integrable Infinite-Dimensional Hamiltonian SystemsThe book is devoted to partial differential equations of Hamiltonian form, close to integrable equations. For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly persist in the perturbed one. The theorem is applied to classical nonlinear PDE's with one-dimensional space variable such as the nonlinear string and nonlinear Schr|dinger equation andshow that the equations have "regular" (=time-quasiperiodic and time-periodic) solutions in rich supply. These results cannot be obtained by other techniques. The book will thus be of interest to mathematicians and physicists working with nonlinear PDE's. An extensivesummary of the results and of related topics is provided in the Introduction. All the nontraditional material used is discussed in the firstpart of the book and in five appendices. |
Contents
Statement of the main theorem and its consequences | 13 |
53 | 83 |
Appendix A Interpolation theorem | 91 |
Copyright | |
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Common terms and phrases
A₁ Am+1 antiselfadjoint applicable assumptions of Theorem asymptotics Banach space Borel set Borel subset C₁ Cauchy estimate coefficients convergence d₁ define depend Dirichlet boundary conditions domain of definition eigenvalues equal equation 1.3 exists a Borel function Hamiltonian equation hamiltonian Hm Hamiltonian systems Hilbert basis Hm+1 holds Hom+1 homological equations infinite-dimensional integrable invariant tori isomorphism K₁ L2-space Lemma Let us consider Let us denote Let us suppose linear equation linear operator Lyapunov exponents M₁ Math metric space n-tori nondegenerate nonlinear Schrödinger equation norm numbers Om+1 one-dimensional parameter partially hyperbolic PDE's phase space potential proof prove quasiperiodic solutions satisfies selfadjoint Sobolev space spectrum statement strong solution symplectic Hilbert scale symplectic structure Theorem 1.1 time-quasiperiodic solutions torus transformed hamiltonian variables vector vector-field weak solution zero