The method of Poincaré normal forms in problems of integrability of equations of evolution type. (English. Russian original) Zbl 0632.35026
Russ. Math. Surv. 41, No. 5, 63-114 (1986); translation from Usp. Mat. Nauk 41, No. 5(251), 109-152 (1986).
The method of the Poincaré normal forms is applied to an evolution equation in a Banach space \(U: du/dt=Au+\Phi (u).\)
The operator A generates a strongly continuous semigroup and \(\Phi\) : \(U\to U\) is a nonlinear operator. The main result is an infinite dimensional analogue of Siegel’s Theorem for reduction of a system of ordinary differential equations to a linear normal form. Various applications to the Schrödinger and Korteweg-de Vries equations are given.
The operator A generates a strongly continuous semigroup and \(\Phi\) : \(U\to U\) is a nonlinear operator. The main result is an infinite dimensional analogue of Siegel’s Theorem for reduction of a system of ordinary differential equations to a linear normal form. Various applications to the Schrödinger and Korteweg-de Vries equations are given.
Reviewer: I.Ginchev
MSC:
| 35G10 | Initial value problems for linear higher-order PDEs |
| 35K25 | Higher-order parabolic equations |
| 47D03 | Groups and semigroups of linear operators |
| 35Q99 | Partial differential equations of mathematical physics and other areas of application |
