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On the Korteweg-de Vries equation and KAM theory. (English) Zbl 1033.35101

Hildebrandt, Stefan (ed.) et al., Geometric analysis and nonlinear partial differential equations. Berlin: Springer (ISBN 3-540-44051-8/hbk). 397-416 (2003).
From the text: We give an overview of results concerning the Korteweg-de Vries equation \(u_t= -u_{xxx}+ 6uu_x\) and small perturbations of it. All the technical details are contained in the authors’ book [KdV and KAM, Springer, Berlin (2003; Zbl 1032.37001)].
Our purpose here is to study small Hamiltonian perturbations of the KdV equation with periodic boundary conditions. In the unperturbed system all solutions are periodic, quasi-periodic, or almost periodic in time. The aim is to show that large families of periodic and quasi-periodic solutions persist under such perturbations. This is true not only for the KdV equation itself, but in principle for all equations in the KdV hierarchy. As an example, the second KdV equation is also considered.
For the entire collection see [Zbl 1007.00073].

MSC:

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

Citations:

Zbl 1032.37001
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