KAMTheorem with Normal Frequencies of FiniteLimit‐Pointsfor Some Shallow Water Equations

Abstract

By constructing an infinite‐dimensional KAM theorem of the normal frequencies being dense at a finite point, we show that some shallow water equations such as the Benjamin‐Bona‐Mahony equation and the generalized d‐dimensional Pochhammer‐Chree equation subject to some boundary conditions possess many (a family of initial values of positive Lebesgue measure of finite dimension) smooth solutions that are quasi‐periodic in time . © 2020 Wiley Periodicals LLC

Response solutions for elliptic-hyperbolic equations with nonlinearities and periodic external forces

“…To tackle this issue, the KAM approach for finite or infinite dimensional dynamic systems is frequently employed. There have been numerous instances in the PDE contexts, here we just provide [23,28,33,33,53,58,61] for references. Another efficient approach is the CWB method, developed by Craig-Wayne [16] and Bourgain [6,7].…”

A note on stability of Eliasson-Kuksin's KAM tori for the nonlinear Schrödinger equation

“…Moreover, due to the size of the blocks grows much faster than quadratically along the iterations, they need to take sufficiently many normal form computations at each KAM step to obtain a much faster iteration scheme. See also [6] and [15] for the KAM approach on the space-multidimensional beam equation and some shallow-water equations, respectively. But the problem on the linear stability of the KAM tori for space-multidimensional nonlinear wave equation is still open and requires special attention.…”