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Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces. (English) Zbl 1260.37045

In this paper, the authors prove the existence of Cantor families of small amplitude time-periodic solutions for the nonlinear wave equation \(u_{tt} -\Delta u+\mu u=f(x,u)\) and the nonlinear Schrödinger equation \(iu_{t} -\Delta u+\mu u=f(x,|u|^2 )u\), where the spatial variable \(x\) ranges over an arbitrary compact Lie group \(M\) or, more generally, over an arbitrary compact homogeneous space \(M\) (for instance, the sphere \(S^n\)). The operator \(\Delta\) is the Laplace-Beltrami operator on \(M\) with respect to a Riemannian metric compatible with the group structure or action, the mass \(\mu\) is positive, and the nonlinearity is finitely differentiable and vanishes at \(u=0\) at least quadratically. The proof of the existence of these Cantor families of solutions is based on an abstract Nash-Moser implicit function theorem developed by them.

MSC:

37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
58C15 Implicit function theorems; global Newton methods on manifolds
58J45 Hyperbolic equations on manifolds
35Q55 NLS equations (nonlinear Schrödinger equations)
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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