Burq, N.; Gérard, P.; Tzvetkov, N. An instability property of the nonlinear Schrödinger equation on \(S^d\). (English) Zbl 1003.35113 Math. Res. Lett. 9, No. 2-3, 323-335 (2002). Summary: We consider the nonlinear Schrödinger equation on spheres. We describe the nonlinear evolutions of the highest weight spherical harmonics. As a consequence, in contrast with the flat torus, we obtain that the flow map fails to be uniformly continuous for Sobolev regularity above the threshold suggested by a simple scaling argument. Cited in 58 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 37L50 Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 35B35 Stability in context of PDEs 35R25 Ill-posed problems for PDEs 58J35 Heat and other parabolic equation methods for PDEs on manifolds Keywords:eigenfunctions; dispersive equations; Laplace-Beltrami operator; nonlinear Schrödinger equation PDFBibTeX XMLCite \textit{N. Burq} et al., Math. Res. Lett. 9, No. 2--3, 323--335 (2002; Zbl 1003.35113) Full Text: DOI