An instability property of the nonlinear Schrödinger equation on \(S^d\). (English) Zbl 1003.35113

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.


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
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