P. Bizon, D. Hunik-Kostyra, and D.E. Pelinovsky
Stationary states of the cubic conformal flow on S3
Discrete Continuous Dynamical Systems A 40 (2020), 1-32
Abstract:
We consider the resonant system of amplitude equations for the conformally
invariant cubic wave equation on the three-sphere. Using the local bifurcation theory,
we characterize all stationary states that bifurcate from the first two eigenmodes.
Thanks to the variational formulation of the resonant system and
energy conservation, we also determine variational characterization and stability of the bifurcating states.
For the lowest eigenmode, we obtain two orbitally stable families of the bifurcating stationary states:
one is a constrained maximizer of energy and the other one is a constrained minimizer of the energy,
where the constraints are due to other conserved quantities of the resonant system.
For the second eigenmode, we obtain two constrained minimizers of the energy, which are also
orbitally stable in the time evolution. All other bifurcating states are
saddle points of energy under these constraints
and their stability in the time evolution is unknown.
Keywords:
Cubic wave equation on a sphere, bifurcation of stationary states,
variational characterization of stationary states, Lyapunov-Schmidt reductions, normal forms.