Stable Directions for Degenerate Excited States of Nonlinear Schrödinger Equations

We consider the nonlinear Schrödinger equations i∂tψ = H0ψ + λ|ψ|ψ in R× [0,∞) where H0 = −∆ +V and λ = ±1. Assume that the potential V is radial and decays sufficiently fast at infinity. Assume also that the linear Hamiltonian H0 has only two discrete eigenvalues e0 < e1 < 0 where e0 is simple and e1 has multiplicities 3. We show that there exist three branches of nonlinear excited states and for certain finite codimesion subset in the space of initial data, we construct solutions ψ converging to these excited states in both non-resonant and resonant cases. This is the joint work with Stephen Gustafson.