Equipartition of Mass in Nonlinear Schr\"odinger / Gross-Pitaevskii Equations

We study the infinite time dynamics of a class of nonlinear Schr\"odinger / Gross-Pitaevskii equations. In our previous paper, we prove the asymptotic stability of the nonlinear ground state in a general situation which admits degenerate neutral modes of arbitrary finite multiplicity, a typical situation in systems with symmetry. Neutral modes correspond to purely imaginary (neutrally stable) point spectrum of the linearization of the Hamiltonian PDE about a critical point. In particular, a small perturbation of the nonlinear ground state, which typically excites such neutral modes and radiation, will evolve toward an asymptotic nonlinear ground state soliton plus decaying neutral modes plus decaying radiation. In the present article, we give a much more detailed, in fact quantitative, picture of the asymptotic evolution. Specificially we prove an equipartition law: The asymptotic soliton which emerges has a mass which is equal to the initial soliton mass plus one half the mass contained in the initially perturbing neutral modes.