Mass-based asymptotic selection for perturbed unstable cubic-quintic ground states

For $\omega<\omega_c$, consider initial data of the form

$$u_{0}(x)=Q_{\omega}(x)+\epsilon(|x|), \ \text{ with $\|\epsilon\|_{H^{1}}\ll 1$.}$$

Here $Q_\omega$ is a cubic-quintic ground state, $M$ denotes mass, and $\phi_{\underline\omega}(t,x)=e^{i\underline\omega t}Q_{\underline\omega}(x)$ is a solitary wave with $Q_{\underline\omega}$ a stable ground state. Mass-based selection conjecture. (i) If $M(u_0)<M(Q_\omega)$, then the solution $u$ to the nonlinear Schrödinger equation is purely dispersive. (ii) If $M(u_0)>M(Q_\omega)$, then the solution converges, as $t\to+\infty$, to $\phi_{\underline\omega}(t,x)$ plus radiation, where $Q_{\underline\omega}$ is a stable ground state satisfying

$$M(Q_{\underline\omega})<M(Q_\omega).$$

These numerical observations propose a selection criterion for the stable asymptotic state after perturbing an unstable ground state. The supplied status evidence explicitly says that finding a possible selection criterion for $\underline\omega$ remains an open question.