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We consider the cubic-quintic nonlinear Schr\"odinger equation: \[i\partial_t u = -\Delta u - |u|^2u + |u|^4u. \]  In the first part of the paper, we analyze the one-parameter family ofground-state solitons associated to this equation with particular attention tothe shape of the associated mass/energy curve. Additionally, we are able tocharacterize the kernel of the linearized operator about such solitons and todemonstrate that they occur as optimizers for a one-parameter family ofinequalities of Gagliardo--Nirenberg type. Building on this work, in the latterpart of the paper we prove that scattering holds for solutions belonging to theregion $\mathcal{R}$ of the mass/energy plane where the virial is positive. Weshow this region is partially bounded by solitons but also by rescalings ofsolitons (which are not soliton solutions in their own right). The discovery ofrescaled solitons in this context is new and highlights an unexpectedlimitation of any virial-based methodology.

Solitons and scattering for the cubic-quintic nonlinear Schr\"odinger equation on $\mathbb{R}^3$