Soliton Dynamics for the Cubic-Quintic Nonlinear Schrödinger Equation with a Potential

We study the semiclassical dynamics of solitary waves for the three-dimensional cubic–quintic nonlinear Schrödinger equation with an external potential. For initial data given by a modulated free ground state $P_{\omega }$, we prove that the wave-packet center follows the associated classical Hamiltonian flow. For $\omega \in \mathcal{I}$, we obtain qualitative persistence near the soliton orbit. Under the additional positive-slope condition $\omega \in {\mathcal{I}}_{+}$, we establish an $O(\text{eps})$ approximation in the scaled $H^1$ norm on every fixed time interval. We also derive $O({\text{eps}}^2)$ concentration estimates for the mass, momentum, and center of mass.