A Weakly Turbulent solution to the cubic Nonlinear Harmonic Oscillator on $\mathbb{R}^2$ perturbed by a real smooth potential decaying to zero at infinity

We build a smooth real potential $V(t,x)$ on $(t_0,+\infty)\times \mathbb{R}^2$ decaying to zero as $t\to \infty$ and a smooth solution to the associated perturbed cubic Nonlinear Harmonic Oscillator whose Sobolev norms blow up logarithmically as $t\to \infty$. Adapting the method of Faou and Raphael for the linear case, we modulate the solitons associated to the Nonlinear Harmonic Oscillator by time-dependent parameters solving a quasi-Hamiltonian dynamical system whose action grows up logarithmically, thus yielding logarithmic growth for the Sobolev norm of the solution.