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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 theassociated perturbed cubic Nonlinear Harmonic Oscillator whose Sobolev normsblow up logarithmically as $t\to \infty$. Adapting the method of Faou andRaphael for the linear case, we modulate the solitons associated to theNonlinear Harmonic Oscillator by time-dependent parameters solving aquasi-Hamiltonian dynamical system whose action grows up logarithmically, thusyielding logarithmic growth for the Sobolev norm of the solution.

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