A weakly turbulent solution to the cubic nonlinear harmonic oscillator on ℝ2 perturbed by a real smooth potential decaying to zero at infinity.

We build a smooth real potential V(t, x) on (t 0 , + ∞) × R 2 decaying to zero as t → ∞ and a smooth solution to the associated perturbed cubic noninear harmonic oscillator whose Sobolev norms blow up logarithmically as t → ∞ . 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. [ABSTRACT FROM AUTHOR]