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

Abstract

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 noninear 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.

Keywords:

• Nonlinear harmonic oscillator
• weak turbulence
• solitons
• modulations
• perturbing potential
• backward integration

Acknowledgments

The author wishes to express her deepest thanks to Professor Pierre Germain and Professor Pierre Raphael, as the former helped greatly with formatting the present paper through proofreading and the latter inspired deeply the proof. The author expresses her warmest thanks to the anonymous reviewer as their report was very enlightening.

Disclosure statement

The author reports there are no competing interests to declare.

Additional information

Funding

The author is financially supported as a master’s student by the Ecole Normale Supérieure (Paris, France).