A Weakly Turbulent solution to the cubic Nonlinear Harmonic Oscillator on ℝ² perturbed by a real smooth potential decaying to zero at infinity

2023/06/20 by Chabert, Maxine · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2306.11524

Abstract

We build a smooth real potential V(t,x) on (t₀,+∞)× ℝ² decaying to zero as t→ ∞ and a smooth solution to the associated perturbed cubic Nonlinear 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.

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• Long-time behavior of resonant time-dependent perturbations of periodic transport equations on ℝ
• On Growth of Sobolev norms for cubic Schrödinger equation with harmonic potential in dimensions d=2,3

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