On the energy transfer to high frequencies in the damped/driven nonlinear Schrödinger equation

Guan Huang (1) , Sergei Kuksin (2, 3)

1 THU - Tsinghua University [Beijing]
2 UPD7 UFR Mathématiques - Université Paris Diderot - Paris 7 - UFR Mathématiques
3 Shandong University

We consider a damped/driven nonlinear Schrödinger equation in R n , where n is arbitrary, Eut − ν∆u + i u 2 u = √ νη(t, x), ν > 0, under odd periodic boundary conditions. Here η(t, x) is a random force which is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy u(t) 2 m ≤ Cν −m , uniformly in t ≥ 0 and ν > 0. In this work we prove that for small ν > 0 and any initial data, with large probability the Sobolev norms u(t, ⋅) m with m > 2 become large at least to the order of ν −κn,m with κn,m > 0, on time intervals of order O(1 ν). It proves that solutions of the equation develop short space-scale of order ν to a positive degree, and rigorously establishes the (direct) cascade of energy for the equation.

Domaines: Equations aux dérivées partielles [math.AP]; Science non linéaire [physics]

Guan Huang, Sergei Kuksin. On the energy transfer to high frequencies in the damped/driven nonlinear Schrödinger equation. Stochastics and Partial Differential Equations: Analysis and Computations, In press, ⟨10.1007/s40072-020-00187-2⟩. ⟨hal-03112246⟩