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

• Published in: Stochastic partial differential equations : analysis and computations Vol. 9; no. 4; pp. 867 - 891
• Main Authors: Huang, Guan, Kuksin, Sergei
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• Subjects: Analysis of PDEs; Boundary conditions; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Energy transfer; Mathematics; Mathematics and Statistics; Nonlinear Sciences; Norms; Numerical Analysis; Partial Differential Equations; Probability Theory and Stochastic Processes; Schrodinger equation; Statistical Theory and Methods; NLS; Energy cascading; Sobolev norms
• ISSN: 2194-0401; 2194-041X
• DOI: 10.1007/s40072-020-00187-2

We consider a damped/driven nonlinear Schrödinger equation in R n , where n is arbitrary, E u t - ν Δ u + i | u | 2 u = ν η ( 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 ) ‖ m 2 ≤ C ν - 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 - κ n , m with κ n , 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.