Stability of the Cubic Nonlinear Schrodinger Equation on an Irrational Torus

Abstract

© 2020 Society for Industrial and Applied Mathematics Publications. All rights reserved. A characteristic of the defocusing cubic nonlinear Schrödinger equation (NLSE), when defined so that the space variable is the multidimensional square (hence, rational) torus, is that there exist solutions that start with arbitrarily small Sobolev norms and evolve to develop arbitrarily large modes at later times; this phenomenon is recognized as a weak energy transfer to high modes for the NLSE [Colliander et al., Invent. Math., 181 (2010), pp. 39{113] and [R. Carles and E. Faou, Discrete Contin. Dyn. Syst., 32 (2012), pp. 2063{2077]. In this paper, we show that when the system is considered on an irrational torus, energy transfer is more difficult to detect.