Long time stability of some small amplitude solutions in nonlinear Schrödinger equations
The author considers small perturbations of the Zakharov-Shabat nonlinear Schrödinger equation on \([0,2M]\) with vanishing or periodic boundary conditions in \([0,M]\). It is given a bound on the long time diffusion of the actions for initial data in open subsets of the phase space in the spirit of Nekhoroshev's theorem. The result restricts to solutions which (1) are of small amplitude, and (2) correspond to initial data close to the majority of finite dimensional invariant tori of the linearized system, namely those fulfilling a suitable (generic) nonresonance condition, with the above limitations. The difficulties associated to the actual integration of the ZS equation are avoided, and, the result concerns precisely the dynamics in a neighbourhood of nonresonant tori.
small perturbations of the Zakharov-Shabat nonlinear Schrödinger equation
scientific article; zbMATH DE number 1121194