Quasi-linear dynamics in nonlinear Schr\" odinger equation with periodic boundary conditions

TL;DR

This paper demonstrates that many finite energy initial conditions in the nonlinear Schrödinger equation with periodic boundaries evolve almost linearly, revealing a new class of solutions distinct from known types like solitons.

Contribution

It introduces a novel class of solutions exhibiting quasi-linear dynamics, expanding understanding beyond traditional perturbative solutions in nonlinear Schrödinger equations.

Findings

Large subset of initial data evolve nearly linearly

New solutions are not perturbations of known solutions

Results apply to finite energy initial conditions

Abstract

It is shown that a large subset of initial data with finite energy ($L^2$ norm)evolves nearly linearly in nonlinear Schr\" odinger equation with periodic boundary conditions. These new solutions are not perturbations of the known ones such as solitons, semiclassical or weakly linear solutions.