Nonlinearity Management in Higher Dimensions

TL;DR

This paper proves that averaged nonlinear Schrödinger equations in higher dimensions do not support blow-up solutions, clarifying the effects of nonlinearity management strength and the limitations of the averaging method.

Contribution

It demonstrates that the averaged nonlinear Schrödinger equation prevents blow-up in higher dimensions regardless of nonlinearity strength, resolving previous contradictions.

Findings

Averaged NLS does not support blow-up in higher dimensions.

Strong nonlinearity management aligns with previous results.

Weak nonlinearity management causes divergence in averaging.

Abstract

In the present short communication, we revisit nonlinearity management of the time-periodic nonlinear Schrodinger equation and the related averaging procedure. We prove that the averaged nonlinear Schrodinger equation does not support the blow-up of solutions in higher dimensions, independently of the strength in the nonlinearity coefficient variance. This conclusion agrees with earlier works in the case of strong nonlinearity management but contradicts those in the case of weak nonlinearity management. The apparent discrepancy is explained by the divergence of the averaging procedure in the limit of weak nonlinearity management.