On the Modulational Instability of the Nonlinear Schr\"odinger Equation with Dissipation

TL;DR

This paper investigates how even minimal dissipation affects the modulational instability in the nonlinear Schrödinger equation, revealing suppression over time but potential short-term growth, with a generalized instability criterion supported by analytical and numerical results.

Contribution

It introduces a generalized modulational instability criterion accounting for dissipation effects in the nonlinear Schrödinger equation, supported by analytical derivations and numerical simulations.

Findings

Weak dissipation suppresses long-term instability.

Short-term instability growth can occur despite dissipation.

The generalized MI criterion applies to any power-law dissipation.

Abstract

The modulational instability of spatially uniform states in the nonlinear Schr\"odinger equation is examined in the presence of higher-order dissipation. The study is motivated by results on the effects of three-body recombination in Bose-Einstein condensates, as well as by the important recent work of Segur et al. on the effects of linear damping in NLS settings. We show how the presence of even the weakest possible dissipation suppresses the instability on a longer time scale. However, on a shorter scale, the instability growth may take place, and a corresponding generalization of the MI criterion is developed. The analytical results are corroborated by numerical simulations. The method is valid for any power-law dissipation form, including the constant dissipation as a special case.