Taming of Modulation Instability by Spatio-Temporal Modulation of the Potential

TL;DR

This paper demonstrates that spatio-temporal modulation of a potential can control, suppress, or alter pattern-forming instabilities in nonlinear systems, with broad applications including lasers, reaction-diffusion systems, and Bose-Einstein condensates.

Contribution

It introduces a method to manipulate modulation instability via spatio-temporal potential modulation, enabling stabilization and spectrum control of patterns in complex systems.

Findings

Spatio-temporal potential modulation can suppress or alter pattern-forming instabilities.

The method can transform long wave instability into short wave instability.

Complete elimination of instability can stabilize otherwise unstable systems.

Abstract

Spontaneous pattern formation in a variety of spatially extended nonlinear system always occurs through a modulation instability: homogeneous state of the system becomes unstable with respect to growing modulation modes. Therefore, the manipulation of the modulation instability is of primary importance in controlling and manipulating the character of spatial patterns initiated by that instability. We show that the spatio-temporal periodic modulation of the potential of the spatially extended system results in a modification of its pattern forming instability. Depending on the modulation character the instability can be partially suppressed, can change its spectrum (for instance the long wave instability can transform into short wave instability), can split into two, or can be completely eliminated. The latter result is of especial practical interest, as can be used to stabilize the intrinsically unstable system. The result bears general character, as it is shown here on a universal model of Complex Ginzburg-Landau equations in one and two spatial dimension (and time). The physical mechanism of instability suppression can be applied to a variety of intrinsically unstable dissipative systems, like self-focusing lasers, reaction-diffusion systems, as well as in unstable conservative systems, like attractive Bose Einstein condensates.