On the Emergence of Unstable Modes in an Expanding Domain for Energy-Conserving Wave Equations

TL;DR

This paper investigates how the expansion or contraction of a domain can induce instability in energy-conserving wave equations, such as the nonlinear Schrödinger equation, with applications to Bose-Einstein condensates.

Contribution

It introduces a mechanism for destabilizing stable solutions in energy-conserving wave equations via domain size modulation, extending instability analysis to inhomogeneous settings.

Findings

Domain expansion can trigger modulational instability in nonlinear Schrödinger equations.

Real and Fourier diagnostics effectively monitor instability thresholds.

Inhomogeneous potentials modify the manifestation of the instability mechanism.

Abstract

Motivated by recent work on instabilities in expanding domains in reaction-diffusion settings, we propose an analog of such mechanisms in energy-conserving wave equations. In particular, we consider a nonlinear Schr{\"o}dinger equation in a finite domain and show how the expansion or contraction of the domain, under appropriate conditions, can destabilize its originally stable solutions through the modulational instability mechanism. Using both real and Fourier spacediagnostics, we monitor and control the crossing of the instability threshold and, hence, the activation of the instability. We also consider how the manifestation of this mechanism is modified in a spatially inhomogeneous setting, namely in the presence of an external parabolic potential, which is relevant to trapped Bose-Einstein condensates.