Emergence of unstable modes in an expanding domain for energy-conserving wave equations

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 Schrödinger 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 space diagnostics, 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.