Incoherent localized structures and hidden coherent solitons from the gravitational instability of the Schr\"odinger-Poisson equation

TL;DR

This paper investigates how gravitational interactions in the Schrödinger-Poisson system lead to the formation of incoherent localized structures containing hidden coherent solitons, revealing new stabilization mechanisms through phase-space analysis.

Contribution

It introduces a coupled theoretical framework describing both incoherent and coherent components, unveiling hidden solitons stabilized by an effective trapping potential in gravitational wave systems.

Findings

Hidden coherent solitons exist within incoherent structures.

Incoherent structures stabilize solitons via an effective trapping potential.

The theory applies broadly to long-range wave systems with algebraic decay.

Abstract

The long-term behavior of a modulationally unstable conservative nonintegrable system is known to be characterized by the soliton turbulence self-organization process. We consider this problem in the presence of a long-range interaction in the framework of the Schr\"odinger-Poisson (or Newton-Schr\"odinger) equation accounting for the gravitational interaction. By increasing the amount of nonlinearity, the system self-organizes into a large-scale incoherent localized structure that contains "hidden" coherent soliton states: The solitons can hardly be identified in the usual spatial or spectral domains, while their existence is unveiled in the phase-space representation (spectrogram). We develop a theoretical approach that provides the coupled description of the coherent soliton component (governed by an effective Schr\"odinger-Poisson equation) and of the incoherent component (governed by a wave turbulence Vlasov-Poisson equation). The theory shows that the incoherent structure introduces an effective trapping potential that stabilizes the hidden coherent soliton, a mechanism that we verify by direct numerical simulations. The theory characterizes the properties of the localized incoherent structure, such as its compactly supported spectral shape. It also clarifies the quantum-to-classical correspondence in the presence of gravitational interactions. This study is of potential interest for self-gravitating Boson models of fuzzy dark matter. Although we focus our paper on the Schr\"odinger-Poisson equation, we show that our results are general for long-range wave systems characterized by an algebraic decay of the interacting potential. This work should stimulate nonlinear optics experiments in highly nonlocal nonlinear (thermal) media that mimic the long-range nature of gravitational interactions.