Nonlinear equations are ubiquitous in physics. In optics [

1], they serve as back reaction models of the media onto the propagating beam. In many-body systems, mutual interactions give rise to effective nonlinear potentials within the mean field approximation [

2,

3]. Especially intriguing are nonlocal nonlinear problems arising, for instance, from long-range forces such as gravity. In this paper we study a quantum dynamical model for dark matter which is of recent interest in cosmology, see [

4,

5,

6,

7,

8], as it recovers known features of dynamically cold dark matter (CDM) on sufficiently large spatial scales while alleviating shortcomings of the classical description on small scales due to the quantum nature of the model.

The rich set of hierarchical structures we observe in the universe, ranging from kiloparsec galaxies up to the cosmic web visible on gigaparsec scales, is understood as an evolution snapshot of all gravitationally interacting matter starting from dynamically cold initial conditions [

9]. Observational data [

10] suggests that the main driver of this large-scale structure formation process is dark matter, a non-baryonic matter component of yet undetermined character and origin that only interacts gravitationally. The full fledged dynamics of CDM, i.e., the evolution of its probability distribution in phase space is often modeled by means of a Boltzmann equation. Since the expected number of interacting particles is immense, relaxation times are extremely large and thus one considers structure formation only as collisionless problem [

11]. A brief overview of the classical CDM description is given in

Section 1.1.

An alternative approach in modeling large scale structure formation, first suggested in this context by [

12], is to model the temporal evolution of dark matter as a complex scalar field

$\psi (\mathit{x},t)$ governed by Schrödinger’s equation with a nonlocal, nonlinear self-interaction potential given as the solution to Poisson’s field equation. Depending on the point of view, one can either interpret this model as a distinct description of dark matter known as fuzzy dark matter [

13], in which structure formation is driven by a cosmic Bose–Einstein condensate trapped in its own gravitational potential and where the expansion of space sets the time dependent interaction strength; or as a field theoretical approximation to the classical Boltzmann description [

14,

15,

16,

17]. A more detailed discussion about both the Bose–Einstein condensate and the Boltzmann–Schrödinger correspondence is given in

Section 1.2.

We note in passing that the application of the Schrödinger–Poisson model as a nonlinear equation system can also be found in other fields, such as fundamental quantum mechanics, see e.g., [

18], and nonlinear optics [

19]. In the former it acts as model for quantum collapse theory. The latter studies light propagation in nonlinear media in which steady state heat transfer due to the absorption of light along the light path induces a change in the refractive index which is formally identical to the type of nonlocal nonlinearity we encounter in our gravitational problem.

#### 1.1. Classical Description of Cold Dark Matter

A theoretical description of the temporal evolution of CDM can be developed in the framework of classical Hamiltonian mechanics yielding a set of coupled differential equations known as the Vlasov–Poisson or collisionless Boltzmann equation which in flat space reads:

In this description, dark matter is modeled as the smooth probability density

$f(\mathit{x},\mathit{u},t)$ trapped in its own gravitational potential and evolving in phase space spanned by comoving position

$\mathit{x}$ and conjugate velocity

$\mathit{u}$. The latter quantities factor out the cosmological expansion of space as measured by the dimensionless scale factor

$a\left(t\right)$.

$a\left(t\right)$ depends on the cosmological model, in particular on the present day dimensionless matter density parameter

${\mathsf{\Omega}}_{m0}$ and dark energy density

${\mathsf{\Omega}}_{\mathsf{\Lambda}0}$. Radiation contributions to the total energy budget can typically be neglected. Thus, we restrict ourselves to models with

$1={\mathsf{\Omega}}_{\mathsf{m}0}+{\mathsf{\Omega}}_{\mathsf{\Lambda}0}$. Further details are given in

Appendix A. Integrating

f over velocity space yields the dark matter particle density

$n(\mathit{x},t)$ that consists of a spatially homogeneous background contribution and a density fluctuation associated with the peculiar motion of the particles due to gravity. Hence, the bracket term in Equation (

1) measures the relative excess density with respect to the background. It is called density contrast

$\delta (\mathit{x},t)$. Both the analytical and numerical treatment of Equation (

1) is challenging due its nonlocal nonlinearity and the large number of degrees of freedom in

f. Hence, simplifications are in order.

Analytically, one often resorts to a hydrodynamical model of Equation (

1) by integrating over velocity or position space to obtain equations of motion for the marginal distributions of

f. Assuming a dynamically cold distribution with vanishing velocity dispersion of the form of a phase space sheet, one obtains:

where

${\varphi}_{u}$ denotes the potential of the irrotational velocity flow. Although simple in its formulation, this model breaks down at shell crossing, the moment in the evolution of Equation (

2) when the initial phase space sheet becomes perpendicular to the spatial axis. This so called dust model is incapable of describing multiple matter streams or virialized matter structures as we expect them.

Sampling the true distribution function by means of Newtonian test particles and only following their motion and mutual interaction gives rise to

N-Body simulations [

20,

21] which proved to be an invaluable tool in cosmology. Although successful in predicting the correct large scale features of the cosmic matter distribution,

N-Body simulations produce density profiles of collapsed structures with cuspy core regions which are not supported by observational data. Whether this cusp-core problem is a prediction of the CDM model or an artifact of

N-Body simulations lacking crucial physical effects (e.g., baryonic feedback) is still open to debate. We refer to [

22] for an in depth review of the cusp-core problem as well as other puzzling phenomena collectively often referred to as “small scale crisis”.

#### 1.2. The Schrödinger–Poisson Model (SPM)

An alternative approach in modeling large scale structure formation, first suggested by [

12], is to model the temporal evolution of dark matter as a complex scalar field

$\psi (\mathit{x},t)$ governed by Schrödinger’s equation with a nonlocal and nonlinear self-interaction potential given as the solution to Poisson’s field equation:

Here the potential is already divided by the particle mass. We briefly remark on the different physical interpretations of Equation (

3) that exist in the literature: One may regard Equation (

3) as an alternative approximation to Equation (

1), numerically compared to

N-Body simulations

and analytically competing against the dust-model, see e.g., [

14,

15,

16,

17]. Here, both wavefunction and

$\mu =\frac{\hslash}{m}$ do not carry any quantum mechanical meaning and are best understood as a classical complex field and a free model parameter. In the semiclassical limit,

$\mu \to 0$, we expect good correspondence between observables arising from Equations (

1) and (

3), respectively. For instance, the numerical study in [

16] suggests convergence of the wavefunction’s gravitational potential to the classical result of Equation (

1) as

${\mu}^{2}$.

In fact, if Equation (

3) is augmented with Husimi’s quasi probability distribution,

a smoothed version of Wigner’s distribution,

It was shown in [

14] that the phase space distribution

${f}_{H}$ associated with the dynamics of the scalar field

$\psi $ approximates the dynamics of the smoothed Vlasov distribution

$\overline{f}$, see Equation (

4), in the following sense:

Moreover, higher order moments such as the velocity dispersion are non-vanishing and can be readily computed with only the knowledge of the wavefunction. The non-vanishing hierarchy of distribution moments induces a rich set of phenomena such as shell crossings, multi-streaming, and relaxation into the equilibrium state that can all be studied within the SPM [

14,

15,

17].

Section 3.1 illustrates these prototypical evolution stages by studying the gravitational collapse of a sinusoidal density perturbation in phase space by means of Husimi’s distribution, Equation (

4).

Alternatively, one can interpret Equation (

3) as a distinct model for dark matter competing against the established CDM paradigm. In this fuzzy dark matter picture [

13], Equation (

3) arises as the nonrelativistic weak field limit of the Klein–Gordon–Einstein equation. This equation describes ultralight scalar bosons—so called axions—that constitute a cosmic Bose–Einstein condensate trapped in its own gravitational potential. The boson mass is expected to be of the order of

$m\approx {10}^{-22}\phantom{\rule{3.33333pt}{0ex}}\mathrm{eV}$ and, in fact, state of the art 3 + 1 dimensional simulations of axion dark matter [

6] set

$m=8\times {10}^{-23}\phantom{\rule{3.33333pt}{0ex}}\mathrm{eV}$ by fitting simulated halo density profiles against observational data. The remarkable feature of this model is that due to the minuscule axion mass the de-Broglie wavelength takes on macroscopic kiloparsec values such that Heisenberg’s uncertainty principle acts on cosmic scales [

23] and washes out the small scale structure that the CDM paradigm struggles to model correctly. To see the regularizing nature of fuzzy dark matter at small scales one can examine its associated fluid description. Using the Madelung transformation [

24],

as ansatz for the wavefunction, yields a modified Euler equation for the velocity field

$\mathit{u}=\nabla {\varphi}_{u}$:

This equation contains an additional pressure term compared to the dust model, see discussion around Equation (

2), that counteracts the gravitational collapse. More specifically, in

$3+1$ dimensions each collapsed and virialized matter structure contains a flat solitonic core embedded in a dark matter halo of universal shape, known as the Navarro–Frenk–White (NFW) profile [

25]. We refer to [

6,

23] for cosmological simulations in

$3+1$ dimensions and refer to [

4,

5,

7,

8] for numerical studies carried out in a static spacetime.

The present work focuses on a systematic analysis of the

$1+1$ case of Equation (

3). The paper is structured as follows: In

Section 2 we provide details of the numerical procedure used to integrate Equation (

3) in time as well as a convergence study for synthetic initial conditions.

Section 3 then focuses on recovering key characteristics of the fuzzy dark matter model known from

$3+1$ dimensions in only one spatial dimension:

Section 3.1 investigates the phase space evolution of spatially non-localized initial conditions in an expanding spacetime by means of Husimi’s distribution, Equation (

4). To assess both qualitative and quantitative features of the dynamical equilibrium state

Section 3.2 specializes to spatially localized initial conditions. In particular,

Section 3.2.1 illustrates the existence of flat density cores. Contrary to the

$3+1$ case, the core region is not stationary but oscillates in time, which we interpret as a nonlinear superposition of a solitonic ground state and higher order nonlinear modes also known as breathers [

26]. Finally,

Section 3.2.2 investigates the halo of a collapsed structure in static spacetime and recovers the classical CDM expectation of a matter-density obeying a power-law scaling combined with a distinct cutoff radius, see e.g., [

27,

28]. We conclude in

Section 4.