New Explicit and Approximate Solutions of the Newton-Schrödinger System

In this paper, we consider the Newton-Schrödinger system ∇2Ψ=γΦ+a(x)Ψ,∇2Φ=∣Ψ∣2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{rcl} \nabla ^{2} \Psi & =& \left( \gamma \Phi +a(x)\right) \Psi ,\\ \nabla ^{2} \Phi & =& \mid \Psi \mid ^{2}, \end{array} \right. \end{aligned}$$\end{document}which arises in certain quantum transport and chemistry problems. Explicit analytic solutions, which contain an auxiliary parameter, are obtained. An existence and uniqueness theorem to this nonlinear system subject to the boundary conditions is proved. Also, we introduce approximate solutions to the modified Newton-Schrödinger system in the case of spherically-symmetric stationary and time-independence by the Adomian decomposition method.


Introduction
The Newton-Schrödinger equation, occasionally named Schrödinger-Poisson or Gross-Pitaevskii-Newton equation, was first presented to describe self-gravitating scalar particles as a non-relativistic approximation to boson stars [1]. Later then, it was used in very various physical contexts. For illustration, it has been applied in the basics of quantum mechanics to model wave function collapse [2,3], in particular circumstances of cold boson condensates with long-range interactions [4], or for fermion gases in magnetic fields [5]. On the other hand, in cosmology, the Newton-Schrödinger equation governs the two different dark-matter pictures, i.e., those of quantum chromo dynamic axions [6] and scalar field dark matter [7]. Further, in nonlinear optics, it can characterize the propagation of light in liquid nematic crystals [8] or thermo-optical media [9].
Moreover, the Newton-Schrödinger equation has also attracted the attention of experimentalists. They use it as the theoretical background to describe the interaction of quantum matter with classical gravity [10], e.g., the production of solitons in condensed cold atoms [11] or the exploration of rogue waves in the ocean [12]. Gravity optics analogies have been investigated for Newtonian gravity, features of general relativity, and even concerns associated with quantum gravity [13].
In most cases, the Schrödinger-Newton system is a coupling of the Schrö-dinger and Poisson equations. For a single particle, this coupled system consists of the Schrödinger equation for the particle moving in its own gravitational field, which is produced by its own probability density through the Poisson equation. In fact, the Newton-Schrödinger equations are expressed as where ∶ ℝ n × ℝ ⟶ ℂ is the (time-dependent) wave function, the gravitational potential, E(x) is an ordinary potential, is the usual Laplacian operator, and > 0 is a parameter. The system can be reformulated under the standing wave ansatz ( All terms in (1.1) and (1.2) have been rescaled to carry the forms to their canonical expression, and all quantities are dimensionless.
Different results on the existence and uniqueness of solutions to this system has been presented and discussed in Refs. [14,[16][17][18] for several dimensional cases. In the case when the external potential a(x) is a constant, the authors [14], showed that the solutions of this system subject to

The Explicit Solution to the Newton-Schrödinger System in One Dimension
The one-dimensional Newton-Schrödinger system can be written as: which is similar to the two-wave (symbiotic) solitons of interactions in nonlinear quadratic media [35]. Introduce the new dependent variables (1.5) • If a(x) = a is a constant [14], then based on the rotationally symmetric solutions of 4-dimensional semi-linear elliptic problem with exponential nonlinearity: Δ 2 v − 4 e u = 0, where is a parameter [15]. . It follows that , is an arbitrary constant and > 0.
• Similarly, for a �� (x) = 0, we have a(x) = ax + b, where a and b are two constants, then

The Explicit Solution to the Newton-Schrödinger System in Two Dimensions
Proceeding as before, introducing the new dependent variables Thus the system (1.1) reduces to Applying the Laplace operator ▿ 2 to both sides of the second equation of (3.2), we have Let's start off by assuming that a(x, y) is a harmonic function, that is ▿ 2 a = 0 and all solutions of (3.4) are of the form where k > 0, and m are two constants to be determined.
We obtain ▿ 2 v = 2m ( +x+y) 2 and | ∇v | 2 = 2m 2 ( +x+y) 2 . Thus, a simple computation leads to k = 12 √ and m = 2. Consequently, The solutions obtained in Theorem 3 are valid for system (1.1) in ℝ n . The function a(x) is in fact represented in terms of the external potential E(x). When dealing with the quantization of the gravitational field, the external potential E(x) will be a harmonic oscillator or quadratic potential in this situation. Macroscopic quantum systems in a harmonic trap potential [36,37] have been used in particular to test the Schrödinger-Newton equation. Furthermore, the external potential can be assumed as a quadratic function [37] where the behavior of the wave function can be given in terms of the Hermite polynomial.

An Existence and Uniqueness Theorem
To establish a result on the existence and uniqueness theorem, we attach to this nonlinear system (2.1) the following boundary conditions: and where is a given constant. Further, the wave function must satisfy the normalization condition We have the following lemmas.
Integrating the second equation of (2.1) from x to ∞, and using the BC Φ � (∞) = 0, we obtain Rewrite now the first equation of (2.1) in the form In view of Lemma 4, we can eliminate Φ from the RLH of the first equation of (2.1), that is Thus and Thus we have proved

Lemma 5 where and
In the theory of differential equations, there are a lot of methods to establish the existence and uniqueness of solutions. Theorems concerning the existence and properties of fixed points are known as fixed-point theorems. Such theorems are the most important tools for proving the existence and uniqueness of the solution. For more details about the application of the fixed-point theorem in proving the uniqueness of the solutions to ODEs, we can refer to any book or research, e.g., (Section 2.3: Uniqueness [38]).
In order to make use of this theorem, it is sufficient to prove the following lemma.     Proof To prove the existence and uniqueness of the solution, we suppose that (Ψ 1 , Φ 1 ) and (Ψ 2 , Φ 2 ) are two solutions of (2.1) subject to (4.1)-(4.3) such that (Ψ 1 , Φ 1 ) ≠ (Ψ 2 , Φ 2 ). Thus and So that

Approximate Solutions of Spherically-Symmetric Stationary Solutions
In the case of spherical symmetry and time independence, the system (1.5) becomes [19,20] (4.29) (1 + s)( + a(s))ds. In Ref. [20], the authors used (a shooting method) which is a numerical technique to obtain the stationary states based on fourth-order Runge-Kutta integration, starting at r = 0 and integrating outwards towards infinity). The initial values are picked so that U(0) = 1 and other values of S(0) are chosen numerically to fit to the first and second bound-state wave functions.

Consequently,
It remains now to apply the second boundary condition S → 0 as r → ∞ to the function S(r). To compute the constants S(0) and U(0). It is clear that this boundary condition cannot be applied directly to the series (5.19). Recall that it is customary to combine the series solutions obtained by the decomposition method with the Padé approximants to provide an effective tool to treat boundary value problems on an infinite or semi-infinite interval [31].
To minimize the size of calculations, we substitute x = r 2 in the series to obtain The [2/2]

approximant in this case is defined by
To determine the three coefficients of the two polynomials in the numerator and the denominator, we set We find

and
So that the Padé approximant is or The constants S(0) and U(0) can be evaluated by establishing the Padé approximants by setting the coefficient of r of highest power in the numerator polynomial to 0. In view of this, we obtain the following relation or In the beginning, we calculate the numerical solutions of the system (5.1), using the Cash-Karp fourth-fifth order Runge-Kutta method with degree four interpolant, which improves the numerical solutions that are found in Ref. [20]. We have chosen the same initial condition U 0 = 1 but, the values of S 0 are modified to get the correct solutions. The numerical solution of the first and second bound-state wave functions are plotted in Fig. 1. In the same way, we have presented in Fig. 2 the numerical solutions with the initial conditions S 0 = 0.4 and U 0 = 1 , which shows decay oscillations of the wave function S(r) with the radial position r. In Fig. 3 we compare the approximated solutions (5.19) of the system (5.1) and the numerical solutions of the function S(r) in terms of the radial position r. The left panel represents the solutions S(r), while the right one exhibits the function U(r). In this case, the conditions are restrained to the relationship (5.28) with U 0 = 1 . It is clear, the perfect coincidence at near the neighbor of the origin where the approximated solutions are expended. Thus, the result is still the same as what Kibble [39] had already noted this shows that understanding the wave-function collapse is necessary to comprehend how gravity and quantum mechanics are related. Since the Newton-Schrödinger system illustrates how classical gravity and quantum matter are coupled and how a gravitational measurement collapses the wave function, it does not appear to be in direct conflict with the fundamental physics principles.

Conclusion
In this work, we deal with interesting nonlinear problems that play a fundamental role in many areas of physics, and the nonlinear Schrödinger equation is one of the typical models for their description under different circumstances. In  particular, the Schrödinger-Newton equations are at the heart of these nonlinear problems and are explored here. The system is formulated within the framework of two coupled nonlinear differential equations, which can be solved numerically with specific boundary conditions. But the generalization to any kind of boundary condition constitutes a heavy task. Thus, with conformity to actual physical problems, we have investigated the explicit solution for the Newton-Schrödinger system in one and two dimensions. Further, we have examined the uniqueness theorem in the case of one dimension. On the other hand, the approximate solutions of spherically symmetric stationary solutions are established with suitable initial conditions using an important treatment, which is based on the Adomian decomposition technique. Also, we get in a simple manner the relationship between the initial conditions associated with the problem. We compared the approximate solutions to the numerical solutions, and we find a good agreement between them. These vital results open a new class of explorations, which include more nonlinear terms with more coupled differential equations to give rise to new physical concepts.