Symmetry-Driven Multi-Soliton Dynamics in Bose–Einstein Condensates in Reduced Dimensions

Abstract

We present a theoretical and numerical study of soliton formation and dynamics in a Bose–Einstein condensate (BEC) confined within a symmetric harmonic trap, subjected to an external barrier potential. Our investigation focuses on the role of symmetry in the system, particularly highlighting how the interplay between the harmonic confinement and the barrier shape governs the generation and evolution of multi-soliton states. Employing a reduction of the 3D Gross–Pitaevskii equation to lower-dimensional regimes, we analyze the behavior of dark solitons in 2D and 1D configurations using, for the former, exact solutions constructed from the Hirota’s bilinear formalism. We observe that the number of generated solitons exhibits a plateau-like dependence on the height of the potential barrier, reflecting the system’s symmetry and nonlinearity. Furthermore, we break the central symmetry by translating the barrier, leading to asymmetrical soliton patterns and novel dynamical behaviors. These findings underline the fundamental role of symmetry in the formation and stability of solitons in confined quantum gases, offering new perspectives on soliton engineering in trapped BECs.

1. Introduction

Quantum mechanics was originally established to describe the behavior of particles on the microscopic scale, such as atoms and molecules. However, under certain conditions, quantum phenomena can manifest on the macroscopic scale, with superconductivity and superfluidity being two prominent examples. One of the most remarkable macroscopic quantum phenomena is Bose–Einstein condensation (BEC), where a dilute gas of particles occupies the same quantum state below a critical temperature, forming a global coherent matter wave. This phenomenon was predicted theoretically by Bose and Einstein in the 1920s, but it took more than seven decades for it to be observed experimentally. The realization of BEC in ultra-cold gases was achieved in 1995 by Cornell, Wieman, and Ketterle using dilute alkali atoms such as rubidium and sodium [1]. This achievement opened a new era in quantum physics and earned them the Nobel Prize in Physics in 2001 [2]. Since then, studies on ultracold atomic gases have flourished theoretically and experimentally, with BECs being realized in various atomic species. These systems provide an exceptional platform for investigating quantum many-body physics, due to the high degree of experimental control over particle interactions, trapping geometries, and external potentials [3,4]. In particular, BECs confined in harmonic traps have become the standard experimental setup, allowing the exploration of collective excitations, quantum vortices, and nonlinear wave phenomena [5,6,7].

Among the most intriguing nonlinear phenomena in BECs are solitons. Solitons are localized wavefunctions that maintain their shape during propagation because of the balance between dispersion and nonlinearity. Solitons are robust features of nonlinear systems and have been studied extensively in diverse fields, including fluid dynamics, nonlinear optics, and plasma physics [8,9,10,11,12]. The search for exact expressions of such waves is of great importance in order to understand the interactions and dynamics of solitons. Several techniques have been presented, such as the inverse scattering transform [13] and the Hirota bilinear formalism [14]. The latter is an algebraic method that enables one to generate a random number of solitons by modeling each of the solitons as an independent exponential function. In BECs, a dark soliton corresponds to localized density dips on a uniform or trapped background, resulting from a phase jump across their core. They can be generated experimentally through phase imprinting techniques or by density perturbations [10,13,15,16,17]. Solitons in 1D systems exhibit remarkable stability, whereas in higher dimensions (2D and 3D), they are subject to the so-called snake instability, which causes transverse modulations and fragmentation into vortex pairs [18,19,20].

Although the dynamics of solitons in harmonic traps have been extensively investigated, much attention has recently turned to the role of engineered potentials in controlling soliton generation and interactions [19,21,22]. In fact, optical potentials, created by using laser fields, offer a powerful means of shaping the trapping environment beyond simple harmonic confinement. For example, Laguerre–Gaussian (LG) beams can impart orbital angular momentum to atoms and create ring-shaped or more complex potentials [23]. On the other hand, a complementary and practically simpler approach involves the introduction of local barrier potentials within a harmonic trap, which can serve as a trigger for the formation of solitons upon their removal [24].

In this work, we first establish soliton solutions in 2D and 1D configurations by reducing the 3D Gross–Pitaevskii equation (GPE) to lower dimensions. This reduction is valid when the system is strongly confined in one spatial dimension, freezing its dynamics, and allowing description by a 1D or 2D equation. We then focus on the controlled generation of dark solitons in BECs confined in symmetric harmonic trap, using Gaussian or constant barrier potentials. Our goal is to understand how the height, width, and spatial displacement of the barrier influence the number, the trajectories, and the stability of the generated solitons. Through numerical simulations of GPE, we demonstrate that the number of solitons does not vary continuously with the barrier height but rather exhibits a stepwise or plateau behavior, indicating that the soliton number is quantized in certain height intervals. This plateau structure represents an interesting feature of soliton formation in confined systems and suggests that barrier engineering can serve as a reliable tool for tailoring multi-soliton states.

Moreover, we show that while Gaussian barriers require removal to generate solitons, constant barriers lead to soliton formation whether they are removed or not. We also explore the impact of breaking the symmetry of the barrier position within the harmonic trap, revealing that off-centered barriers lead to asymmetric soliton distributions and modified collision patterns. These results emphasize the impact of external potential shaping in controlling solitons and their dynamics in trapped BECs, with potential applications in matter-wave interferometry, precision measurements, and the study of soliton gases.

In contrast to our earlier investigation [21], where we explored the impact of the trap potential on the generation and dynamics of solitons in BECs, shaping the trapping potential has been performed by using the customized potentials created by two crossed Laguerre–Gaussian (LG) beams. In this study, we rigorously derive the exact analytical expression of the solution of the N-soliton using the Hirota bilinear formalism for an arbitrary integer $N>0$, thus generalizing the framework to arbitrary soliton numbers. In addition, we introduce a spatially displaced constant potential barrier and systematically quantify its role in the dynamics of soliton generation. Through numerical simulations, we demonstrate that the number of emergent solitons saturates nonlinearly with increasing barrier height. In particular, we discover a pronounced spatial asymmetry in the trajectories of the soliton induced by the displacement of the barrier, a phenomenon absent in symmetric potential configurations. These findings advance our understanding of soliton interactions in inhomogeneous media, with implications for nonlinear wave control in photonic and quantum control.

The paper is structured as follows. In Section 2, we describe the model and the setup of the proposed experiment. Section 3 details the 2D reduction of the model and details the exact expressions of multi-soliton solutions. The 2D analysis of our results is provided in Section 4. Section 5 presents the 1D reduction of the Gross–Pitaevskii equation and studies the effects of the barrier potential on the number of generated solitons. In Section 6, we discuss the numerical results, focusing on the role of barrier displacement and the effect of the asymmetry. Finally, in Section 7, we conclude with a summary of our findings and an outlook on future directions.

2. The Model

In our model, the BEC is achieved in a fully optical trap formed by two intersecting Laguerre–Gaussian beams, both with orbital angular momentum index $l=1$. One beam propagates along the z-axis, while the other one is perpendicular, creating a three-dimensional harmonic trapping potential (see Figure 1). We consider atomic species like rubidium (${}^{87}\mathrm{R}\mathrm{b}$). This optical approach allows precise control over the potential geometry, surpassing the limitations of conventional harmonic traps [25,26]. Here, we consider the case $\mathit{\ell }=1$ to exhibit exact expressions for soliton solutions in the case of harmonic trapping potential. We refer the interested reader to the numerical interpretation of solitons for $\mathit{\ell }\ne 1$ in [21].

Assuming that two LG beams intersect orthogonally and that $\rho =\sqrt{x^2+y^2}<<w_0$, the potential has the harmonic form

$$V(\rho ,z)=U_{\rho }{\rho }^2+U_z{z}^2,$$

(1)

where $U_{\rho }$ and $U_z$ are the potential depths along the respective axes. These depths may be expressed using the characteristics of the laser such as the power $P_0$, the laser detuning $\delta$, and the waist $w_0$ of the two beams [25].

3. 2D-BEC Analysis

The dynamics of the BEC at zero temperature are governed by the Gross–Pitaevskii equation (GPE), a nonlinear Schrödinger equation that describes the macroscopic wave function of the condensate. The GPE is given by

$$i\hslash {\displaystyle \frac{\partial \psi (\mathbf{r},t)}{\partial t}}=\left(-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2+V(\mathbf{r},t)+Ng{\left|\psi (\mathbf{r},t)\right|}^2\right)\psi (\mathbf{r},t).$$

(2)

In this equation, $\psi (\mathbf{r},t)$ represents the condensate wave function and $\mathbf{r}=(x,y,z)$. The term ${\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2$ corresponds to the kinetic energy of the particles, where ℏ is the reduced Planck constant, and m is the atomic mass. The external potential $V(\mathbf{r},t)$ can vary spatially and temporally, influencing the behavior and dynamics of the condensate. The interaction term ${Ng\left|\psi (\mathbf{r},t)\right|}^2$ accounts for the mean-field interactions between the particles in the condensate, with N being the number of particles and $g=4\pi {\hslash }^2{a}_s/m$ representing the interaction strength characterized by the length of s-wave scattering $a_s$.

To explore the behavior of solitons in BEC, we investigate soliton solutions in reduced spatial dimensional spaces. We start from the 3D GPE (2) and make some confinement assumptions in order to reduce the $3D$ model to $1D$ and $2D$ equations. This reduction technique is justified when the system exhibits strong spatial confinement in one dimension, “freezing” the dynamics along it. This allows one to modify the original equation to lower-dimensional equations.

3.1. Dimension Reduction

For the 2D reduction, we assume that ${\displaystyle \frac{U_{\rho }}{U_z}}<<1$ in the potential V given in Equation (1). This assumption implies that the motion of atoms in the z direction is frozen to the ground state, and that the system can be viewed as two-dimensional.

Let us assume that the wave function is composed of a constant Gaussian axial part ${\varphi }_0$, which is the ground state of the axial harmonic potential, and a time-varying transverse component f as

$$\psi (\mathbf{r},t)=f(x,y,t){\varphi }_0\left(z\right),|{\varphi }_0{\left(z\right)|}^2={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{e}^{-{\displaystyle \frac{z^2}{2{\sigma }^2}}}.$$

(3)

The parameter $\sigma$ will be chosen later. In the harmonic setting, we choose the potential V in Equation (2) as

$$V(\mathbf{r},t)=V\left(\mathbf{r}\right)=V_{2D}\left(\rho \right)+{\displaystyle \frac{1}{2}}m{\omega }_z^2{z}^2,\mathrm{where}{V}_{2D}\left(\rho \right)=U_{\rho }{\rho }^2={\displaystyle \frac{1}{2}}m{\omega }_{\perp }^2(x^2+y^2).$$

(4)

Introducing ansatz (3) into Equation (2) and integrating over z yields an equation for f:

$$i\hslash f_t=-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }_{2D}^{2}f+{\displaystyle \frac{\hslash {\omega }_z}{2}}f+V_{2D}f+Ng\eta {\left|f\right|}^{2}f,$$

(5)

for ${\sigma }^2={\displaystyle \frac{\hslash }{2m{\omega }_z}}$ and

$$\eta ={\int }_{\mathbb{R}}{\left|{\varphi }_0\left(z\right)\right|}^4\mathrm{dz}={\displaystyle \frac{1}{2\sqrt{\pi }\sigma }}=\sqrt{{\displaystyle \frac{m{\omega }_z}{2\pi \hslash }}}.$$

(6)

The term ${\displaystyle \frac{\hslash {\omega }_z}{2}}f$ can be viewed as a constant barrier potential and can be factored out using the transformation

$$f(x,y,t)=e^{i\nu t}F(x,y,t),\nu =-{\displaystyle \frac{{\omega }_z}{2}}.$$

(7)

This transformation is invariant under time translation $t⟶t+{\displaystyle \frac{2\pi }{\nu }}$ and satisfies ${\left|f\right|}^2={\left|F\right|}^2$ for all t, letting the probability density function ${\left|\psi \right|}^2$ remain unchanged. We obtain the desired equation for F:

$$i\hslash F_t=-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }_{2D}^{2}F+V_{2D}F+Ng\eta {\left|F\right|}^{2}F.$$

(8)

Hence, to resume,

$${\left|\psi (\mathbf{r},t)\right|}^2=\sqrt{{\displaystyle \frac{m{\omega }_z}{\pi \hslash }}}{\left|F(x,y,t)\right|}^2\times exp\left(-{\displaystyle \frac{m{\omega }_z}{\hslash }}{z}^2\right)$$

(9)

is the probability density function in the $2D$ reduction case.

Gaussian and Constant Barriers

It has been shown that adding a Gaussian barrier to the harmonic potential and releasing it at a fixed positive time is an efficient technique to generate soliton solutions. Explicitly speaking, one can consider the following external potential:

$${\tilde{V}}_{2D}(\rho ,t)=V_{2D}\left(\rho \right)+V_{\rho }\left(t\right)e^{-{\rho }^2/2{\sigma }_{\rho }^2},$$

(10)

where $V_{\rho }\left(t\right)=V_0$ is a non-zero constant for $t<0.02$ s, setting $V_{\rho }\left(t\right)=0$ for time $t\ge 0.02$ s.

In our previous analysis [21] and in this paper, we explore, throughout numerical simulations, the effects of changing the height $V_0$ and the width ${\sigma }_{\rho }$ on the generation of solitons. As will be shown, soliton states in 2D exhibit the so-called snake instability. Analytically, it is difficult to obtain exact solutions of the equation with this Gaussian barrier. However, when ${\sigma }_{\rho }$ tends to infinity, we obtain a constant barrier $V_0$. In this particular case, we retrieve a similar equation such as Equation (5), where the constant barrier may be factored out by using a transformation of the form (7). In the next sections, numerical simulations are made on releasing the constant barrier at fixed time t. We stress here that this type of experience is feasible. In the analytical case, the constant barrier is not removed in order to obtain exact solutions which would be difficult otherwise.

3.2. Hirota Bilinear Formalism

In order to find exact soliton solutions of Equation (8) with harmonic potential $V_{2D}$ given in Equation (4), one can use the Hirota bilinear method [14]. This algebraic method consists of bilinearizing a partial differential equation using a transformation of the dependent variable from which multiple solitons can be obtained from a sum of independent exponential functions. This allows one to obtain explicit expressions for a random number N of solitons and obtain interaction coefficients which can be varied in order to study interactions between such waves.

Let us consider the following transformation [27] for the function F in Equation (8):

$$F(x,y,t)=H(X,Y,T)\times exp\left(r+iq(X^2+Y^2)\right),$$

(11)

where $r=r\left(T\right)$ and

$$X=e^{r}x,Y=e^{r}y,T={\omega }_{\perp }t,q=-{\displaystyle \frac{m{\omega }_{\perp }}{2\hslash }}{e}^{-2r}\dot{r}.$$

(12)

Using the ansatz (11) and taking r to satisfy

$$\ddot{r}={\left(\dot{r}\right)}^2+1\mathrm{with}\dot{r}={\displaystyle \frac{\mathrm{d}r}{\mathrm{d}T}},$$

(13)

allows one to transform Equation (8) into a Schrödinger-type equation:

$$i\hslash {\omega }_{\perp }{H}_T=e^{2r}\left(-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }_{2D}^{2}H+Ng\eta {\left|H\right|}^{2}H\right).$$

(14)

The differential equation satisfied by r can be integrated to obtain the following exact expression:

$$r\left(T\right)=-\ln |\cos (T+{\alpha }_1)|+{\alpha }_2=-\ln \left|\cos ({\omega }_{\perp }t+{\alpha }_1)\right|+{\alpha }_2,$$

(15)

where ${\alpha }_1$ and ${\alpha }_2$ are constants of integration. These theoretical calculations highlight time translation invariance $t⟶t+{\displaystyle \frac{2\pi }{{\omega }_{\perp }}}$. The time period ${\displaystyle \frac{2\pi }{{\omega }_{\perp }}}$ implies fewer collisions between solitons as ${\omega }_{\perp }$ decreases.

Schrödinger-type equations have been largely studied and have been shown to possess multi-soliton solutions. Indeed, using the Hirota bilinear transformation [14],

$$H={\displaystyle \frac{C}{R}},$$

(16)

where C and R are, respectively, complex and real-valued functions. This Hirota change of variable bilinearizes Equation (14) into

$$\begin{array}{c}\hfill \left(i{\mathcal{D}}_T+{\displaystyle \frac{\hslash }{2m{\omega }_{\perp }}}{e}^{2r}({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)\right)(C·R)=0,\end{array}$$

(17)

$$\begin{array}{c}\hfill {\displaystyle \frac{{\hslash }^2}{2m}}({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)(R·R)+Ng\eta {\left|C\right|}^2=0,\end{array}$$

(18)

where the Hirota bilinear derivative $\mathcal{D}$ is defined as

$${\mathcal{D}}_z^n(f_1·f_2)={({\partial }_{z_1}-{\partial }_{z_2})}^n{f}_1\left(z_1\right)f_2\left(z_2\right){|}_{z_1=z_2=z}.$$

(19)

Below, we construct the multi-soliton solution and study the effects of the physical constants on their dynamics.

In order to illustrate these solitons, one needs to plot the function ${\left|\psi \right|}^2$. From the above transformations, we have that

$${\left|\psi \right|}^2=|{\varphi }_0{|}^2{\left|F\right|}^2=e^{2r}{\left|{\varphi }_0\right|}^2{\displaystyle \frac{{\left|C\right|}^2}{R^2}}.$$

(20)

Using the second Equation (18) of the bilinear system, we can show that the above function may be re-written in terms of the real-valued function R as

$$\begin{array}{ccc}\hfill {\left|\psi (X,Y,z,T)\right|}^2& =& -{\displaystyle \frac{1}{\alpha }}{e}^{2r\left(T\right)}{\left|{\varphi }_0\left(z\right)\right|}^2{\nabla }_{2D}^{2}ln\left(R(X,Y,T)\right)\hfill \\ & =& -{\displaystyle \frac{1}{\alpha }}{\left|{\varphi }_0\left(z\right)\right|}^2{\nabla }_{2D}^{2}ln\left(R(e^{r\left({\omega }_{\perp }t\right)}x,e^{r\left({\omega }_{\perp }t\right)}y,{\omega }_{\perp }t)\right).\hfill \end{array}$$

(21)

The parameter $\alpha$ will play an important role in soliton interactions. We thus obtain explicitly

$$\alpha ={\displaystyle \frac{Ng\eta m}{{\hslash }^2}}.$$

(22)

Returning to Equation (14), we can observe that this parameter $\alpha$ is proportional to the quotient between the interaction coefficient and the kinetic one. This balance between nonlinearity and dispersion plays an important role in the generation of solitons [13,14].

3.2.1. The 1-Soliton Solution

An $ϵ$-expansion is used around the ground state solution $C=0$ and $R=1$ to construct a N-soliton solution from the bilinear Equations (17) and (18). For $N=1$, corresponding to a single soliton, we suppose

$$C=ϵC_1\mathrm{and}R=1+{ϵ}^2{R}_2,$$

(23)

where $ϵ$ is a non-zero real parameter, $C_1$ is a complex-valued function, and $R_2$ a real-valued function. Introducing these components into Equations (17) and (18) yields a system of equations by setting each power of $ϵ$ to zero:

$$\begin{array}{c}\hfill i{\partial }_T{C}_1+{\displaystyle \frac{\hslash }{2m{\omega }_{\perp }}}{e}^{2r}{\nabla }_{2D}^2{C}_1=0,\end{array}$$

(24)

$$\begin{array}{c}\hfill \left(i{\mathcal{D}}_T+{\displaystyle \frac{\hslash }{2m{\omega }_{\perp }}}{e}^{2r}({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)\right)(C_1·R_2)=0,\end{array}$$

(25)

$$\begin{array}{c}\hfill {\displaystyle \frac{{\hslash }^2}{m}}{\nabla }_{2D}^2{R}_2+Ng\eta {\left|C_1\right|}^2=0,\end{array}$$

(26)

$$\begin{array}{c}\hfill ({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)(R_2·R_2)=0.\end{array}$$

(27)

For a single soliton solution, we suppose that

$$C_1=e^{\Lambda },\Lambda ={\kappa }_{X}X+{\kappa }_{Y}Y+\lambda \left(T\right),$$

(28)

where ${\kappa }_X$ and ${\kappa }_Y$ are constants, and $\lambda$ is a pure imaginary function of T. Introducing this exponential form for $C_1$ into Equation (24) yields a constraint for $\lambda$:

$$i\dot{\lambda }+{\displaystyle \frac{\hslash }{2m{\omega }_{\perp }}}{e}^{2r}\left({\kappa }_X^2+{\kappa }_Y^2\right)=0⟺\lambda \left(T\right)=i{\displaystyle \frac{\hslash }{2m{\omega }_{\perp }}}{e}^{2{\alpha }_2}\left({\kappa }_X^2+{\kappa }_Y^2\right)tan(T+{\alpha }_1).$$

(29)

Introducing $C_1$ in Equation (26) gives the explicit form for $R_2$:

$$R_2=\xi e^{\Lambda +{\Lambda }^{*}},\xi =-{\displaystyle \frac{\alpha }{4\left({\kappa }_X^2+{\kappa }_Y^2\right)}}.$$

(30)

The expressions for $C_1$ and $R_2$ are sufficient for systems (24)–(27) to be satisfied.

To resume, we obtain an exact solution $\psi$ where ${\left|\psi \right|}^2$ is given by

$${\left|\psi \right|}^2={ϵ}^2\sqrt{{\displaystyle \frac{m{\omega }_z}{\pi \hslash }}}\times {\displaystyle \frac{exp\left(\Lambda +{\Lambda }^{*}+2r-\frac{{\omega }_{z}m}{\hslash }{z}^2\right)}{{\left(1+{ϵ}^2\xi exp(\Lambda +{\Lambda }^{*})\right)}^2}}.$$

(31)

In order to study the interaction between N solitons, one supposes the following $ϵ$-expansions for functions C and R:

$$C=\sum _{k=1}^N{ϵ}^{2k-1}{C}_{2k-1}\mathrm{and}R=1+\sum _{k=1}^N{ϵ}^{2k}{R}_{2k},$$

(32)

where $C_j$ and $R_j$ are, respectively, complex and real-valued. Introducing these forms into Equations (17) and (18) yields a system of equations for each power of $ϵ$. These equations can then be solved using a sum of independent exponential functions.

3.2.2. The 2-Soliton Solution

To obtain the 2-soliton solution, we take $N=2$ in Equation (32). We thus suppose that

$$C=ϵC_1+{ϵ}^3{C}_3\mathrm{and}R=R_0+{ϵ}^2{R}_2+{ϵ}^4{R}_4,$$

(33)

where $R_0=1$. Introducing these components into Equations (17) and (18) yields a system of equations for each power of $ϵ$:

$$\begin{array}{c}\hfill \left(i{\mathcal{D}}_T+{\displaystyle \frac{\hslash }{2m{\omega }_{\perp }}}{e}^{2r}({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)\right)\left(C_1·1\right)=0,\end{array}$$

(34)

$$\begin{array}{c}\hfill {\displaystyle \frac{{\hslash }^2}{m}}({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)\left(R_2·1\right)+Ng\eta {\left|C_1\right|}^2=0,\end{array}$$

(35)

$$\begin{array}{c}\hfill \left(i{\mathcal{D}}_T+{\displaystyle \frac{\hslash }{2m{\omega }_{\perp }}}{e}^{2r}({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)\right)\left(C_1·R_2+C_3·1\right)=0,\end{array}$$

(36)

$$\begin{array}{c}\hfill {\displaystyle \frac{{\hslash }^2}{2m}}({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)\left(2R_4·1+R_2·R_2\right)+Ng\eta (C_1^{*}{C}_3+C_1{C}_3^{*})=0,\end{array}$$

(37)

$$\begin{array}{c}\hfill \left(i{\mathcal{D}}_T+{\displaystyle \frac{\hslash }{2m{\omega }_{\perp }}}{e}^{2r}({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)\right)\left(C_1·R_4+C_3·R_2\right)=0,\end{array}$$

(38)

$$\begin{array}{c}\hfill {\displaystyle \frac{{\hslash }^2}{m}}({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)\left(R_4·R_2\right)+Ng\eta {\left|C_3\right|}^2=0\end{array}$$

(39)

$$\begin{array}{c}\hfill \left(i{\mathcal{D}}_T+{\displaystyle \frac{\hslash }{2m{\omega }_{\perp }}}{e}^{2r}({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)\right)\left(C_3·R_4\right)=0,\end{array}$$

(40)

$$\begin{array}{c}\hfill ({\mathcal{D}}_X^2+{\mathcal{D}}_Y^2)\left(R_4·R_4\right)=0.\end{array}$$

(41)

For the 2-soliton solution, we suppose that

$$C_1=e^{{\Lambda }_1}+e^{{\Lambda }_2},$$

(42)

where ${\Lambda }_j={\kappa }_j(X+Y)+{\lambda }_j\left(T\right)$. In this configuration, ${\kappa }_j$ are constants such that ${\kappa }_1\ne {\kappa }_2$ (two distinct solitary waves) and ${\lambda }_j$ are pure imaginary functions of the variable T for $j=1,2$. Introducing the expression for $C_1$ into Equation (34) yields the constraints

$$i{\dot{\lambda }}_j+{\displaystyle \frac{\hslash }{m{\omega }_{\perp }}}{e}^{2r}{\kappa }_j^2=0⟺{\lambda }_j\left(T\right)=i{\displaystyle \frac{\hslash }{m{\omega }_{\perp }}}{e}^{2{\alpha }_2}{\kappa }_j^{2}tan(T+{\alpha }_1),$$

(43)

for $j=1,2$. The explicit expression for $R_2$ is then recovered using Equation (35):

$$R_2={\xi }_m^n{e}^{{\Lambda }_m+{\Lambda }_n^{*}}={\xi }_1^1{e}^{{\Lambda }_1+{\Lambda }_1^{*}}+{\xi }_1^2{e}^{{\Lambda }_1+{\Lambda }_2^{*}}+{\xi }_2^1{e}^{{\Lambda }_2+{\Lambda }_1^{*}}+{\xi }_2^2{e}^{{\Lambda }_2+{\Lambda }_2^{*}},$$

(44)

where the repeated indices m and n are meant to be summed over $m,n=1,2$. The coefficients ${\xi }_m^n$ are given as

$${\xi }_m^n=-{\displaystyle \frac{\alpha }{2{({\kappa }_m+{\kappa }_n)}^2}}.$$

(45)

The function $C_3$ is recovered in a similar fashion by Equation (36). We obtain

$$C_3={\xi }_{12}^p{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_p^{*}}={\xi }_{12}^1{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_1^{*}}+{\xi }_{12}^2{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_2^{*}},$$

(46)

where

$${\xi }_{12}^k=-{\displaystyle \frac{2}{\alpha }}{\xi }_1^k{\xi }_2^k{({\kappa }_1-{\kappa }_2)}^2=-{\displaystyle \frac{\alpha }{2}}{\displaystyle \frac{{({\kappa }_1-{\kappa }_2)}^2}{{({\kappa }_1+{\kappa }_k)}^2{({\kappa }_2+{\kappa }_k)}^2}}.$$

(47)

To obtain the explicit expression for $R_4$, we take Equation (37). We obtain

$$R_4={\xi }_{12}^{12}{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_1^{*}+{\Lambda }_2^{*}}\mathrm{with}{\xi }_{12}^{12}={\xi }_{12}^1{\xi }_{12}^2={\displaystyle \frac{{\alpha }^2}{64}}{\displaystyle \frac{{({\kappa }_1-{\kappa }_2)}^4}{{\kappa }_1^2{\kappa }_2^2{({\kappa }_1+{\kappa }_2)}^4}}.$$

(48)

With these expressions, we can show that the system of equations is solved.

We can thus observe that when ${\kappa }_1={\kappa }_2$, the interaction coefficients ${\xi }_{12}^k$ and ${\xi }_{12}^{12}$ vanish. This is to be expected since each soliton is characterized by the value of $\kappa$. For ${\kappa }_1\ne {\kappa }_2$, the interaction coefficients are non-zero, and we can study the influence of each parameter to understand the impact of physical quantities on the dynamic of solitons. Furthermore, the parameter $\alpha$ defined in Equation (22) plays a crucial role in the interaction coefficients. Indeed, a small parameter $\alpha$ implies weak interaction and almost mimics two non-interacting solitary waves.

3.2.3. The 3-Soliton Solution

The exact solutions of integrable models have been of large intensive study in the past and recent years. Mathematicians have established different integrability criteria such as the existence of a Lax Pair, an inverse scattering transformation [13], the existence of a 3-soliton solution [28], and others. Although mathematicians have not properly defined this concept, the proof of the existence of a 3-soliton solution is believed to ensure the existence of a N-soliton solutions for all $N\in {\mathbb{N}}^{*}$.

To obtain the 3-soliton solution, we take $N=3$ in the expressions (32). We thus suppose that

$$C=ϵC_1+{ϵ}^3{C}_3+{ϵ}^5{C}_5\mathrm{and}R=R_0+{ϵ}^2{R}_2+{ϵ}^4{R}_4+{ϵ}^6{R}_6,$$

(49)

where $R_0=1$. For the 3-soliton solution, we suppose that

$$C_1=e^{{\Lambda }_1}+e^{{\Lambda }_2}+e^{{\Lambda }_3},$$

(50)

where ${\Lambda }_j={\kappa }_j(X+Y)+{\lambda }_j\left(T\right)$, ${\kappa }_j$ are constants such that ${\kappa }_1\ne {\kappa }_2\ne {\kappa }_3$ and ${\lambda }_j$ are pure imaginary functions of the variable T for $j=1,2,3$. As for the 2-soliton solution, the functions ${\lambda }_j$ satisfy the differential equation

$$i\dot{{\lambda }_j}+{\displaystyle \frac{\hslash }{m{\omega }_{\perp }}}{e}^{2r}{\kappa }_j^2=0⟺{\lambda }_j\left(T\right)=i{\displaystyle \frac{\hslash }{m{\omega }_{\perp }}}{e}^{2{\alpha }_2}{\kappa }_j^{2}tan(T+{\alpha }_1),$$

(51)

for $j=1,2,3.$ Proceeding as for the 2-soliton solution, one can find the unknown functions $C_{2j-1}$ and $R_{2j}$ for $j=1,2,3$. After thorough symbolic computations, one obtains the explicit expressions for these functions:

$$\begin{array}{ccc}\hfill R_2& =& \hfill {\xi }_m^n{e}^{{\Lambda }_m+{\Lambda }_n^{*}}={\xi }_1^1{e}^{{\Lambda }_1+{\Lambda }_1^{*}}+{\xi }_1^2{e}^{{\Lambda }_1+{\Lambda }_2^{*}}+{\xi }_1^3{e}^{{\Lambda }_1+{\Lambda }_3^{*}}+{\xi }_2^1{e}^{{\Lambda }_2+{\Lambda }_1^{*}}+{\xi }_2^2{e}^{{\Lambda }_2+{\Lambda }_2^{*}}\\ & +& {\xi }_2^3{e}^{{\Lambda }_2+{\Lambda }_3^{*}}+{\xi }_3^1{e}^{{\Lambda }_3+{\Lambda }_1^{*}}+{\xi }_3^2{e}^{{\Lambda }_3+{\Lambda }_2^{*}}+{\xi }_3^3{e}^{{\Lambda }_3+{\Lambda }_3^{*}},\hfill \end{array}$$

(52)

$$\begin{array}{ccc}\hfill C_3& =& {\xi }_{\left(mn\right)}^p{e}^{{\Lambda }_m+{\Lambda }_n+{\Lambda }_p^{*}}={\xi }_{12}^1{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_1^{*}}+{\xi }_{12}^2{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_2^{*}}+{\xi }_{12}^3{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_3^{*}}\hfill \\ & +& {\xi }_{13}^1{e}^{{\Lambda }_1+{\Lambda }_3+{\Lambda }_1^{*}}+{\xi }_{13}^2{e}^{{\Lambda }_1+{\Lambda }_3+{\Lambda }_2^{*}}+{\xi }_{13}^3{e}^{{\Lambda }_1+{\Lambda }_3+{\Lambda }_3^{*}}+{\xi }_{23}^1{e}^{{\Lambda }_2+{\Lambda }_3+{\Lambda }_1^{*}}\hfill \\ & +& {\xi }_{23}^2{e}^{{\Lambda }_2+{\Lambda }_3+{\Lambda }_2^{*}}+{\xi }_{23}^3{e}^{{\Lambda }_2+{\Lambda }_3+{\Lambda }_3^{*}},\hfill \end{array}$$

(53)

$$\begin{array}{ccc}\hfill R_4& =& {\xi }_{\left(mn\right)}^{\left(pq\right)}{e}^{{\Lambda }_m+{\Lambda }_n+{\Lambda }_p^{*}+{\Lambda }_q^{*}}={\xi }_{12}^{12}{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_1^{*}+{\Lambda }_2^{*}}+{\xi }_{12}^{13}{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_1^{*}+{\Lambda }_3^{*}}+{\xi }_{12}^{23}{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_2^{*}+{\Lambda }_3^{*}}\hfill \\ & +& {\xi }_{13}^{12}{e}^{{\Lambda }_1+{\Lambda }_3+{\Lambda }_1^{*}+{\Lambda }_2^{*}}+{\xi }_{13}^{13}{e}^{{\Lambda }_1+{\Lambda }_3+{\Lambda }_1^{*}+{\Lambda }_3^{*}}+{\xi }_{13}^{23}{e}^{{\Lambda }_1+{\Lambda }_3+{\Lambda }_2^{*}+{\Lambda }_3^{*}}+{\xi }_{23}^{12}{e}^{{\Lambda }_2+{\Lambda }_3+{\Lambda }_1^{*}+{\Lambda }_2^{*}}\hfill \\ & +& +{\xi }_{23}^{13}{e}^{{\Lambda }_2+{\Lambda }_3+{\Lambda }_1^{*}+{\Lambda }_3^{*}}+{\xi }_{23}^{23}{e}^{{\Lambda }_2+{\Lambda }_3+{\Lambda }_2^{*}+{\Lambda }_3^{*}},\hfill \end{array}$$

(54)

$$\begin{array}{ccc}\hfill C_5& =& {\xi }_{123}^{\left(pq\right)}{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_3+{\Lambda }_p^{*}+{\Lambda }_q^{*}}={\xi }_{123}^{12}{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_3+{\Lambda }_1^{*}+{\Lambda }_2^{*}}+{\xi }_{123}^{13}{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_3+{\Lambda }_1^{*}+{\Lambda }_3^{*}}\hfill \\ & +& {\xi }_{123}^{23}{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_3+{\Lambda }_2^{*}+{\Lambda }_3^{*}},\hfill \end{array}$$

(55)

$$\begin{array}{ccc}\hfill R_6& =& {\xi }_{123}^{123}{e}^{{\Lambda }_1+{\Lambda }_2+{\Lambda }_3+{\Lambda }_1^{*}+{\Lambda }_2^{*}+{\Lambda }_3^{*}},\hfill \end{array}$$

(56)

where repeated indices are meant to be summed, and $\left(mn\right)$ signifies strictly ordered pairs of the numbers $1,2$ and 3: $\left(12\right)$, $\left(13\right)$, and $\left(23\right)$. The different coefficients may be explicitly calculated. One obtains

$$\begin{array}{ccc}\hfill {\xi }_m^n& =& {\xi }_n^m=-{\displaystyle \frac{\alpha }{2{({\kappa }_m+{\kappa }_n)}^2}},A_{mn}={({\kappa }_m-{\kappa }_n)}^2,\hfill \end{array}$$

(57)

$$\begin{array}{ccc}\hfill {\xi }_{\left(mn\right)}^p& =& \left(-{\displaystyle \frac{2}{\alpha }}\right){\xi }_m^p{\xi }_n^p{A}_{mn},\hfill \end{array}$$

(58)

$$\begin{array}{ccc}\hfill {\xi }_{\left(mn\right)}^{\left(pq\right)}& =& {\left(-{\displaystyle \frac{2}{\alpha }}\right)}^2{\xi }_m^p{\xi }_m^q{\xi }_n^p{\xi }_n^q{A}_{mn}{A}_{pq},\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill {\xi }_{123}^{\left(pq\right)}& =& {\left(-{\displaystyle \frac{2}{\alpha }}\right)}^4{\xi }_1^p{\xi }_1^q{\xi }_2^p{\xi }_2^q{\xi }_3^p{\xi }_3^q{A}_{12}{A}_{13}{A}_{23}{A}_{pq},\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill {\xi }_{123}^{123}& =& {\left(-{\displaystyle \frac{2}{\alpha }}\right)}^6{\xi }_1^1{\xi }_2^2{\xi }_3^3{\left({\xi }_1^2\right)}^2{\left({\xi }_1^3\right)}^2{\left({\xi }_2^3\right)}^2{A}_{12}^2{A}_{13}^2{A}_{23}^2.\hfill \end{array}$$

(61)

The above coefficients are dependent on the parameter $\alpha$ as defined in Equation (22). Again, as $\alpha$ tends to zero, these coefficients annihilate, and the three solitons propagate as three non-interacting solitary waves. On the other hand, as $\alpha$ grows (meaning that the interacting coefficient grows), the contribution of these coefficients becomes significant, and the solitons exhibit strong interactions.

3.2.4. The N-Soliton Solution

As we already know, the square norm of the wave function can be expressed in terms of R uniquely (see Equation (21)). In the N-soliton context, it is in the form

$$R=1+{ϵ}^2{R}_2+{ϵ}^4{R}_4+\cdots +{ϵ}^{2N}{R}_{2N}.$$

(62)

Thus, one can generalize to a N-soliton solution for $N\ge 2$. Indeed, the coefficients may be explicitly calculated to obtain

$${\xi }_{(j_1{j}_2\dots j_P)}^{(l_1{l}_2\dots l_M)}={\left(-{\displaystyle \frac{2}{\alpha }}\right)}^{\left({\displaystyle \genfrac{}{}{0pt}{}{P}{2}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{M}{2}}\right)}\prod _{j,l}{\xi }_j^l\prod _{\left(j_{\mu }{j}_{\nu }\right),\left(l_{\beta }{l}_{\gamma }\right)}{A}_{j_{\mu }{j}_{\nu }}{A}_{l_{\beta }{l}_{\gamma }}.$$

(63)

Using these coefficients, we obtain

$$\begin{array}{ccc}\hfill R_2& =& {\xi }_m^n{e}^{{\Lambda }_m+{\Lambda }_n^{*}},m,n\in \left[[1,N]\right],\hfill \end{array}$$

(64)

$$\begin{array}{ccc}\hfill R_{2k}& =& {\xi }_{(j_1{j}_2\dots j_k)}^{(l_1{l}_2\dots l_k)}exp\left(\sum _{p=1}^k{\Lambda }_{j_p}+{\Lambda }_{l_p}^{*}\right),\hfill \end{array}$$

(65)

for $k\in \left[\right[2,N\left]\right]$.

4. Results

To simulate the dynamics of the BEC, we employ a combination of numerical techniques to solve the time-dependent GPE (2). Using the Thomas–Fermi approximation, the BEC wave function is initialized by neglecting the kinetic term in the GPE [29]. The initial density distribution is then given by

$$n\left(\mathbf{r}\right)=max\left({\displaystyle \frac{\mu -V\left(\mathbf{r}\right)}{g}},0\right)$$

(66)

where $\mu$ is the chemical potential of the BEC. The GPE (2) is solved using the split-step Fourier method [30]. This approach alternates between position space for potential and interaction terms and momentum space for kinetic terms. To obtain the ground state of the system, we implement the imaginary-time relaxation technique, evolving the wave function in imaginary time by Wick rotation. The wave function is normalized to maintain a constant particle number, and the process continues until the energy and wave function converge to the ground state. Following ground state calculation, we switch back to real-time evolution to observe solitons generation and dynamics. The split-step Fourier method is applied in each time step, with the ground state serving as the initial condition. The spatial grid typically consists of $2^{12}$ points in each dimension, with time steps $dt={10}^{-5}$ ms for a total dynamics time of $t_f=0.2$ ms.

In the 2D configuration, soliton solutions are exhibited that, while exact theoretically, are found to be rapidly unstable over time or present discontinuities in their expressions due to the cosine contribution in the phase (15). This instability is well known experimentally in the 2D setting as the snake instability [20]. In fact, from Equation (21), we have

$${\left|\psi \right|}^2=-{\displaystyle \frac{1}{\alpha }}{\left|{\varphi }_0\left(z\right)\right|}^2{\nabla }_{2D}^{2}lnR\left({\displaystyle \frac{x}{|cos({\omega }_{\perp }t\left)\right|}},{\displaystyle \frac{y}{|cos({\omega }_{\perp }t\left)\right|}},{\omega }_{\perp }t\right).$$

(67)

For the purpose of this discussion, we choose ${\alpha }_1={\alpha }_2=0$ without lost of generality. We can observe that for

$$t_n=(2n+1){\displaystyle \frac{\pi }{2{\omega }_{\perp }}},n\in \mathbb{N},$$

(68)

the exact solutions present discontinuities.

We investigated the generation of solitons through the introduction and subsequent removal of the external barrier potential as in [21]. Surprisingly, our numerical simulations reveal a striking difference between two types of barriers. First, when the Gaussian barrier is not removed (panel (a) of Figure 2), no soliton is generated and the condensate remains largely unperturbed, split into two parts. Here, we set the height $V_0=100$ and the width ${\sigma }_{\rho }=0.6$ in the Gaussian barrier (10). However, for a constant barrier of the same height $V_0=100$, solitons are generated regardless of whether the barrier is removed (panel (b) of Figure 2) or left in place (panel (d) of Figure 2). The resulting solitons exhibit identical dynamics in both cases. Within the framework of the $2D$ reduction, we note that Equation (5) introduces a constant energy term ${\displaystyle \frac{\hslash {\omega }_z}{2}}$, which is subsequently eliminated through the transformation (7) preserving the probability density. This observation shows that the generation of solitons is independent of the constant barrier potential. Furthermore, the density probability functions ${\left|f\right|}^2$ and ${\left|F\right|}^2$ remain invariant under time-translation symmetry, reflecting their insensitivity to temporal shifts. These observations explain, indeed, this same behavior observed in panels (b) and (d) of Figure 2.

As can be seen in Figure 2, the analysis of $2D$ BEC reveals complex dynamics and instabilities of the soliton structures. Although exact multi-soliton solutions can be constructed using integrable techniques such as Hirota’s bilinear formalism, these solutions exhibit inherent dynamical instabilities [18]. As said above, dark soliton solutions in 2D systems are prone to decay over time due to the so-called snake instability, a well-established phenomenon in nonlinear systems and BECs [19]. The snake instability arises from the transverse modulation of the soliton, causing it to break into vortex pairs or more complex nonlinear structures due to phase discontinuities or singularities introduced by the cosine terms in the analytical expressions (see details above). These artifacts are not just mathematical curiosities but correspond to the physical instabilities observed in experimental and numerical studies.

5. 1D-BEC Analysis

In contrast, as we will see in the 1D BEC setting, solitons exhibit remarkable stability over time. The reduced dimensionality suppresses the snake instability, allowing dark solitons to propagate robustly within a quasi-1D trap. This stability makes 1D solitons a natural platform for exploring controlled soliton generation and interactions.

In this section, we start by showing the 1D reduction of the GPE (2), and then investigate the generation of solitons in a 1D BEC by introducing and subsequently removing the external barrier, modeled as a Gaussian or constant potential. Through systematic numerical simulations, we show that the number of generated solitons depends critically on both the height and the shape of the barrier. Our goal is to establish a relationship between the geometry of the barrier and the number of generated solitons, enabling precise control over the number of generated solitons. Such control is crucial for tailoring multi-soliton states in trapped BECs, with potential applications in matter-wave interferometry and nonlinear quantum dynamics.

5.1. Dimension Reduction

For the 1D reduction, we assume that ${\displaystyle \frac{U_z}{U_{\rho }}}<<1$ in the trap potential V given in Equation (1). This assumption implies that the motion of atoms in the $(x,y)$ plane direction is frozen into the ground state, allowing the system to be treated as one-dimensional.

Let us assume that the wave function is composed of a constant Gaussian transverse part ${\varphi }_0$, which is the ground state of the transverse harmonic potential, and a time-varying axial component f as

$$\psi (\mathbf{r},t)=f(z,t){\varphi }_0\left(x\right){\varphi }_0\left(y\right),$$

(69)

where $|{\varphi }_0{\left(\zeta \right)|}^2$ is given in Equation (3) for $\zeta =x,y$. The parameter $\sigma$ will be chosen in future steps. For the 1D reduction of the GPE (2), we suppose that the potential V is given as in Equation (4) where we write

$$V_{1D}\left(z\right)=U_z{z}^2,U_z={\displaystyle \frac{1}{2}}m{\omega }_z^2.$$

(70)

Introducing the factorization (69) into the 3D GPE (2) and integrating over x and y yields the following equation:

$$i\hslash f_t=-{\displaystyle \frac{{\hslash }^2}{2m}}{f}_{zz}+\hslash {\omega }_{\perp }f+V_{1D}f+Ng\eta {\left|f\right|}^{2}f,$$

(71)

where ${\sigma }^2={\displaystyle \frac{\hslash }{2m{\omega }_{\perp }}}$ and

$$\eta ={\int }_{{\mathbb{R}}^2}|{\varphi }_0{\left(x\right)|}^4{\left|{\varphi }_0\left(y\right)\right|}^4\mathrm{d}x\mathrm{d}y={\displaystyle \frac{m{\omega }_{\perp }}{2\pi \hslash }}.$$

(72)

The term $\hslash {\omega }_{\perp }f$ can be viewed as a constant barrier potential and can be factored out using a time rotation substitution:

$$f(z,t)=e^{i\nu t}F(z,t),\nu =-{\omega }_{\perp }.$$

(73)

It is important to note that this transformation leaves the density probability function ${\left|\psi \right|}^2$ unchanged. Furthermore, we observe a time translation invariance $t⟶t+{\displaystyle \frac{2\pi }{{\omega }_{\perp }}}$. This observation can be noticed in Figure 3 We obtain the desired equation applying this substitution in Equation (71):

$$i\hslash F_t=-{\displaystyle \frac{{\hslash }^2}{2m}}{F}_{zz}+V_{1D}F+Ng\eta {\left|F\right|}^{2}F.$$

(74)

Hence, to resume,

$${\left|\psi (\mathbf{r},t)\right|}^2={\displaystyle \frac{m{\omega }_{\perp }}{\pi \hslash }}{\left|F(z,t)\right|}^2\times exp\left(-{\displaystyle \frac{m{\omega }_{\perp }}{\hslash }}{\rho }^2\right)$$

(75)

is the probability density function in the $1D$ reduction case.

In the 1D setting, soliton solutions can be dynamically generated by the sudden removal of an external barrier potential applied along one spatial direction. We consider the following external potential in the axial z-direction:

$$\tilde{V}(z,t)=V_{1\mathrm{D}}\left(z\right)+V_z\left(t\right)e^{-{\displaystyle \frac{{(z-z_0)}^2}{2{\sigma }_z^2}}},$$

(76)

where $V_{1\mathrm{D}}\left(z\right)$ is given in (70), $V_z\left(t\right)=V_0$ for $t<0.02$ s is the height of the Gaussian barrier, ${\sigma }_z$ determines its width, and $z_0$ is a spatial offset representing the center of the barrier. This potential allows us to interpolate between a localized Gaussian barrier and a uniform constant potential as ${\sigma }_z\to \infty$, leading to a flat barrier of height $V_0$.

Our numerical simulations in the $1D$ configuration with ${\displaystyle \frac{U_z}{U_{\rho }}}=0.04$ in Equation (1), reveal that, unlike the two-dimensional case, the solitons are perfectly stable and exhibit remarkably regular dynamics on long time scales. Surprisingly, we still observe a clear distinction between the behavior induced by Gaussian barriers (76), with $V_0=400$, ${\sigma }_z=0.6$ and $z_0=0$, and the constant barriers. When the Gaussian barrier is not removed (see panel (a) of Figure 3), no soliton is generated, and the BEC remains largely undisturbed. In contrast, when a constant barrier is used, solitons are generated regardless of whether the barrier is removed (panel (d) of Figure 3) or left in place (panel (b) of Figure 3). Now, in the $1D$ reduction, we observe that Equation (71) introduces a constant barrier $\hslash {\omega }_{\perp }$, which is factored out using the probability density invariant transformation (73). This observation shows that, as in the $2D$ case, the generation of solitons is independent of the addition or removal of the constant barrier potential. Furthermore, we have shown that the density probability functions are left invariant under an arbitrary time translation. These reasons explain the similar dynamic behavior observed in panels (b) and (d) of Figure 3. The resulting solitons display identical and stable propagation in both cases, indicating that the constant barrier efficiently seeds the formation of robust soliton states. Finally, when the Gaussian barrier is removed after the holding time $t=0.02$ s (panel (c) of Figure 3), solitons are generated and propagated globally in a stable manner, exhibiting characteristic dynamics similar to those observed with the constant barrier. These results confirm that the formation of solitons in $1D$ is not only robust but also highly sensitive to the nature and temporal evolution of the barrier potential.

5.2. Number of Solitons Versus the Height of the Barrier

Our primary focus is to investigate, through numerical simulations, the influence of the constant barrier height $V_0$ on the soliton generation process and their subsequent dynamics and interactions. The barrier is held in place until the end of the dynamics.

Systematic numerical simulations reveal that the number of generated solitons depends nonlinearly on the barrier height $V_0$ which has the energy dimension of $\hslash {\omega }_{\perp }$. For small values of $V_0$, few or no solitons are created, while increasing the barrier height progressively leads to the production of multiple solitons. Interestingly, this process exhibits a plateau structure, wherein certain ranges of barrier heights correspond to a fixed number of solitons. This behavior reflects the interplay between the potential energy introduced by the barrier and the inherent nonlinearity of the condensate.

The numerical results are summarized in Figure 4, which shows the number of solitons N as a function of the barrier height $V_0$. The figure reveals a series of plateau regions, indicating that for specific intervals of $V_0$, the number of generated solitons remains constant before increasing in discrete steps. Notably, we observe that the maximum number of solitons produced is equal to 14, attained for large barrier heights $V_0\gtrsim {10}^6$. This saturation suggests a limit to the number of stable solitons that can be induced within the harmonic trap configuration, likely dictated by the available energy and the condensate’s healing length. Furthermore, we observe that only even numbers of solitons are generated. This can be explained by the point symmetry $z⟶-z$ of Equation (74) which suggests a reflection symmetry along the z-axis.

A similar analysis was made for the Gaussian barrier in [24]. The authors showed that the number of solitons varies in a continuous way, rather than in discrete plateaus, as the barrier width ${\sigma }_z$ in (76) increases. Authors observed that as ${\sigma }_z$ increases, the Gaussian barrier asymptotically approaches a constant potential, leading to a saturated soliton count analogous to our constant barrier results. Our work explicitly quantifies these saturated states by taking the limit as ${\sigma }_z⟶+\infty$ in Equation (76) in order to obtain a constant barrier potential. These saturated states are thus quantified in Figure 4.

6. Symmetry and Asymmetry

In this section, we explore the role of the spatial shifting of the barrier off-center. We break the symmetry of the system, leading to asymmetric soliton distributions. One should recall that in the $2D$ case, soliton dynamics exhibit inherent instabilities as shown in Figure 2 that disrupt coherent symmetry/asymmetry in their motion, making their behavior highly sensitive to perturbation and numerically/experimentally transient. For clarity, we focus on the $1D$ case to emphasize the role of dimensionality in soliton stability and invariance. In Figure 5, we show, in panel (b), the dynamics of the solitons by centering the constant barrier and fixing it at $V_0=1000$. We notice the generation of 10 solitons (see Figure 4) that are perfectly symmetric and colliding periodically in the center of the BEC. This period, as depicted in Equation (73), is related to the width of the trap and can be explicitly calculated to be ${\displaystyle \frac{2\pi }{{\omega }_{\perp }}}$. In panels (a) and (c), we break the spatial symmetry by displacing the constant barrier, respectively, on the left side of the BEC at $z_0=-8.27$ μm or on the right side at $z_0=+8.27$ μm. As one could notice, the solitons in panel (a) start by moving to the right side and collide once at around $t_1=0.1$ s, and then they move to the left side and collide at around $t_2=0.2$ s. For panel (c), the dynamics is perfectly asymmetric to panel (a). This highlights the sensitivity of robust soliton generation to both the barrier height and its spatial positioning, emphasizing the richness of soliton dynamics in confined Bose–Einstein condensates.

The ability to control the number, the symmetry, and the dynamics of solitons in a confined 1D BEC offers several advantages. First, this tunability allows for the exploration of controlled soliton collisions, which are of fundamental interest in the study of nonlinear integrable systems and non-equilibrium quantum dynamics. Second, solitons in BECs can serve as matter-wave interferometers, with applications in precision measurements and atom interferometry. The periodic soliton oscillations observed in the symmetric configuration could provide a platform for implementing interferometric schemes based on solitons collisions. Moreover, the sensitivity of the solitons’ trajectories to the barrier position could be exploited for sensing purposes, as small perturbations to the potential or external forces would lead to measurable deviations in the solitons’ paths. Finally, barrier engineering opens pathways toward the preparation of tailored multi-soliton states, which could serve as a basis for studying soliton gases, quantum turbulence, and other complex nonlinear excitations in ultra-cold quantum gases.

7. Conclusions

In this work, we have investigated theoretically and numerically the generation of solitons in 1D and 2D BECs. The theoretical approach in this paper has exhibit exact expressions for the N-soliton solutions in 2D BECs using the Hirota bilinear method. We have shown the dependence of these solitons on the parameter $\alpha$ defined in Equation (22) confirming the fundamental balance between nonlinearity and dispersion in the existence of such waves. Furthermore, dimensional reduction has demonstrated time translation invariance of solutions in 1D and 2D BECs, exhibiting periodicity in soliton collisions. We have explored the formation and dynamics of dark solitons in BECs confined in a harmonic trap, emphasizing the role of external barrier potentials in controlling multi-soliton states. In contrast to the two-dimensional case, where soliton solutions are prone to snake instability and rapid decay, we confirmed that in 1D, solitons are stable over long time periods and exhibit robust oscillatory dynamics within the trap. We demonstrated, as well, that the number of generated solitons depends on both the height and the shape of the barrier potential. A plateau structure emerges as the barrier height increases, indicating that the number of solitons remains constant over specific intervals of potential heights before increasing in discrete steps. This plateau behavior reflects the quantized nature of soliton generation in confined condensates.

Additionally, we observed a fundamental distinction between Gaussian and constant barriers. While Gaussian barriers require removal to trigger soliton formation, constant barriers lead to soliton generation regardless of whether they are removed or left in place. The resulting solitons display identical dynamics in both cases. We further investigated the effect of spatial displacement of the barrier from the trap center, breaking the spatial symmetry of the system. This displacement resulted in asymmetric soliton distributions and modified collision patterns, demonstrating the high sensitivity of soliton trajectories to the initial potential setup. The ability to engineer multi-soliton states through barrier shaping and spatial displacement opens up several promising applications. The controlled oscillations and periodic collisions of solitons offer a potential platform for matter-wave interferometry, while the sensitivity of soliton trajectories to external perturbations suggests applications in precision sensing.