1. Introduction
The phenomenon of a marked increase in wave-like behavior occurs in atomic gases as temperatures decrease, leading to special physical properties such as superfluidity [1–3], superconductivity [4, 5], and quantum entanglement [6–8], is referred to as Bose–Einstein condensates (BECs). With the observation of spinor BECs in the gas of spin-1 Na23 atoms [9], interests in related theories are sparked, which drive the exploration of various new phenomena, including spin configurations [10, 11], quantum magnetization [12, 13], and the second-order Zeeman effect [14–16]. In recent years, methods involving time-dependent nonlinearity manipulated by Feshbach-resonance techniques have sparked research interest [17–20]. For instance, Feshbach-resonance is utilized to control scattering lengths, and numerical methods are employed to establish the complete stability of solitons [17]. Simultaneously, physical techniques enable control over the collapse and expansion in BECs [18]. Furthermore, methods for modulation of BEC systems involving external potentials with spatiotemporal dependencies have also been investigated [21–23]. Specifically, numerous studies focus on solitons in BECs or spinor BECs, with a primary focus on scenarios that only involve real external potential while a fraction further introduce gain-and-loss distribution manifested as imaginary component [24–26]. Moreover, there has been relatively scant research on modulating stable solitons in spinor BECs based on the concept of ${ \mathcal P }{ \mathcal T }$-symmetry [27–30], i.e. external potential and gain-and-loss distribution must satisfy the parity conditions. ${ \mathcal P }{ \mathcal T }$-symmetry was first proposed by Bender et al in 1998, and it has found significant applications in various fields including optics [31–34], matter waves [35, 36], magnetics [37, 38], acoustics [39, 40], and electronics [41, 42]. In dissipative systems, the ${ \mathcal P }{ \mathcal T }$-symmetry of complex potentials is a necessary condition for the existence of soliton families [43]. Despite the presence of gain and loss, the linear spectrum of the system can still be entirely real [44], allowing all linear modes to exhibit regular wave behavior as in conservative systems. Various ${ \mathcal P }{ \mathcal T }$-symmetric potentials have been applied to BEC research since Klaiman et al proposed considering BECs in double-well potentials in 2008 [45], leading to an increasing number of studies on solitons in BECs [46–48].
The spinor BECs has 2F + 1 components. Governed by the variational principle, the evolution of the wave function in quasi-one-dimensional spin-1 BECs can be expressed using the three-coupled nonlinear Gross–Pitaevskii equations (GPEs) in the following form [49–53]:
$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}{\hslash }{\partial }_{\tau }{{\rm{\Psi }}}_{+} & = & -\displaystyle \frac{{{\hslash }}^{2}}{2m}{\partial }_{\zeta }^{2}{{\rm{\Psi }}}_{+}+\hat{{V}_{+}}(\zeta ){{\rm{\Psi }}}_{+}+({b}_{0}+{b}_{2})(| {{\rm{\Psi }}}_{+}{| }^{2}\\ & & +| {{\rm{\Psi }}}_{0}{| }^{2}){{\rm{\Psi }}}_{+}+({b}_{0}-{b}_{2})| {{\rm{\Psi }}}_{-}{| }^{2}{{\rm{\Psi }}}_{+}+{b}_{2}{{\rm{\Psi }}}_{0}^{2}{{\rm{\Psi }}}_{-}^{* },\\ {\rm{i}}{\hslash }{\partial }_{\tau }{{\rm{\Psi }}}_{0} & = & -\displaystyle \frac{{{\hslash }}^{2}}{2m}{\partial }_{\zeta }^{2}{{\rm{\Psi }}}_{0}+\hat{{V}_{0}}(\zeta ){{\rm{\Psi }}}_{0}+({b}_{0}+{b}_{2})(| {{\rm{\Psi }}}_{+}{| }^{2}\\ & & +| {{\rm{\Psi }}}_{-}{| }^{2}){{\rm{\Psi }}}_{0}+{b}_{0}| {{\rm{\Psi }}}_{0}{| }^{2}{{\rm{\Psi }}}_{0}\,+\,2{b}_{2}{{\rm{\Psi }}}_{0}^{* }{{\rm{\Psi }}}_{+}{{\rm{\Psi }}}_{-},\\ {\rm{i}}{\hslash }{\partial }_{\tau }{{\rm{\Psi }}}_{-} & = & -\displaystyle \frac{{{\hslash }}^{2}}{2m}{\partial }_{\zeta }^{2}{{\rm{\Psi }}}_{-}+\hat{{V}_{-}}(\zeta ){{\rm{\Psi }}}_{-}+({b}_{0}+{b}_{2})(| {{\rm{\Psi }}}_{0}{| }^{2}\\ & & +| {{\rm{\Psi }}}_{-}{| }^{2}){{\rm{\Psi }}}_{-}+({b}_{0}-{b}_{2})| {{\rm{\Psi }}}_{+}{| }^{2}{{\rm{\Psi }}}_{-}+{b}_{2}{{\rm{\Psi }}}_{0}^{2}{{\rm{\Psi }}}_{+}^{* },\end{array}\end{eqnarray}$
where ${\boldsymbol{\Psi }}={\left({{\rm{\Psi }}}_{+},{{\rm{\Psi }}}_{0},{{\rm{\Psi }}}_{-}\right)}^{{\rm{T}}}$, $\Psi$j(ζ, τ) (j = + , 0, −) is the wave function of ζ, τ which describe the transverse coordinate and propagation time respectively. And the asterisk stands for the complex conjugate, ℏ is the reduced Planck constant, m is the atomic mass and $\hat{{V}_{j}}(\zeta )$ (j = + , 0, −) stands for the real external trap potential. Here, the mean-field coupling constants b0 and b2, which denote the interactions among the three components $\Psi$j(ζ, τ) (j = + , 0, −) responsible for spin-independent and spin-dependent collisions between identical spin-1 bosons, taking the form b0 = πℏ2(a0 + 2a2)/3m and b2 = πℏ2(a2 − a0)/3m where a0 and a2 represent the s-wave scattering lengths within the symmetric channels characterized by the total spin of the colliding atoms, F = 0 and F = 2, respectively [54–56].In this paper, considering the transformation $({{\rm{\Psi }}}_{+},{{\rm{\Psi }}}_{0},{{\rm{\Psi }}}_{-})\,\to ({\psi }_{+},\sqrt{2}{\psi }_{0},{\psi }_{-})$, $x=\zeta \sqrt{2{\rm{m}}}/{\hslash }$, t = τ/ℏ, the dimensionless three-component GPEs are formed as:3 ) transforms into the classical Scarf-II potential [65–67]. As the case of c0 = c2 < 0 has been investigated in the references [68, 69], yielding analytical solutions of equations (1 ) under ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential and ${ \mathcal P }{ \mathcal T }$-symmetric harmonic-Hermitian-Gaussian potential, along with results on different types of soliton collisions, this paper primarily focuses on the scenarios where c2 = 0.1 (antiferromagnetic) and c2 = − 0.5 (ferromagnetic) under ${ \mathcal P }{ \mathcal T }$-symmetric generalized Scarf-II potentials.
$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}{\partial }_{t}{\psi }_{+} & = & -{\partial }_{x}^{2}{\psi }_{+}+{U}_{+}(x){\psi }_{+}+({c}_{0}+{c}_{2})(| {\psi }_{+}{| }^{2}+2| {\psi }_{0}{| }^{2}){\psi }_{+}\\ & & +\,({c}_{0}-{c}_{2})| {\psi }_{-}{| }^{2}{\psi }_{+}+2{c}_{2}{\psi }_{0}^{2}{\psi }_{-}^{* },\\ {\rm{i}}{\partial }_{t}{\psi }_{0} & = & -{\partial }_{x}^{2}{\psi }_{0}+{U}_{0}(x){\psi }_{0}+({c}_{0}+{c}_{2})(| {\psi }_{+}{| }^{2}\\ & & +\,| {\psi }_{-}{| }^{2}){\psi }_{0}\,+\,2{c}_{0}| {\psi }_{0}{| }^{2}{\psi }_{0}\,+\,2{c}_{2}{\psi }_{0}^{* }{\psi }_{+}{\psi }_{-},\\ {\rm{i}}{\partial }_{t}{\psi }_{-} & = & -{\partial }_{x}^{2}{\psi }_{-}+{U}_{-}(x){\psi }_{-}+({c}_{0}+{c}_{2})(| {\psi }_{-}{| }^{2}+2| {\psi }_{0}{| }^{2}){\psi }_{-}\\ & & +\,({c}_{0}-{c}_{2})| {\psi }_{+}{| }^{2}{\psi }_{-}+2{c}_{2}{\psi }_{0}^{2}{\psi }_{+}^{* },\end{array}\end{eqnarray}$
where Uj(x) = Vj(x) + iWj(x) (j = + , 0, −) is the ${ \mathcal P }{ \mathcal T }$-symmetric generalized Scarf-II potential [57, 58] which can modulate the degree of the energy distribution of matter-wave solitons presented under the external potential well and Wj(x) (j = + , 0, −) represents the gain-and-loss distribution. Since the introduction of the concept of ${ \mathcal P }{ \mathcal T }$-symmetry, achieving a balance between gain and loss in quantum systems has been a long-standing goal in quantum mechanics, and it has only recently been realized experimentally [59]. Such ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potentials in a spinor BEC can possibly be realized by creating arbitrary optical potentials in time average [60] and also the experimental realization of localized loss [61–63] and of localized gain [64]. The concrete forms of the ${ \mathcal P }{ \mathcal T }$-symmetric generalized Scarf-II potential are shown as: $\begin{eqnarray}\begin{array}{rcl}{V}_{j}(x) & = & {v}_{j}{{\rm{sech}} }^{2}({x})+{\alpha }_{{j}}{{\rm{sech}} }^{2n}({x}),\\ {W}_{{j}}({x}) & = & {w}_{{j}}{\rm{sech}} ({x})\tanh ({x}),\quad {j}=+,\,0,-,\end{array}\end{eqnarray}$
with vj, αj, wj (j = + , 0, −), and n > 0 being all real-valued potential parameters, which can modulate the structure of ${ \mathcal P }{ \mathcal T }$-symmetric potential. Clearly, when αj = 0 or n = 1 (vj + αj ≠ 0, j = + , 0, −), the complex potential (The subsequent sections of this paper are organized as follows. In section 2 , we explore the soliton states of equations (2 ), obtaining its families of analytical soliton states under ${ \mathcal P }{ \mathcal T }$-symmetric generalized Scarf-II potentials. We conduct linear stability analysis, distinguishing stable and unstable regions. The results are validated through dynamic evolution with 5% perturbations. In section 3 , we utilize the power-conserving squared-operator methods [44] to generate families of numerical soliton states for equations (2 ). A detailed investigation of points on the curves is carried out, obtaining the evolution of corresponding stable and unstable numerical solitons. Some analytical soliton states, numerical single-hump soliton states, and numerical double-hump soliton states are chosen to interact with exotic Gaussian solitons. Finally, our conclusions are presented in section 4 .
2. Analytical soliton states
With the stationary states transformation ${\psi }_{j}(x,t)={\phi }_{j}(x){{\rm{e}}}^{{\rm{i}}{\mu }_{j}t}$ (j = + , 0, −), the three-component GPEs can be derived as4 ), we utilize the constraints w+ + w− = 2w0, μ+ = μ0 = μ− = n2, ${A}_{0}^{2}={A}_{+}{A}_{-}$, α+ = α0 = α−,
$\begin{eqnarray}\begin{array}{l}-{\mu }_{+}{\phi }_{+}+{\phi }_{+}^{{\prime\prime} }-[{V}_{+}(x)+{\rm{i}}{W}_{+}(x)]{\phi }_{+}-({c}_{0}+{c}_{2})(| {\phi }_{+}{| }^{2}\\ \quad +2| {\phi }_{0}{| }^{2}){\phi }_{+}-({c}_{0}-{c}_{2})| {\phi }_{-}{| }^{2}{\phi }_{+}-2{c}_{2}{\phi }_{0}^{2}{\phi }_{-}^{* }=0,\\ \quad -{\mu }_{0}{\phi }_{0}+{\phi }_{0}^{{\prime\prime} }-[{V}_{0}(x)+{\rm{i}}{W}_{0}(x)]{\phi }_{0}-({c}_{0}+{c}_{2})(| {\phi }_{+}{| }^{2}\\ \quad +| {\phi }_{-}{| }^{2}){\phi }_{0}-2{c}_{0}| {\phi }_{0}{| }^{2}{\phi }_{0}-2{c}_{2}{\phi }_{0}^{* }{\phi }_{+}{\phi }_{-}=0,\\ \quad -{\mu }_{-}{\phi }_{-}+{\phi }_{-}^{{\prime\prime} }-[{V}_{-}(x)+{\rm{i}}{W}_{-}(x)]{\phi }_{-}-({c}_{0}+{c}_{2})(| {\phi }_{-}{| }^{2}\\ \quad +2| {\phi }_{0}{| }^{2}){\phi }_{-}-({c}_{0}-{c}_{2})| {\phi }_{+}{| }^{2}{\phi }_{-}-2{c}_{2}{\phi }_{0}^{2}{\phi }_{+}^{* }=0,\end{array}\end{eqnarray}$
where μj (j = + , 0, −) represents the real chemical potential. For obtaining the soliton states of equations ( $\begin{eqnarray*}{v}_{j}=-{\left(\displaystyle \frac{{w}_{j}}{2n+1}\right)}^{2}-{n}^{2}-n,\qquad j=+,\,0,-,\end{eqnarray*}$
and derive $\begin{eqnarray}\begin{array}{l}\,{\phi }_{j}(x)={A}_{j}{{\rm{sech}} }^{n}({x})\ {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(x)},\\ \,{\theta }_{j}(x)=-\displaystyle \frac{2{w}_{j}}{2n+1}\arctan \left(\tanh \displaystyle \frac{x}{2}\right),\qquad {j}=+,0,-,\\ \,{A}_{+}+{A}_{-}=\sqrt{\displaystyle \frac{{\alpha }_{+}}{-({c}_{0}+{c}_{2})}}.\end{array}\end{eqnarray}$
To demonstrate the stability of soliton states, we perform the linear stability analysis on the obtained soliton solutions $\Psi$j(x, t) (j = + , 0, −) of Equation (2 ). Substituting the perturbed solutions (ε ≪ 1)2 ) yields an eigenvalue problem LY = δY, where δ stands for the eigenvalue, fj(x), gj(x) (j = + , 0, −) are components of the corresponding eigenfunction, ${\boldsymbol{Y}}\,={\left({f}_{1},{g}_{1},{f}_{2},{g}_{2},{f}_{3},{g}_{3}\right)}^{{\rm{T}}}$, and the specific form of L is shown in appendix.
$\begin{eqnarray}\begin{array}{rcl}{\psi }_{j}(x,t) & = & {\phi }_{j}(x){{\rm{e}}}^{{\rm{i}}{\mu }_{j}t}+\,\epsilon \left[{f}_{j}(x){{\rm{e}}}^{{\rm{i}}\delta t}\right.\\ & & \left.+{g}_{j}^{* }(x){{\rm{e}}}^{-{\rm{i}}{\delta }^{* }t}\right]{{\rm{e}}}^{{\rm{i}}{\mu }_{j}t},\quad j=+,0,-,\end{array}\end{eqnarray}$
into equations (For the given eigenvalue problem, we discretize the differential operator into a matrix using spectral methods [70, 71] and obtain the numerical values for δ. Considering the inherent errors in numerical methods, when the maximum of $| \mathrm{Im}(\delta )| $ is sufficiently small ($| \mathrm{Im}(\delta )| \lt {10}^{-5}$), δ can be approximated as a real number. In this case, the solutions $\Psi$j(x, t) (j = + , 0, −) can be considered linearly stable. This theoretical assumption is further validated through numerical evolution with 5% perturbations.
As a result, when we choose the system parameters as c0 = 1, c2 = − 0.5, the curve of analytical soliton family for n concerning the atomic densities of soliton can be obtained, as illustrated in figure 1(a1). The atomic densities are defined as Pj = ∫∣$\Psi$j(x)∣2 dx (j = + , 0, −), and the energy flux can be described by introducing the Poynting vector ${S}_{j}\,={\rm{i}}({\phi }_{j}{\phi }_{j,x}^{* }-{\phi }_{j}^{* }{\phi }_{j,x})/2$ (j = + , 0, −) [72]. Due to constraints on the system parameters for analytical soliton states, only scenarios under the special condition where P+ = P0 = P− = P are studied in this paper. Through linear stability analysis, we distinguish the stable and unstable portions of the curve. The unstable regions are indicated by the red dashed lines, while the stable regions are represented by the blue solid lines. We select one point from each of the stable and unstable sections on the curve for further investigation. Point A in figure 1(a1) is selected from the unstable region, corresponding to P = 1.6000 and n = 0.5. In figure 1(b1)–(b3), we present its evolutionary results after undergoing 5% perturbations as initial conditions and the corresponding eigenvalue distribution is reflected in figure 1(b4). Point B in figure 1(a1) is chosen from the stable region, with P = 0.8000 and n = 1.5. Similarly, in figure 1(c1)-(c4), we illustrate its evolutionary results with 5% perturbations, along with the corresponding eigenvalue distribution. Clearly, the soliton states corresponding to point A is unstable, while that of point B is stable, confirming the accuracy of linear stable analysis. Moreover, we observe that as the value of n increases from 0.5 to 1.5, the atomic density decreases by just half, with αj remaining constant and ∣vj∣ (j = + , 0, −) increasing. This suggests that higher values of ∣vj∣ (j = + , 0, −) may correspond to more stable soliton states.
Figure 1. Families of analytical soliton states, the progression of stable/unstable analytical solitons and their eigenvalues for linear stability, associated with c2 = − 0.5 and c2 = 0.1 respectively. The parameters are chosen as: c0 = 1, wj = 0.6, Aj = 0.5 (j = + , 0, −). (b1)-(b4) correspond to point A in (a1) where vj = − 0.8400, αj = − 0.5 (j = + , 0, −) and P = 1.6000, n = 0.5. (c1)-(c4) correspond to point B in (a1) where vj = − 3.7725, αj = − 0.5 and P = 0.8000, n = 1.5. (d1)-(d4) correspond to point C in (a2) where vj = − 0.8400, αj = − 1.1 (j = + , 0, −) and P = 1.6000, n = 0.5. |
When the value of c2 is changed from −0.5 to 0.1, the curve of the obtained analytical solution family exhibit the same shape and position, but the stable region is significantly enlarged, as shown in figure 1(a2). From the curve in figure 1(a2), we select point C with the same coordinates as point A for further investigation. In figure 1(d1)-(d4), we present its evolutionary results after undergoing 5% perturbations as initial conditions and the corresponding eigenvalue distribution. However, the evolution corresponding to point C is stable, indicating that increasing c2 from −0.5 to 0.1 can decrease the value of αj, and smaller αj (j = + , 0, −) values may correspond to stable solitons.
3. Numerical soliton states
Obtaining the analytical solution of equations (2 ) demands stringent system parameter specifications, rendering the implementation of BECs experiments challenging. Consequently, to surmount these limitations and manipulate the propagation coefficient, numerical methods are employed in our research endeavors.
In our numerical investigation, we first focus on the case when c0 = 1 and c2 = − 0.5. With the system parameters chosen as wj = 2.5, vj = − 3.8, and αj = − 0.4 (j = + , 0, −), we present families of single-hump numerical soliton states in figure 2(a1)-(a3). Clearly, as n grows, the entire curve relocates towards smaller μ values. We further investigate points D and E selected from the curves shown in figure 2(a1)-(a3). Point D corresponds to P+ = 3.6600, P0 = 4.6321, P− =5.3079, μ+ = 1.8563, μ0 = 1.8555, μ− = 1.8571; point E corresponds to P+ = 4.0000, P0 = 5.0624, P− = 4.5376, μ+ = 1.8724, μ0 = 1.7685, μ− = 2.0890. By performing linear stability analysis on the soliton states corresponding to these three points, we ascertain that the soliton states corresponding to point D is stable, and the stability of the soliton states corresponding to point E is unstable. Similarly, we validate these findings by conducting numerical evolution with 5% perturbations.
Figure 2. Families of numerical single-hump soliton states, the progression of stable/unstable numerical single-hump solitons and their eigenvalues for linear stability, associated with different n. The parameters are chosen as: c0 = 1, c2 = − 0.5, w+ = w0 = w− = 2.5, v+ = v0 = v− = − 3.8, α+ = α0 = α− = − 0.4. |
Next, we investigate the numerical families of double-hump soliton states when c0 = 1 and c2 = 0.1. Given the constancy of other conditions, the numerical double-hump solution families across various n values exhibit significant overlap, hence in figure 3, we only show the scenario for n = 2. We change the values of wj from 2.5 to 0.8, vj from −3.8 to −6, while keeping the values of αj (j = + , 0, −) unchanged. Similarly, we further investigate points F and G selected from the curves shown in figure 3(a1)-(a3). Figure 3(b1)-(b4) and figure 3(d1)-(d4) depict the progression of stable soliton states respectively, along with their associated eigenvalue distributions. Of note, as shown in figure 3(b1)-(b3), the double-hump soliton states corresponding to point F exhibit periodic oscillations while conserving the energy during propagation, which can be seen as a form of energy exchange between soliton states. Furthermore, compared to previous studies where all three soliton states exhibited periodic oscillations, the phenomenon of energy exchange only existing between $\Psi$+ and $\Psi$0, with $\Psi$− remaining constant, might be relatively infrequent. To further investigate the oscillatory behavior of the double-hump soliton states, the total number energy flux from the numerical solutions is investigated. Point G from figure 3(a1)-(a3) is selected for study, compared to point F, the corresponding oscillation period is longer, and $\Psi$− also begins to exhibit slight oscillations, as shown in figure 3(c1)-(e4). This suggests that the oscillation of the double-hump soliton states may be related to the magnitude of the energy and indicates the dynamic stability of the double-hump soliton against small perturbations.
Figure 3. Families of numerical double-hump soliton states, the progression of stable numerical double-hump solitons and their energy flux associated with n = 2. The parameters are chosen as: c0 = 1, c2 = 0.1, w+ = w0 = w− = 0.8, v+ = v0 = v− = − 6, α+ = α0 = α− = − 0.4. And their maximum of $| \mathrm{Im}(\delta )| $ for linear stability are all 10−7. |
Next, we turn our attention to studying the interaction between Gaussian solitons and the solitons depicted in figure 1 to figure 3. We consider the following initial conditions7 ); figure 4(b1)-(b3) correspond to the case of numerical single-hump solitons (point D) with P+ = 3.6600, P0 = 4.6321, P− = 5.3079, μ+ = 1.8563, μ0 = 1.8555, μ− = 1.8571, μ0 = 1.7329, μ− = 1.7262, and n = 0.5, satisfying the initial condition (7 ); figure 4(c1)-(c3) correspond to the case of numerical double-hump solitons (point F) with P+ = 0.0400, P0 = 0.0506, P− = 13.5547, μ+ = 0.7273, μ0 = 0.6493, μ− = 0.6556, and n = 2, satisfying the initial condition (7 ); figure 4(d1)-(d3) correspond to the case of numerical double-hump solitons (point G) with P+ = 0.3200, P0 = 0.4050, P− = 12.8750, μ+ = 0.7083, μ0 = 0.6319, μ− = 0.6497, and n = 2, satisfying the initial condition (8 ). For figure 4(a1)-(a3), we choose q+ =q0 = q− = 0.2, for figure 4(b1)-(b3), we choose q+ = q0 = 0.05, q− = 0.3, while for figure 4(d1)-(d3), we choose q+ = q0 = 0.1, q− = 0.5. We notice that after the interaction between the stable soliton states and the exotic Gaussian solitons, both of them maintain their original shapes and continue to propagate forward, as shown in figure 4(a1)-(c3). This indicates that an elastic collision occurs between the soliton states and the exotic Gaussian solitons. Moreover, some studies suggest that interactions between solitons may trigger energy flow among the three components. However as shown in figure 4(d1)-(d2), we observe a specific scenario where the energy of $\Psi$+ decreases, and the energy of $\Psi$0 increases, while the energy of $\Psi$− remains unchanged after the interaction between the exotic Gaussian solitons and the double-hump solitons. This can be regarded as a special form of energy flow involving only two components. In order to explore more properties about the interaction, the corresponding energy flux are checked. As illustrated in figure 5, the Poynting vectors of $\Psi$j(x, t) all keep invariant. Moreover, the energy flux of the soliton states has the same characteristics as that $\Psi$j(x, t) propagate, which further verifies the correctness of figure 4.
$\begin{eqnarray}\begin{array}{rcl}\overline{{\psi }_{j}}(x,0) & = & {\psi }_{j}(x,0)+{q}_{j}{{\rm{e}}}^{-0.05{\left(x+23\right)}^{2}+3{\rm{i}}x}\\ & & +\,{q}_{j}{{\rm{e}}}^{-0.05{\left(x-23\right)}^{2}-3{\rm{i}}x},\quad j=+,0,-,\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}\overline{{\psi }_{j}}(x,0)={\psi }_{j}(x,0)+{q}_{j}{{\rm{e}}}^{-0.05{\left(x+23\right)}^{2}+4{\rm{i}}x}\\ \quad +{q}_{j}{{\rm{e}}}^{-0.05{\left(x-23\right)}^{2}-4{\rm{i}}x},\\ \quad j=+,0,-,\end{array}\end{eqnarray}$
where qj is the maximum amplitude of the exotic Gaussian soliton. Figure 4(a1)-(a3) correspond to the case of analytical solitons with c0 = 1, c2 = − 0.5, n = 0.5, wj = 0.6, Aj = 0.5, P = 1.6000, vj = − 3.7725, and αj = − 0.5 (j = + , 0, −), satisfying the initial condition (
Figure 4. Interactions of single-hump and double-hump soliton states with exotic Gaussian solitons. (a1)-(a3) correspond to analytical soliton states at c0 = 1, c2 = − 0.5, n = 0.5, wj = 0.6, Aj = 0.5, P = 1.6000, vj = − 3.7725 and αj = − 0.5 (j = + , 0, −). (b1)-(b3) correspond to numerical single-hump soliton states at n = 0.5. (c1)-(d3) correspond to numerical double-hump soliton states at n = 2. |
4. Conclusions
In conclusion, we introduce ${ \mathcal P }{ \mathcal T }$-symmetric generalized Scarf-II potentials into three-coupled Gross–Pitaevskii equations with two different scenarios of inter-component interaction parameter values c0 and c2, and obtain corresponding analytical/numerical soliton states and soliton families. In both scenarios of parameter configurations, the analytical soliton families exhibit identical curve shapes, but the stable region of the analytical soliton family curves significantly increases under the condition of c2 = 0.1. With the condition of c2 = − 0.5 (ferromagnetic), the single-hump solution families obtained through iterative algorithms shift towards smaller values of μ as n increases. Besides, in the case of c2 = 0.1 (antiferromagnetic), we obtain point F whose corresponding double-hump soliton states exhibit periodic oscillations while conserving the energy during propagation, within the double-hump soliton families. Notably, the phenomenon of energy exchange only existing between $\Psi$+ and $\Psi$0, with $\Psi$− remaining constant, is observed. Moreover, after the interaction between the double-hump soliton states and the exotic Gaussian solitons, the energy of $\Psi$+ decreases and the energy of $\Psi$0 increases, while the energy of $\Psi$− unchanged. This phenomenon can be seen as collisions triggering the flow of energy among the three components. These findings could contribute to a deeper understanding of solitons in Bose–Einstein condensates with ${ \mathcal P }{ \mathcal T }$-symmetric potentials and potentially aid in designing relevant physics experiments.
Acknowledgments
We express our sincere thanks to the editor, referees, and all the members of our discussion group for their valuable comments. This work was supported by NSFC under Grant No. 12272403, and Beijing Training Program of Innovation under Grant No. S202410019024. The funding body plays an important role in the design of the study, in analysis, calculation, and in writing of the manuscript.
Declarations of interest
The authors declare that they have no conflict of interest.
Appendix
The matrix L of the linear eigenvalue problem, which is derived during linear stability analysis, has a specific form as follows:
$\begin{eqnarray}{\boldsymbol{L}}=\left(\begin{array}{cccccc}{L}_{1}\qquad & {L}_{12}\qquad & {L}_{13}\qquad & {L}_{14}\qquad & {L}_{15}\qquad & {L}_{16}\\ {L}_{21}\qquad & -{L}_{1}^{* }\qquad & {L}_{23}\qquad & {L}_{24}\qquad & {L}_{25}\qquad & {L}_{26}\\ {L}_{31}\qquad & {L}_{32}\qquad & {L}_{2}\qquad & {L}_{34}\qquad & {L}_{35}\qquad & {L}_{36}\\ {L}_{41}\qquad & {L}_{42}\qquad & {L}_{43}\qquad & -{L}_{2}^{* }\qquad & {L}_{45}\qquad & {L}_{46}\\ {L}_{51}\qquad & {L}_{52}\qquad & {L}_{53}\qquad & {L}_{54}\qquad & {L}_{3}\qquad & {L}_{56}\\ {L}_{61}\qquad & {L}_{62}\qquad & {L}_{63}\qquad & {L}_{64}\qquad & {L}_{65}\qquad & -{L}_{3}^{* }\end{array}\right),\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{L}_{1} & = & {\partial }_{x}^{2}-2({c}_{0}+{c}_{2}){\phi }_{+}{\phi }_{+}^{* }-2({c}_{0}+{c}_{2}){\phi }_{0}{\phi }_{0}^{* }\\ & & -({c}_{0}-{c}_{2}){\phi }_{-}{\phi }_{-}^{* }-{U}_{+}(x)-{\mu }_{+},\\ {L}_{2} & = & {\partial }_{x}^{2}-({c}_{0}+{c}_{2}){\phi }_{+}{\phi }_{+}^{* }-4{c}_{0}{\phi }_{0}{\phi }_{0}^{* }\\ & & -({c}_{0}+{c}_{2}){\phi }_{-}{\phi }_{-}^{* }-{U}_{0}(x)-{\mu }_{0},\\ {L}_{3} & = & {\partial }_{x}^{2}-({c}_{0}-{c}_{2}){\phi }_{+}{\phi }_{+}^{* }-2({c}_{0}+{c}_{2}){\phi }_{0}{\phi }_{0}^{* }\\ & & -2({c}_{0}+{c}_{2}){\phi }_{-}{\phi }_{-}^{* }-{U}_{-}(x)-{\mu }_{-},\\ {L}_{12} & = & -({c}_{0}+{c}_{2}){\phi }_{+}^{2},\\ {L}_{14} & = & -2({c}_{0}+{c}_{2}){\phi }_{+}{\phi }_{0},\\ {L}_{15} & = & -({c}_{0}-{c}_{2}){\phi }_{+}{\phi }_{-}^{* },\\ {L}_{21} & = & ({c}_{0}+{c}_{2}){\phi }_{+}^{* 2},\\ {L}_{23} & = & 2({c}_{0}+{c}_{2}){\phi }_{+}^{* }{\phi }_{0}^{* },\\ {L}_{26} & = & ({c}_{0}-{c}_{2}){\phi }_{+}^{* }{\phi }_{-},\\ {L}_{32} & = & -({c}_{0}+{c}_{2}){\phi }_{+}{\phi }_{0},\\ {L}_{36} & = & -({c}_{0}+{c}_{2}){\phi }_{0}{\phi }_{-},\\ {L}_{41} & = & ({c}_{0}+{c}_{2}){\phi }_{+}^{* }{\phi }_{0}^{* },\\ {L}_{45} & = & ({c}_{0}+{c}_{2}){\phi }_{0}^{* }{\phi }_{-}^{* },\\ {L}_{51} & = & -({c}_{0}-{c}_{2}){\phi }_{+}^{* }{\phi }_{-},\\ {L}_{54} & = & -2({c}_{0}+{c}_{2}){\phi }_{0}{\phi }_{-},\\ {L}_{56} & = & -({c}_{0}+{c}_{2}){\phi }_{-}^{2},\\ {L}_{62} & = & ({c}_{0}+{c}_{2}){\phi }_{+}{\phi }_{-}^{* },\\ {L}_{63} & = & 2({c}_{0}+{c}_{2}){\phi }_{0}^{* }{\phi }_{-}^{* },\\ {L}_{65} & = & ({c}_{0}+{c}_{2}){\phi }_{-}^{* 2},\\ {L}_{13} & = & -2({c}_{0}+{c}_{2}){\phi }_{+}{\phi }_{0}^{* }+4{c}_{2}{\phi }_{0}{\phi }_{-}^{* },\\ {L}_{16} & = & -({c}_{0}-{c}_{2}){\phi }_{+}{\phi }_{-}-2{c}_{2}{\phi }_{0}^{2},\\ {L}_{24} & = & 2({c}_{0}+{c}_{2}){\phi }_{+}^{* }{\phi }_{0}+4{c}_{2}{\phi }_{0}^{* }{\phi }_{-},\\ {L}_{25} & = & ({c}_{0}-{c}_{2}){\phi }_{+}^{* }{\phi }_{-}^{* }+2{c}_{2}{\phi }_{0}^{* 2},\\ {L}_{31} & = & -({c}_{0}+{c}_{2}){\phi }_{+}^{* }{\phi }_{0}-2{c}_{2}{\phi }_{0}^{* }{\phi }_{-},\\ {L}_{34} & = & -2{c}_{0}{\phi }_{0}^{2}-2{c}_{2}{\phi }_{+}{\phi }_{-},\\ {L}_{35} & = & -({c}_{0}+{c}_{2}){\phi }_{0}{\phi }_{-}^{* }-2{c}_{2}{\phi }_{+}{\phi }_{0}^{* },\\ {L}_{42} & = & ({c}_{0}+{c}_{2}){\phi }_{+}{\phi }_{0}^{* }+4{c}_{2}{\phi }_{0}{\phi }_{-}^{* },\\ {L}_{43} & = & 2{c}_{0}{\phi }_{0}^{* 2}+2{c}_{2}{\phi }_{+}^{* }{\phi }_{-}^{* },\\ {L}_{46} & = & ({c}_{0}+{c}_{2}){\phi }_{0}^{* }{\phi }_{-}+2{c}_{2}{\phi }_{+}^{* }{\phi }_{0},\\ {L}_{52} & = & -({c}_{0}-{c}_{2}){\phi }_{+}{\phi }_{-}-2{c}_{2}{\phi }_{0}^{2},\\ {L}_{53} & = & -2({c}_{0}+{c}_{2}){\phi }_{0}^{* }{\phi }_{-}-4{c}_{2}{\phi }_{+}^{* }{\phi }_{0},\\ {L}_{61} & = & ({c}_{0}-{c}_{2}){\phi }_{+}^{* }{\phi }_{-}^{* }+2{c}_{2}{\phi }_{0}^{* 2},\\ {L}_{64} & = & 2({c}_{0}+{c}_{2}){\phi }_{0}{\phi }_{-}^{* }+4{c}_{2}{\phi }_{+}{\phi }_{0}^{* }.\end{array}\end{eqnarray*}$