Stability analysis of nonlinear localized modes in the coupled Gross-Pitaevskii equations with PT -symmetric Scarf-II potential

We study the nonlinear localized modes in two-component Bose-Einstein condensates with parity-time-symmetric Scarf-II potential, which can be described by the coupled Gross-Pitaevskii equations. Firstly, we investigate the linear stability of the nonlinear modes in the focusing and defocusing cases, and get the stable and unstable domains of nonlinear localized modes. Then we validate the results by evolving them with 5% perturbations as an initial condition. Finally, we get stable solitons by considering excitations of the soliton via adiabatical change of system parameters. These findings of nonlinear modes can be potentially applied to physical experiments of matter waves in Bose-Einstein condensates.

In this paper, we investigate the coupled Gross-Pitaevskii equations with complex PT -symmetric potentials [28]: where a j represent the assumed-equal intra-component and inter-component interactions, U j (x) are the complex PT -symmetric potentials, and the imaginary parts of U j (x) stand for the gain or loss term from the thermal clouds.
The present paper is built up as follows.In Sect.2, we consider the analytic bright-soliton solution in the coupled Gross-Pitaevskii equations with complex PT -symmetric Scarf-II potentials; the linear stability analysis and the numerical evolution results corroborating the analytical solitons are presented in Sect.3; In Sect.4, we perform numerical simulations for the excitation and evolution of nonlinear modes via adiabatical change of system parameters; Finally, conclusions and discussions are given in Sect. 5.

Localized modes in coupled Gross-Pitaevskii equations
We concentrate on stationary solutions of Eq. ( 3) in the form where ν j are the real propagation constant.The complex solutions φ j (x) satisfy the following condition which can be solved for the given potentials U j (x) and real propagation constant ν j .
For the PT -symmetric potentials U j (x) are all chosen as the well-known Scarf-II potentials as We have the analytic bright-soliton solution as [37] φ j (x) = A j sech(x) e iϕj , j = 1, 2 , with the phase is under the constraints of 18 and ν j = 1.
For the nonlinear modes given in Eq. ( 7), the power of the solutions is defined as . The power flows from left (right) to right (left) at x 0 when S(x 0 ) > 0 (S(x 0 ) < 0).

Linear Stability analysis
In this section, we investigate the linear stability of the nonlinear modes, which is a standard protocol to show the stability of nonlinear localized modes.We consider the perturbed solution ψ j (x, t), in the form where ǫ ≪ 1, which is the small perturbation on the solution.f j (x) and g j (x) are the perturbation eigenfunctions of the linearized eigenvalue problem.By substituting Eq. ( 10) into Eq.( 5) and linearizing with respect to ǫ, we can drive the following linear eigenvalue problem: where The imaginary part of δ measures the growth rate of the perturbation instability.If |Im(δ)| > 0, then the perturbation will grow exponentially with t, and the solutions are unstable; otherwise, the solutions are stable.In our numerical simulation, we use the Fourier collocation method to discretize the associated differential operator as a matrix to solve the eigenvalue problem [49].To further verify the stability of the solitons, we numerically investigate the stability by evolving them with 5% perturbations as the initial condition to simulate the random white noise (i.e., ψ(x, 0) = φ(x)(1 + ξ) and ξ represents 5% perturbations).In our numerical simulations, the second-order spatial differential is carried out by using Fourier spectral collocation method, and the integration in time is carried out by using the explicit fourth-order Runge-Kutta method [50].
Firstly, under the constraint of we consider the focusing case a 1 = a 2 = 1 and the defocusing case a 1 = a 2 = −1, respectively.Then we get the stable (blue) and unstable (red) domains of nonlinear localized modes in (V 1 , W 1 ) space [see Figs . 1].They are determined by the maximum absolute value of imaginary parts of the linearized eigenvalue δ in Eq. (11).We find that solitons tend to be unstable with the increase of |W 1 | in the focusing case.It is worth noting that when |W 1 | = 3, solitons are stable in the defocusing case.
Since the above situations are obtained in the case of A 1 = 0.5, and A 2 is obtained by Eq. 9. Next, we consider the case of A 1 = A 2 .The relationships between P 2 and the parameter W 1 are shown in Figs. 2. We can find that when other parameters are fixed, P 2 and |W 1 | are positively correlated in the focusing case, while they are negatively correlated in the defocusing case.In addition, for both cases A 1 = 0.5 and A 1 = A 2 , the intervals of W 1 have no difference when the solutions are stable.
In particular, for the fixed parameters The parameters are chosen as: and (a-c) Furthermore, in the focusing case, the amplitude of the nonlinear mode is periodically oscillating when V 1 and W 1 are sufficiently small, and it experiences more than 5 periods within 1200 ≤ t ≤ 1500 [see Figs.5].

Adiabatic excitation for the nonlinear modes
In this section, we consider excitations of the above-mentioned solitons via adiabatical changes of system parameters.We change the parameters as the functions of t.To modulate the system parameters smoothly, we consider the following "switch-on" function: where ζ (ini) , ζ (end) respectively represent the real initial-state and final-state parameters [38,45].Adiabatic excitation includes two stages: excitation stage (0 < t < 500) and propagation stage (500 ≤ t ≤ 1500).We   consider two cases of excitations by setting a 1 , V 1 and V 2 to be functions of t, that is To facilitate the display of power changes over time, we set = 2, a 2 = 0.1, and the power of nonlinear modes is reduced [see Figs.6a-6c].Then, we set = 1, a 2 = 0.0033, and the total power of nonlinear modes is to decrease and then increase [see Figs.6d-6f].The above results mean that the power is not conserved during adiabatic excitations, and it has a correlation with the initial and final state potential parameters.

conclusion
In conclusion, we study the nonlinear modes in two-component Bose-Einstein condensates with PT -symmetric Scarf-II potential, which can be described by the coupled Gross-Pitaevskii equations.We investigate the linear stability of the nonlinear modes and validate the results by evolving them with 5% perturbations as an initial condition.We find that solitons tend to be unstable with the increase of |W 1 | in the focusing case.It is worth noting that when |W 1 | = 3, solitons are stable in the defocusing case.In the focusing case, the amplitude of the nonlinear mode is periodically oscillating when V 1 and W 1 are sufficiently small.Finally, we consider excitations of the solitons via adiabatical changes of system parameters, then we find that the power is not conserved during this adiabatic excitation.These findings of nonlinear modes can be potentially applied to physical experiments of matter waves in Bose-Einstein condensates.
In addition, we can consider other PT -symmetric potentials in the coupled Gross-Pitaevskii equations.Due to the limitations of the parameters in this model, the amplitudes of the two solutions are constrained by a certain relationship.Therefore, we can also consider the case of unequal intra-component and inter-component interactions.

Figure 6 :
Figure 6: Adiabatic excitation of nonlinear mode and its evolution.The parameters are chosen as: A 1 = A 2 = 2.2733, W 1 =