Advertisement
Browse Subject Areas
?

Click through the PLOS taxonomy to find articles in your field.

For more information about PLOS Subject Areas, click here.

  • Loading metrics

Open Access

Peer-reviewed

Research Article

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

  • Jia-Rui Zhang ,

    Roles Conceptualization, Data curation, Formal analysis, Funding acquisition, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing

    jrzhang@cau.edu.cn

    Affiliation College of Science, China Agricultural University, Beijing, China

    ⨯
  • Xia Wang

    Roles Data curation, Formal analysis, Investigation, Software, Visualization, Writing – review & editing

    Affiliation College of Science, China Agricultural University, Beijing, China

    ⨯

Abstract

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.

Citation: Zhang J-R, Wang X (2023) Stability analysis of nonlinear localized modes in the coupled Gross-Pitaevskii equations with -symmetric Scarf-II potential. PLoS ONE 18(11): e0294790. https://doi.org/10.1371/journal.pone.0294790

Editor: Boris Malomed, Tel Aviv University, ISRAEL

Received: June 5, 2023; Accepted: November 7, 2023; Published: November 27, 2023

Copyright: © 2023 Zhang, Wang. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: This article does not report data and the data availability policy is not applicable to this article.

Funding: This work was supported by the National Training Program of Innovation (Grant numbers 202210019045). The funding body plays an important role in the design of the study, in analysis, calculation, and in writing of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

1 Introduction

Bose-Einstein condensate (BEC) [1, 2], as one of the important physical phenomena, has attracted the attention of researchers. The successful observation of solitons in BECs has become one of the research focuses in the fields of condensed matter physics and atom optics [3–5]. Compared with the single-component ones, the multi-component BECs possess the inter-component interactions and have complicated quantum phases and properties [6–13]. Many novel phenomena have been discovered in multi-component BECs [14–22], including symbiotic solitons, soliton trains, soliton pairs, multi-domain walls, and multi-mode collective excitations. As one kind of multi-component BECs, the two-component BECs trapped in a quasi-one-dimensional harmonic potential at zero temperature can be described by the following coupled Gross-Pitaevskii equations [23, 24]: (1a) (1b) where Ψj is the two-component parameter, r = (x, y, z), ℏ is the Planck constant, M is the atomic mass, ∇2 is the Laplacian, stands for the harmonic potentials, gjj is the interactions between atoms, and gj,3−j describes the inter-component interactions (j = 1, 2) [25–28].

If the trap frequencies in the radial directions ωj⊥ are larger than the axial directions ωjx, Eq (1) becomes a quasi-one-dimensional system along the x direction. Through the normalization and transformation Ψj → ψj, , t → (2π/ω1⊥)t, Eq (1) can be written in the form [25–28]: (2a) (2b) where λj = ωjx/ωj⊥, bjj and bj,3−j are related to , trap frequencies in the radial directions ωj⊥, and interactions between atoms gjj or inter-component gj,3−j (j = 1, 2). For the harmonic potentials, the soliton states were studied widely [25–28]. In this paper, we consider the two-component BECs trapped in another potential.

Put forward by Bender and his coworker in 1998 [29–31], parity-time- (-) symmetry behaviors have attracted much attention in both non-Hermitian Hamiltonian systems and nonlinear wave systems [32–34], which make systems with complex potentials possibly support fully-real linear spectra [35] and stable nonlinear modes [36–39]. That is, the potential function U(x) = V(x) + iW(x) satisfies V(x) = V(−x) and W(−x) = −W(x) [29–31]. Over the past few years, various -symmetric potentials have been introduced into the nonlinear Schrödinger equation and the existence of different nonlinear local modes is analytically and numerically investigated [40–51]. To better investigate physical phenomena, it is meaningful to introduce new forms of -symmetric potentials in nonlinear systems.

In this paper, we investigate the coupled Gross-Pitaevskii equations with complex -symmetric potentials [28]: (3a) (3b) where aj represent the intra-component and inter-component interactions while the interactions take equal values when a1 = a2, Uj(x) are the complex -symmetric potentials, and the imaginary parts of Uj(x) stand for the gain or loss term from the thermal clouds (j = 1, 2).

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 -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.

2 Localized modes in coupled Gross-Pitaevskii equations

We concentrate on stationary solutions of Eq (3) in the form (4) where νj are the real propagation constant. The complex solutions ϕj(x) satisfy the following condition (5a) (5b) which can be solved for the given potentials Uj(x) and real propagation constant νj.

For the -symmetric potentials Uj(x) are all chosen as the well-known Scarf-II potentials as (6)

We have the analytic bright-soliton solution as [37] (7) with the phases being (8) under the constraints of (9) and νj = 1.

For the nonlinear modes given in Eq (7), the power of the solutions is defined as , P = P1 + P2, while the Poynting vector . The power flows from left (right) to right (left) at x0 when S(x0) > 0 (S(x0) < 0).

3 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 ψj(x, t), in the form (10) where ϵ ≪ 1, which is the small perturbation on the solution. fj(x) and gj(x) are the perturbation eigenfunctions of the linearized eigenvalue problem. By substituting Eq (10) into Eq (5) and linearizing with respect to ϵ, we can drive the following linear eigenvalue problem: (11) where (12a) (12b)

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 [52]. 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 [53].

Firstly, under the constraint of A1 = 0.5, V1 = V2, W1 = W2, we consider the focusing case a1 = a2 = 1 and the defocusing case a1 = a2 = −1, respectively. Then we get the stable (blue) and unstable (red) domains of nonlinear localized modes in (V1, W1) space [see Fig 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 |W1| in the focusing case. It is worth noting that when |W1| = 3, solitons are stable in the defocusing case.

thumbnail
Download:
Fig 1. Maximal imaginary part of the linearization eigenvalue δ in the (V1, W1)-space (common logarithmic scale), under the constraint of A1 = 0.5, V1 = V2, W1 = W2 and (a) a1 = 1; (b) a1 = −1.

https://doi.org/10.1371/journal.pone.0294790.g001

Since the above situations are obtained in the case of A1 = 0.5, and A2 is obtained by Eq 9. Next, we consider the case of A1 = A2. The relationships between P2 and the parameter W1 are shown in Fig 2. We can find that when other parameters are fixed, P2 and |W1| are positively correlated in the focusing case, while they are negatively correlated in the defocusing case. In addition, for both cases A1 = 0.5 and A1 = A2, the intervals of W1 have no difference when the solutions are stable.

thumbnail
Download:
Fig 2. The relationship between power of nonlinear mode ϕ2 and W1.

The parameters are chosen as: W1 = W2, and (a) V1 = V2 = 1, a1 = 1; (b) V1 = V2 = 8, a1 = −1.

https://doi.org/10.1371/journal.pone.0294790.g002

In particular, for the fixed parameters a1 = a2 = 1, A1 = 0.5, V1 = V2 = 1, Fig 3(a)–3(c) display the stable soliton for W1 = W2 = 0.25 while Fig 3(d)–3(f) display the unstable soliton for W1 = W2 = 0.55; for the fixed parameters a1 = a2 = −1, A1 = 0.5, V1 = V2 = 8, Fig 4(a)–4(c) display the stable soliton for W1 = W2 = 3 while Fig 4(d)–4(f) display the unstable soliton for W1 = W2 = 2.

thumbnail
Download:
Fig 3.

(a, d) The soliton solutions. (b, e) Linear stability eigenvalues. (c, f) Stable or unstable propagations of nonlinear modes. The parameters are chosen as: a1 = 1, A1 = 0.5, V1 = V2 = 1, and (a-c) W1 = W2 = 0.25; (d-f) W1 = W2 = 0.55.

https://doi.org/10.1371/journal.pone.0294790.g003

thumbnail
Download:
Fig 4.

(a, d) The soliton solutions. (b, e) Linear stability eigenvalues. (c, f) Stable or unstable propagations of nonlinear modes. The parameters are chosen as: a1 = −1, A1 = 0.5, V1 = V2 = 8, and (a-c) W1 = W2 = 3; (d-f) W1 = W2 = 2.

https://doi.org/10.1371/journal.pone.0294790.g004

Furthermore, in the focusing case, the amplitude of the nonlinear mode is periodically oscillating when V1 and W1 are sufficiently small, and it experiences more than 5 periods within 1200 ≤ t ≤ 1500 [see Fig 5].

thumbnail
Download:
Fig 5.

(a) The soliton solutions. (b) Linear stability eigenvalues. (c) Stable propagations of nonlinear modes. The parameters are chosen as: a1 = 1, A1 = 0.5, V1 = V2 = 0.01, W1 = W2 = 0.06.

https://doi.org/10.1371/journal.pone.0294790.g005

4 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: (13) where ζ(ini), ζ(end) respectively represent the real initial-state and final-state parameters [38, 45, 54]. Adiabatic excitation includes two stages: excitation stage (0 < t < 500) and propagation stage (500 ≤ t ≤ 1500). We consider two cases of excitations by setting a1, V1 and V2 to be functions of t, that is a1 → a1(t), V1 → V1(t) and V2 → V2(t). To facilitate the display of power changes over time, we set A1 = A2, W1 = W2 = 0.55, , , , . Firstly, we set , , a2 = 0.1, and the power of nonlinear modes is reduced [see Fig 6a–6c]. Then, we set , , a2 = 0.0033, and the total power of nonlinear modes is to decrease and then increase [see Fig 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.

thumbnail
Download:
Fig 6. Adiabatic excitation of nonlinear mode and its evolution.

The parameters are chosen as: A1 = A2 = 2.2733, W1 = W2 = 0.55, , , , and (a-c) , , a2 = 0.1; (d-f) , , a2 = 0.0033.

https://doi.org/10.1371/journal.pone.0294790.g006

5 Conclusion

In conclusion, we study the nonlinear modes in two-component Bose-Einstein condensates with -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 |W1| in the focusing case. It is worth noting that when |W1| = 3, solitons are stable in the defocusing case. In the focusing case, the amplitude of the nonlinear mode is periodically oscillating when V1 and W1 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 -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.

Acknowledgments

We express our sincere thanks to the editor, referees and all the members of our discussion group for their valuable comments.

References

  1. 1. Einstein A., Quantentheorie des einatomigen idealen gases, Sitz. Ber. Kgl. Preuss. Akad. Wiss. (1924) 261–7.
  2. 2. Einstein A., Quantentheorie des einatomigen idealen gases, Sitzungber Preuss 9 (1925) 3–14.
  3. 3. Anderson M. H., Ensher J. R., Matthews M. R., Wieman C. E., Cornell E. A., Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269 (1995) 198–201. pmid:17789847
  4. 4. Davis K. B., Mewes M. O., Andrews M. R., Van Druten N. J., Durfee D. S., Kurn D. M., et al., Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75 (1995) 3969–3973. pmid:10059782
  5. 5. Strecker K. E., Partridge G. B., Truscott A. G., and Hulet R. G., Formation and propagation of matter-wave soliton trains, Nature 417 (2002) 150–153. pmid:11986621
  6. 6. Myatt C. J., Burt E. A., Ghrist R. W., Cornell E. A., Wieman C. E., Production of Two Overlapping Bose-Einstein Condensates by Sympathetic Cooling, Phys. Rev. Lett. 78 (1997) 586.
  7. 7. Pu H., Bigelow N. P., Properties of Two-Species Bose Condensates, Phys. Rev. Lett. 80 (1998) 1130.
  8. 8. Modugno G., Ferrari G., Roati G., Brecha R. J., Simoni A., Inguscio M., Bose-Einstein condensation of potassium atoms by sympathetic cooling, Science 294 (2001) 1320. pmid:11641466
  9. 9. Alon O. E., Streltsov A. I., Cederbaum L. S., Zoo of Quantum Phases and Excitations of Cold Bosonic Atoms in Optical Lattices, Phys. Rev. Lett. 95 (2005) 030405. pmid:16090725
  10. 10. Kapale K. T., Dowling J. P., Vortex Phase Qubit: Generating Arbitrary, Counterrotating, Coherent Superpositions in Bose-Einstein Condensates via Optical Angular Momentum Beams, Phys. Rev. Lett. 95 (2005) 173601. pmid:16383828
  11. 11. Bhongale S. G. and Timmermans E., Phase Separated BEC for High-Sensitivity Force Measurement, Phys. Rev. Lett. 100 (2008) 185301. pmid:18518387
  12. 12. Zawadzki M. E., Griffin P. F., Riis E., Arnold A. S., Spatial interference from well-separated split condensates, Phys. Rev. A 81 (2010) 043608.
  13. 13. van Zoest T. et al., Bose-Einstein Condensation in Microgravity, Science 328 (2010) 1540. pmid:20558713
  14. 14. Pérez-García Víctor M., Vekslerchik Vadym, Soliton molecules in trapped vector nonlinear Schrödinger systems, Phys. Rev. E 67 (2003) 061804. pmid:16241251
  15. 15. Montesinos G. D., Pérez-García Víctor M., Michinel Humberto, Stabilized Two-Dimensional Vector Solitons, Phys. Rev. Lett. 92 (2004) 133901. pmid:15089613
  16. 16. Kasamatsu K., Tsubota M., Multiple Domain Formation Induced by Modulation Instability in Two-Component Bose-Einstein Condensates, Phys. Rev. Lett. 93 (2004) 100402. pmid:15447389
  17. 17. Malomed B. A., Nistazakis H. E., Frantzeskakis D. J., Kevrekidis P. G., Static and rotating domain-wall cross patterns in Bose-Einstein condensates, Phys. Rev. A 70 (2004) 043616.
  18. 18. Brazhnyi V. A., Konotop V. V., Stable and unstable vector dark solitons of coupled nonlinear Schrödinger equations: Application to two-component Bose-Einstein condensates, Phys. Rev. E 72 (2005) 026616. pmid:16196744
  19. 19. Kevrekidis P. G., Susanto H., Carretero-González R., Malomed B. A., Frantzeskakis D. J., Vector solitons with an embedded domain wall, Phys. Rev. E 72 (2005) 066604. pmid:16486075
  20. 20. Pérez-García Víctor M., Belmonte Beitia Juan, Symbiotic solitons in heteronuclear multicomponent Bose-Einstein condensates, Phys. Rev. A 72 (2005) 033620.
  21. 21. Adhikari Sadhan K., Bright solitons in coupled defocusing NLS equation supported by coupling: Application to Bose-Einstein condensation, Phys. Lett. A 346 (2005) 179.
  22. 22. Xue J.-K., Li G.-Q., Zhang A.-X., Peng P., Nonlinear mode coupling and resonant excitations in two-component Bose-Einstein condensates, Phys. Rev. E 77 (2008) 016606. pmid:18351950
  23. 23. Ho T. L., Shenoy V. B., Binary Mixtures of Bose Condensates of Alkali Atoms, Phys. Rev. Lett. 77 (1996) 3276. pmid:10062180
  24. 24. Esry B. D., Greene C. H., Burke J. P., Bohn J. L., Hartree-Fock Theory for Double Condensates, Phys. Rev. Lett. 78 (1997) 3594.
  25. 25. Liu X., Pu H., Xiong B., Liu W. M., Gong J., Formation and transformation of vector solitons in two-species Bose-Einstein condensates with a tunable interaction, Phys. Rev. A 79 (2009) 013423.
  26. 26. Zhang X.-F., Hu X.-H., Liu X.-X., Liu W. M., Vector solitons in two-component Bose-Einstein condensates with tunable interactions and harmonic potential, Phys. Rev. A 79 (2009) 033630.
  27. 27. Yakimenko A. I., Shchebetovska K. O., Vilchinskii S. I., Weyrauch M., Stable bright solitons in two-component Bose-Einstein condensates, Phys. Rev. A 85 (2012) 053640.
  28. 28. Wang Y.-F., Tian B., Wang M., Bell-Polynomial Approach and Integrability for the Coupled Gross-Pitaevskii Equations in Bose-Einstein Condensates, Stud. Appl. Math. 131 (2013) 119–134.
  29. 29. Bender C. M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having symmetry, Phys. Rev. Lett. 80 (1998) 5243–5246.
  30. 30. Bender C. M., Brody D. C., Jones H. F., Must a Hamiltonian be Hermitian?, Am. J. Phys. 71 (2003) 1095–1102.
  31. 31. Bender C. M., Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70 (2007) 947–1018.
  32. 32. Nixon S., Ge L., Yang J., Stability analysis for solitons in -symmetric optical lattices, Phys. Rev. A 85 (2012) 023822.
  33. 33. Lumer Y., Plotnik Y., Rechtsman M. C., Segev M., Nonlinearly Induced Transition in Photonic Systems, Phys. Rev. Lett. 111, (2013) 263901. pmid:24483795
  34. 34. Yang J., Symmetry breaking of solitons in one-dimensional parity-time symmetric optical potentials, Opt. Lett. 39 (2014) 5547. pmid:25360924
  35. 35. Ahmed Z., Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex -invariant potential, Phys. Lett. A. 282 (2001) 343–348.
  36. 36. Musslimani Z. H., Makris K. G., El-Ganainy R., Christodoulides D. N., Optical solitons in Periodic Potentials, Phys. Rev. Lett. 100 (2008) 030402. pmid:18232949
  37. 37. Yan Z., Wen Z., Hang C., Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarf-II potentials, Phys. Rev. E. 92 (2015) 022913.
  38. 38. Yan Z., Wen Z., Konotop V. V., Solitons in a nonlinear Schrödinger equation with -symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes, Phys. Rev. A. 92 (2015) 023821.
  39. 39. Chen Y., Yan Z., Liu W., Impact of near- symmetry on exciting solitons and interactions based on a complex Ginzburg-Landau model, Opt. Express. 26 (2018) 33022–33034. pmid:30645460
  40. 40. Shi Z., Jiang X., Zhu X., Li H., Bright spatial solitons in defocusing Kerr media with -symmetric potentials, Phys. Rev. A 84 (2011) 053855.
  41. 41. Hu S., Hu W., Optical solitons in the parity-time-symmetric Bessel complex potential, J. Phys. B 45 (2012) 225401.
  42. 42. Khare A., Al-Marzoug S. M., Bahlouli H., Solitons in -symmetric potential with competing nonlinearity, Phys. Lett. A 376 (2012) 2880.
  43. 43. Achilleos V., Kevrekidis P. G., Frantzeskakis D. J., Carretero-Gonzalez R., Dark solitons and vortices in -symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear phase transitions, Phys. Rev. A 86 (2012) 013808.
  44. 44. Yan Z. Y., Complex -symmetric nonlinear Schrödinger equation and Burgers equation, Philos. Trans. R. Soc. Lond. A 371 (2013) 20120059. pmid:23509385
  45. 45. Shen Y., Wen Z., Yan Z., Hang C., Effect of symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media, Chaos 28 (2018) 043104. pmid:31906637
  46. 46. Chen Y., Yan Z., Mihalache D., Stable flat-top solitons and peakons in the -symmetric δ-signum potentials and nonlinear media, Chaos 29 (2019) 083108. pmid:31472484
  47. 47. Xu W.-X., Su S.-J., Xu B., Guo Y.-W., Xu S.-L., Zhao Y., et al, Two dimensional spacial soliton in atomic gases with -symmetry potential, Opt. Express 28 (2020) 35297. pmid:33182979
  48. 48. Chen Y., Yan Z., Malomed B. A., Higher-dimensional soliton generation, stability and excitations of the -symmetric nonlinear Schrödinger equations, Physica D 430 (2022) 133099.
  49. 49. Song J., Zhou Z., Weng W., Yan Z., -symmetric peakon solutions in self-focusing/defocusing power-law nonlinear media: Stability, interactions and adiabatic excitations, Physica D 435 (2022) 133266.
  50. 50. Zhong M., Yan Z., Tian S.-F., Stable matter-wave solitons, interactions, and excitations in the spinor F = 1 Bose-Einstein condensates with -and non--symmetric potentials, Commun. Nonlinear Sci. Numer. Simulat. 118 (2023) 107061.
  51. 51. Song J., Yan Z., Malomed B. A., Formations and dynamics of two-dimensional spinning asymmetric quantum droplets controlled by a -symmetric potential, Chaos 33 (2023) 033141. pmid:37003809
  52. 52. Trefethen L. N., Spectral methods in matlab, SIAM, Philadelphia, 2000.
  53. 53. Yang J., Nonlinear Waves in Integrable and Nonintegrable Systems, SIAM, Philadelphia, 2010.
  54. 54. Hatomura T., Kato G., Bounds for nonadiabatic transitions, Phys. Rev. A 102 (2020) 012216.