Skip to main content
SpringerLink
Log in

Non-conservative variational approximation for nonlinear Schrödinger equations

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

In the work of Galley (Phys Rev Lett 110:174301, 2013) an initial value problem formulation of Hamilton’s principle was proposed and applied to non-conservative systems. Here, we explore this formulation for complex partial differential equations of the nonlinear Schrödinger (NLS) type, examining the dynamics of the coherent solitary wave structures of such models by means of a non-conservative variational approximation (NCVA). We compare the formalism of the NCVA to two other variational techniques used in dissipative systems, namely the perturbed variational approximation and a generalization of the so-called Kantorovich method. All three variational techniques produce equivalent equations of motion for the perturbed NLS models studied herein. We showcase the relevance of the NCVA method by exploring test case examples within the NLS setting including combinations of linear and density-dependent loss and gain. We also present an example applied to exciton–polariton condensates that intrinsically feature loss and a spatially dependent gain term.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. We note that the choice between 1, 2, and ± coordinates is arbitrary as the final results using the two different prescriptions are equivalent. In some problems, it is useful to work in the 1, 2 coordinates. For instance, this is the case when integrating out degrees of freedom that generate both conservative and nonconservative interaction terms in \({{\mathcal {R}}}\). In that case, the conservative contribution, including the difference between the two potentials \(V(\mathbf {q}_1) - V(\mathbf {q}_2)\) (see Sect. 2.1), might be more transparent when written in the 1, 2 coordinates rather than in the ± ones.

  2. In principle, one could use a better suited ansatz like the q-Gaussian proposed in Ref. [62] at the expense of obtaining more complicated reduced NCVA ODEs.

References

  1. B.A. Malomed, Prog. Opt. 43, 71 (2002)

    Article  ADS  Google Scholar 

  2. B.A. Malomed, Soliton Management in Periodic Systems (Springer, Berlin, 2006)

    MATH  Google Scholar 

  3. Th Dauxois, M. Peyrard, Physics of Solitons (Cambridge University Press, London, 2003)

    MATH  Google Scholar 

  4. YuS Kivshar, G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, London, 2003)

    Google Scholar 

  5. D.J. Kaup, T.I. Lakoba, J. Math. Phys. 37, 3442 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  6. D.J. Kaup, T.K. Vogel, Phys. Lett. A 362, 289 (2007)

    Article  ADS  Google Scholar 

  7. C.R. Galley, Phys. Rev. Lett. 110, 174301 (2013)

    Article  ADS  Google Scholar 

  8. C.R. Galley, D. Tsang, L.C. Stein, The principle of stationary nonconservative action for classical mechanics and field theories. arXiv:1412.3082 [math-ph]

  9. P.G. Kevrekidis, Phys. Rev. A 89, 010102 (2014)

    Article  ADS  Google Scholar 

  10. A. Hasegawa, Y. Kodama, Solitons in Optical Communications (Oxford University Press, Oxford, 1995)

    MATH  Google Scholar 

  11. L.P. Pitaevskii, S. Stringari, Bose–Einstein Condensation, Oxford University Press (Oxford, 2003) (Bose–Einstein condensation in dilute gases, Cambridge University Press (Cambridge, C.J. Pethick and H. Smith, 2002)

  12. M.J. Ablowitz, B. Prinari, A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems (Cambridge University Press, Cambridge, 2004)

    MATH  Google Scholar 

  13. I.S. Aranson, L. Kramer, Rev. Mod. Phys. 74, 99–143 (2002)

    Article  ADS  Google Scholar 

  14. D.J. Kaup, SIAM J. Appl. Math. 31, 121–133 (1976)

    Article  MathSciNet  Google Scholar 

  15. J.P. Keener, D.W. McLaughlin, Phys. Rev. A 16, 777–790 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  16. Y. Kodama, M.J. Ablowitz Studies, Appl. Math. 64, 225–245 (1981)

    Google Scholar 

  17. V.I. Karpman, V.V. Solev’ev, Physica D 3, 487–502 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  18. D.J. Kaup, Phys. Rev. A 42, 5689 (1991)

    Article  ADS  Google Scholar 

  19. D.J. Kaup, Phys. Rev. A 44, 4582 (1991)

    Article  ADS  Google Scholar 

  20. S.A. Kiselev, A.J. Sievers, G.V. Chester, Physica D 123, 393–402 (1998)

    Article  ADS  Google Scholar 

  21. Y. Kodama, Physica D 123, 255–266 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  22. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, 1980)

    MATH  Google Scholar 

  23. C. Sulem, P.L. Sulem, The Nonlinear Schrödinger Equation (Springer, New York, 1999)

    MATH  Google Scholar 

  24. R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equations (Academic Press, New York, 1983)

    MATH  Google Scholar 

  25. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers (Birkhauser, Boston, 2005)

    Book  MATH  Google Scholar 

  26. D.J. Benney, A.C. Newell, J. Math. Phys. 46, 133–139 (1967)

    Article  MathSciNet  Google Scholar 

  27. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142–144 (1973)

    Article  ADS  Google Scholar 

  28. A. Hasegawa, Optical Solitons in Fibers (Springer, Heidelberg, 1990)

    Book  Google Scholar 

  29. FKh Abdullaev, S.A. Darmanyan, P.K. Khabibullaev, Optical Solitons (Springer, Heidelberg, 1993)

    Book  Google Scholar 

  30. A. Hasegawa, Y. Kodama, Solitons in Optical Communications (Clarendon Press, Oxford, 1995)

    MATH  Google Scholar 

  31. F. Tappert, C.M. Varma, Phys. Rev. Lett. 25, 1108–1111 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  32. Y.H. Ichikawa, T. Imamure, T. Taniuti, Nonlinear wave modulation in collisionless plasma. J. Phys. Soc. Jpn. 34, 189–197 (1972)

    Article  ADS  Google Scholar 

  33. B.D. Fried, Y.H. Ichikawa, J. Phys. Soc. Jpn. 34, 1073–1082 (1973)

    Article  ADS  Google Scholar 

  34. E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge University Press, Cambridge, 1990)

    MATH  Google Scholar 

  35. P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González (eds.) Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment. Springer Series on Atomic, Optical, and Plasma Physics, vol. 45 (2008)

  36. R. Carretero-González, P.G. Kevrekidis, D.J. Frantzeskakis, Nonlinearity 21, R139 (2008)

    Article  Google Scholar 

  37. J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations (American Mathematical Society, Providence, 1999)

    Book  MATH  Google Scholar 

  38. M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981)

    Book  MATH  Google Scholar 

  39. V.E. Zakharov, S.V. Manakov, S.P. Nonikov, L.P. Pitaevskii, Theory of Solitons (Consultants Bureau, New York, 1984)

    Google Scholar 

  40. A.C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985)

    Book  MATH  Google Scholar 

  41. G.P. Agrawal, Nonlinear Fiber Optics, 2nd edn. (Academic Press, San Diego, 1995)

    MATH  Google Scholar 

  42. D. Anderson, Phys. Rev. A 27, 3135 (1983)

    Article  ADS  Google Scholar 

  43. YuS Kivshar, X. Yang, Phys. Rev. E 49, 1657 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  44. A. Scott, Nonlinear Science: Emergence and Dynamics of Coherent Structures (Oxford University Press, Oxford, 2003)

    MATH  Google Scholar 

  45. S. Chávez Cerda, S.B. Cavalcanti, J.M. Hickmann, Eur. Phys. J. D 1, 313 (1998)

    Article  ADS  Google Scholar 

  46. V. Skarka, N.B. Aleksić, Phys. Rev. Lett. 96, 013903 (2006)

    Article  ADS  Google Scholar 

  47. V. Skarka, N.B. Aleksić, H. Leblond, B.A. Malomed, D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010)

    Article  ADS  Google Scholar 

  48. V. Skarka, N.B. Aleksić, M. Lekić, B.N. Aleksić, B.A. Malomed, D. Mihalache, H. Leblond, Phys. Rev. A 90, 023845 (2014)

    Article  ADS  Google Scholar 

  49. H. Deng, H. Haug, Y. Yamamoto, Rev. Mod. Phys. 82, 1489–1537 (2010)

    Article  ADS  Google Scholar 

  50. A. Amo, J. Lefrère, S. Pigeon, C. Abrados, C. Ciuti, I. Carusotto, R. Houdré, E. Giacobino, A. Bramati, Nat. Phys. 5, 805–809 (2009)

    Article  Google Scholar 

  51. K.G. Lagoudakis, M. Wouters, M. Richard, A. Baas, I. Carusotto, R. André, L.S. Dang, B. Deveaud-Plédran, Nat. Phys. 4, 706–710 (2008)

    Article  Google Scholar 

  52. M.D. Fraser, G. Roumpos, Y. Yamamoto, New J. Phys. 11, 113048 (2009)

    Article  ADS  Google Scholar 

  53. G. Roumpos, M.D. Fraser, A. Löffler, S. Höfling, A. Forchel, Y. Yamamoto, Nat. Phys. 7, 129–133 (2011)

    Article  Google Scholar 

  54. L. Dominici, R. Carretero-González, A. Gianfrate, J. Cuevas-Maraver, A.S. Rodrigues, D.J. Frantzeskakis, P.G. Kevrekidis, G. Lerario, D. Ballarini, M. De Giorgi, G. Gigli, D. Sanvitto, Nat. Comm. 9, 1467 (2018)

    Article  ADS  Google Scholar 

  55. A. Amo, D. Sanvitto, F.P. Laussy, D. Ballarini, E. del Valle, M.D. Martin, A. Lemaitre, J. Bloch, D.N. Krizhanovskii, M.S. Skolnick, C. Tejedor, L. Vina, Nature 457, 291–296 (2009)

    Article  ADS  Google Scholar 

  56. A. Amo, T.C.H. Liew, C. Adrados, R. Houdré, E. Giacobino, A.V. Kavokin, A. Bramati, Nat. Photon. 4, 361–366 (2010)

    Article  ADS  Google Scholar 

  57. D. Sanvitto, S. Pigeon, A. Amo, D. Ballarini, M. De Giorgi, I. Carusotto, R. Hivet, F. Pisanello, V.G. Sala, P.S.S. Guimaraes, R. Houdre, E. Giacobino, C. Ciuti, A. Bramati, G. Gigli, Nat. Photon. 5, 610–614 (2011)

    Article  ADS  Google Scholar 

  58. J. Keeling, N.G. Berloff, Phys. Rev. Lett. 100, 250401 (2008)

    Article  ADS  Google Scholar 

  59. M.O. Borgh, J. Keeling, N.G. Berloff, Phys. Rev. B 81, 234302 (2010)

    Article  ADS  Google Scholar 

  60. J. Keeling, N. Berloff, Contemp. Phys. 52, 131–151 (2011)

    Article  ADS  Google Scholar 

  61. J. Cuevas, A.S. Rodrigues, R. Carretero-González, P.G. Kevrekidis, D.J. Frantzeskakis, Phys. Rev. B 83, 245140 (2011)

    Article  ADS  Google Scholar 

  62. A.I. Nicolin, R. Carretero-González, Physica A 387, 6032 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  63. L.A. Smirnov, D.A. Smirnova, E.A. Ostrovskaya, YuS Kivshar, Phys. Rev. B 89, 235310 (2014)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

J.R. gratefully acknowledges the support from the Computational Science Research Center (CSRC) at SDSU, the ARCS foundations and Cymer. R.C.G. gratefully acknowledges the support of NSF-PHY-1603058. P.G.K. gratefully acknowledges the support of NSF-PHY-1602994 and NSF-DMS-1809074. We would also like to thank S. Chávez Cerda for a useful discussion on this topic, as well as for pointing us to the relevant work of Ref. [45].

Author information

Authors and Affiliations

  1. Nonlinear Dynamical System Group, Computational Science Research Center, Department of Mathematics and Statistics, San Diego State University, San Diego, CA, 92182-7720, USA

    J. Rossi & R. Carretero-González

  2. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003-4515, USA

    P. G. Kevrekidis

Authors
  1. J. Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Carretero-González
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. G. Kevrekidis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Carretero-González.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rossi, J., Carretero-González, R. & Kevrekidis, P.G. Non-conservative variational approximation for nonlinear Schrödinger equations. Eur. Phys. J. Plus 135, 854 (2020). https://doi.org/10.1140/epjp/s13360-020-00689-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-020-00689-x

Search

Navigation