Skip to main content
SpringerLink
Log in

Algebraic Method for Group Classification of (1+1)-Dimensional Linear Schrödinger Equations

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We carry out the complete group classification of the class of (1+1)-dimensional linear Schrödinger equations with complex-valued potentials. After introducing the notion of uniformly semi-normalized classes of differential equations, we compute the equivalence groupoid of the class under study and show that it is uniformly semi-normalized. More specifically, each admissible transformation in the class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of this class. This allows us to apply the new version of the algebraic method based on uniform semi-normalization and reduce the group classification of the class under study to the classification of low-dimensional appropriate subalgebras of the associated equivalence algebra. The partition into classification cases involves two integers that characterize Lie symmetry extensions and are invariant with respect to equivalence transformations.

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

Similar content being viewed by others

Notes

  1. See, e.g., [3, 4, 19, 20, 23, 37, 38, 44] and [6, 7, 32, 39] for the application of the preliminary and advanced versions of the algebraic method, respectively, to various classes of differential equations.

  2. A subgroup of the equivalence group can be considered here instead of the entire group.

References

  1. Anderson, R.L., Kumei, S., Wulfman, C.E.: Invariants of the equations of wave mechanics. I. Rev. Mex. Fis. 21, 1–33 (1972)

    MathSciNet  Google Scholar 

  2. Anderson, R.L., Kumei, S., Wulfman, C.E.: Invariants of the equations of wave mechanics. II. One-particle Schroedinger equations. Rev. Mex. Fis. 21, 35–57 (1972)

    MathSciNet  Google Scholar 

  3. Basarab-Horwath, P., Güngör, F., Lahno, V.: Symmetry classification of third-order nonlinear evolution equations. Part I: semi-simple algebras. Acta Appl. Math. 124, 123–170 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Basarab-Horwath, P., Lahno, V., Zhdanov, R.: The structure of Lie algebras and the classification problem for partial differential equations. Acta Appl. Math. 69, 43–94 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bihlo, A., Dos Santos Cardoso-Bihlo, E., Popovych, R.O.: Enhanced preliminary group classification of a class of generalized diffusion equations. Commun. Nonlinear Sci. Numer. Simul. 16, 3622–3638 (2011). arXiv:1012.0297

    Article  MathSciNet  MATH  Google Scholar 

  6. Bihlo, A., Dos Santos Cardoso-Bihlo, E., Popovych, R.O.: Complete group classification of a class of nonlinear wave equations. J. Math. Phys. 53, 123515 (2012). arXiv:1106.4801

    Article  MathSciNet  MATH  Google Scholar 

  7. Bihlo, A., Popovych, R.O.: Group classification of linear evolution equations. J. Math. Anal. Appl. 448, 982–1005 (2017). arXiv:1605.09251

    Article  MathSciNet  MATH  Google Scholar 

  8. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)

    Book  MATH  Google Scholar 

  9. Bender, C.M.: Complex extension of quantum mechanics. In: Proceedings of Fifth International Conference “Symmetry in Nonlinear Mathematical Physics” 23–29 June, 2003, Kyiv. Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 50, pp. 617–628. Institute of Mathematics of NAS of Ukraine, Kyiv (2004)

    Google Scholar 

  10. Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)

    Article  MathSciNet  Google Scholar 

  11. Boyer, C.P.: The maximal ‘kinematical’ invariance group for an arbitrary potential. Helv. Phys. Acta 47, 589–605 (1974)

    MathSciNet  Google Scholar 

  12. Doebner, H.-D., Goldin, G.A.: Properties of nonlinear Schrödinger equations associated with diffeomorphism group representations. J. Phys. A 27, 1771–1780 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dos Santos Cardoso-Bihlo, E., Bihlo, A., Popovych, R.O.: Enhanced preliminary group classification of a class of generalized diffusion equations. Commun. Nonlinear Sci. Numer. Simul. 16, 3622–3638 (2011). arXiv:1012.0297

    Article  MathSciNet  MATH  Google Scholar 

  14. Fushchich, W.I., Moskaliuk, S.S.: On some exact solutions of the nonlinear Schrödinger equation in three spatial dimensions. Lett. Nuovo Cimento 31, 571–576 (1981)

    Article  MathSciNet  Google Scholar 

  15. Gagnon, L., Winternitz, P.: Lie symmetries of a generalised non-inear Schrödinger equation. I. The symmetry group and its subgroups. J. Phys. A 21, 1493–1511 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gagnon, L., Winternitz, P.: Lie symmetries of a generalised non-linear Schrödinger equation. II. Exact solutions. J. Phys. A 22, 469–497 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gagnon, L., Grammaticos, B., Ramani, A., Winternitz, P.: Lie symmetries of a generalised non-linear Schrödinger equation. III. Reductions to third-order ordinary differential equations. J. Phys. A 22, 499–509 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gagnon, L., Winternitz, P.: Exact solutions of the cubic and quintic nonlinear Schrödinger equation for a cylindrical geometry. Phys. Rev. A 39, 296–306 (1989)

    Article  MathSciNet  Google Scholar 

  19. Gagnon, L., Wintenitz, P.: Symmetry classes of variable coefficient nonlinear Schrödinger equations. J. Phys. A 26, 7061–7076 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gazeau, J.P., Wintenitz, P.: Symmetries of variable coefficient Korteweg–de Vries equations. J. Math. Phys. 33, 4087–4102 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lie, S.: Über die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung. Arch. Math. 6, 328–368 (1881). Translation by N.H. Ibragimov: Lie S., On integration of a class of linear partial differential equations by means of definite integrals, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 2, pp. 473–508. CRC Press, Boca Raton (1994)

    MATH  Google Scholar 

  22. Lisle, I.G.: Equivalence transformations for classes of differential equations. PhD. thesis, University of British, Columbia (1992)

  23. Magadeev, B.A.: Group classification of nonlinear evolution equations. Algebra i Analiz 5, 141–156 (1993). (In Russian); English translation in St. Petersburg Math. J. 5, 345–359 (1994)

    MathSciNet  MATH  Google Scholar 

  24. Miller, W.: Symmetry and Separation of Variables. Addison-Wesley, Reading (1977)

    Google Scholar 

  25. Mostafazadeh, A.: A dynamical formulation of one-dimensional scattering theory and its applications in optics. Ann. Phys. 341, 77–85 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nattermann, P., Doebner, H.-D.: Gauge classification, Lie symmetries and integrability of a family of nonlinear Schrödinger equations. J. Nonlinear Math. Phys. 3, 302–310 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Niederer, U.: The maximal kinematical invariance group of the free Schrödinger equation. Helv. Phys. Acta 45, 802–810 (1972)

    MathSciNet  Google Scholar 

  28. Niederer, U.: The maximal kinematical invariance group of the harmonic oscillator. Helv. Phys. Acta 46, 191–200 (1973)

    Google Scholar 

  29. Niederer, U.: The group theoretical equivalence of the free particle, the harmonic oscillator and the free fall. In: Proceedings of the 2nd International Colloquium on Group Theoretical Methods in Physics, University of Nijmegen, The Netherlands (1973)

    Google Scholar 

  30. Niederer, U.: The maximal kinematical invariance groups of Schrödinger equations with arbitrary potentials. Helv. Phys. Acta 47, 167–172 (1974)

    MathSciNet  Google Scholar 

  31. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)

    Book  MATH  Google Scholar 

  32. Opanasenko, S., Bihlo, A., Popovych, R.O.: Group analysis of general Burgers–Korteweg–de Vries equations. J. Math. Phys. 58, 081511 (2017). arXiv:1703.06932

    Article  MathSciNet  MATH  Google Scholar 

  33. Ovsiannikov, L.V.: Group properties of nonlinear heat equation. Dokl. Akad. Nauk SSSR 125, 492–495 (1959). (In Russian)

    MathSciNet  Google Scholar 

  34. Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)

    MATH  Google Scholar 

  35. Ovsjannikov, L.V., Ibragimov, N.H.: Group analysis of the differential equations of mechanics. In: General Mechanics, vol. 2, pp. 5–52. Akad. Nauk SSSR Vsesojuz. Inst. Nauchn. i Tehn. Informacii, Moscow (1975). (In Russian)

    Google Scholar 

  36. Popovych, R.O., Ivanova, N.M.: New results on group classification of nonlinear diffusion–convection equations. J. Phys. A 37, 7547–7565 (2004). arXiv:math-ph/0306035

    Article  MathSciNet  MATH  Google Scholar 

  37. Popovych, R.O., Ivanova, N.M., Eshraghi, H.: Lie symmetries of (1+1)-dimensional cubic Schrödinger equation with potential. Pr. Inst. Mat. Nats. Akad. Nauk Ukr. 50, 219–224 (2004). arXiv:math-ph/0310039

    MATH  Google Scholar 

  38. Popovych, R.O., Ivanova, N.M., Eshragi, H.: Group classification of (1+1)-dimensional Schrödinger equations with potentials and power nonlinearities. J. Math. Phys. 45, 3049–3057 (2004). arXiv:math-ph/0311039

    Article  MathSciNet  MATH  Google Scholar 

  39. Popovych, R.O., Kunzinger, M., Eshragi, H.: Admissible transformations and normalized classes of non-linear Schrödinger equations. Acta Appl. Math. 109, 315–359 (2010). arXiv:math-ph/0611061

    Article  MathSciNet  MATH  Google Scholar 

  40. Popovych, R.O., Kunzinger, M., Ivanova, N.M.: Conservation laws and potential symmetries of linear parabolic equations. Acta Appl. Math. 100, 113–185 (2008). arXiv:0706.0443

    Article  MathSciNet  MATH  Google Scholar 

  41. Vaneeva, O.O., Popovych, R.O., Sophocleous, C.: Enhanced group analysis and exact solutions of variable coefficient semilinear diffusion equations with a power source. Acta Appl. Math. 106, 1–46 (2009). arXiv:0708.3457

    Article  MathSciNet  MATH  Google Scholar 

  42. Vaneeva, O.O., Popovych, R.O., Sophocleous, C.: Extended group analysis of variable coefficient reaction–diffusion equations with exponential nonlinearities. J. Math. Anal. Appl. 396, 225–242 (2012). arXiv:1111.5198

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhdanov, R., Roman, O.: On preliminary symmetry classification of nonlinear Schrödinger equations with some applications to Doebner–Goldin models. Rep. Math. Phys. 45, 273–291 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhdanov, R.Z., Lahno, V.I.: Group classification of heat conductivity equations with a nonlinear source. J. Phys. A 32, 7405–7418 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are pleased to thank Anatoly Nikitin, Olena Vaneeva and Vyacheslav Boyko for stimulating discussions.

Author information

Authors and Affiliations

  1. College of Science and Technology, University of Rwanda, P.O. Box 3900, Kigali, Rwanda

    Célestin Kurujyibwami

  2. Linköping University, 58183, Linköping, Sweden

    Célestin Kurujyibwami & Peter Basarab-Horwath

  3. Wolfgang Pauli Institute, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Roman O. Popovych

  4. Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs’ka Str., 01004, Kyiv, Ukraine

    Roman O. Popovych

Authors
  1. Célestin Kurujyibwami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Basarab-Horwath
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roman O. Popovych
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roman O. Popovych.

Additional information

The research of C.K. was supported by International Science Programme (ISP) in collaboration with East African Universities Mathematics Programme (EAUMP). The research of R.O.P. was supported by the Austrian Science Fund (FWF), project P25064

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kurujyibwami, C., Basarab-Horwath, P. & Popovych, R.O. Algebraic Method for Group Classification of (1+1)-Dimensional Linear Schrödinger Equations. Acta Appl Math 157, 171–203 (2018). https://doi.org/10.1007/s10440-018-0169-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-018-0169-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation