Skip to main content
SpringerLink
Log in

Finding non-local and contact/dynamical symmetries of Riccati chain

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

In this work, we present a new approach to find non-local symmetries and contact symmetries from the admitted Lie point symmetries of the considered system of nonlinear differential equations. By introducing a new function in both the numerator and denominator in the relation which relates the \(\lambda \)-symmetry function and the Lie point symmetry characteristics, we generate non-local symmetries as well as contact symmetries. To do so, we have to define another function \(g_3\) and then we identify two different cases, where the function \(g_3=0\) and \(g_3 \ne 0\). To validate the results, we consider the Ricatti chain as an example and find the non-local and contact symmetries admitted by the first four of the underlying equations. We also find the contact symmetries admitted by the well-known Mathews–Lakshmanan oscillator equation.

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

References

  1. P J Olver, Equivalence, invariants, and symmetry (Cambridge University Press, Cambridge, 1995)

    Book  MATH  Google Scholar 

  2. G W Bluman and S Kumei, Symmetries and differential equations (Springer-Verlag, New York, 1989)

    Book  MATH  Google Scholar 

  3. G W Bluman and S C Anco, Symmetries and integration methods for differential equations (Springer-Verlag, New York, 2002)

    MATH  Google Scholar 

  4. P E Hydon, Symmetry methods for differential equations: A beginnner’s guide (Cambridge University Press, Cambridge, 2000)

    Book  MATH  Google Scholar 

  5. N H Ibragimov, Elementary Lie group analysis and ordinary differential equations (Wiley, New York, 1999)

    MATH  Google Scholar 

  6. W H Steeb, Invertible point transformations and nonlinear differential equations (World Scientific, London, 1993)

    Book  MATH  Google Scholar 

  7. S Lie, Geometrie der Beruehrungstransformationen (Leipzig, Teubner, 1894); Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen (Teubner, Leipzig, 1912)

  8. R Gladwin Pradeep, V K Chandrasekar, R Mohanasubha, M Senthilvelan and M Lakshmanan, Commun. Nonlinear Sci. Numer. Simul. 36, 303 (2016)

    Article  ADS  Google Scholar 

  9. F Schwarz, J. Phys. A. Math. Gen. 16, L133 (1983)

    Article  ADS  Google Scholar 

  10. J M Cerveró and J Villarroel, J. Phys. A. Math. Gen. 17, 1777 (1984)

    Article  ADS  Google Scholar 

  11. A A Adam and F M Mahomed, Nonlinear Dynam. 30, 267 (2002)

    Article  Google Scholar 

  12. B Abraham-Shrauner and A Guo, J. Phys. A: Math. Gen. 25, 5597 (1992)

    Article  ADS  Google Scholar 

  13. R Gladwin Pradeep, V K Chandrasekar, M Senthilvelan and M Lakshmanan, J. Phys. A: Math. Theor. 44, 445201 (2011)

    Article  ADS  Google Scholar 

  14. K S Govinder and P G L Leach, J. Phys. A: Math. Gen. 30, 2055 (1997)

    Article  ADS  Google Scholar 

  15. P G L Leach and K Andriopoulos, Appl. Anal. Disc. Math. 1, 150 (2007)

    Article  Google Scholar 

  16. B Abraham-Shrauner, K S Govinder and P G L Leach, Phys. Lett. A 203, 169 (1995)

    Article  ADS  Google Scholar 

  17. B Abraham-Shrauner and K S Govinder, J. Nonlinear Math. Phys. 13, 612 (2006)

    Article  ADS  Google Scholar 

  18. B Abraham-Shrauner, J. Math. Phys. 34, 4809 (1993)

    Article  ADS  Google Scholar 

  19. R Mohanasubha, M I Sabiya Shakila and M Senthilvelan, Commun. Nonlinear Sci. Numer. Simul. 19, 799 (2014)

    Article  ADS  Google Scholar 

  20. K S Govinder and P G L Leach, J. Phys. A: Math. Gen. 28, 5349 (1995)

    Article  ADS  Google Scholar 

  21. M C Nucci and P G L Leach, J. Math. Anal. Appl. 251, 871 (2000)

    Article  Google Scholar 

  22. F M Mahomed and P G L Leach, Quaestiones Math. 8, 241 (1985)

    Article  Google Scholar 

  23. V K Chandrasekar, M Senthilvelan, A Kundu and M Lakshmanan, J. Phys. A: Math. Gen. 39, 9743 (2006)

    Article  ADS  Google Scholar 

  24. M L Gandarias, Theor. Math. Phys. 159, 778 (2009)

    Article  Google Scholar 

  25. M L Gandarias and M S Bruzon, J. Nonlinear Math. Phys. 18, 123 (2011)

    Article  ADS  Google Scholar 

  26. M S Bruzon, M L Gandarias and M Senthilvelan, Phys. Lett. A 375, 2985 (2011); J. Math. Phys. 53, 023512 (2012)

  27. C Muriel and J L Romero, J. Phys. A: Math. Theor. 42, 365207 (2009)

    Article  Google Scholar 

  28. C Muriel, J L Romero and A Ruiz, IMA J. Appl. Math. 82, 1061 (2017)

    Article  Google Scholar 

  29. C Muriel and J L Romero, SIGMA 8, 106 (2012)

    Google Scholar 

  30. J F Riccati, Actorum Eruditorum quae Lipsiae Publicantur Suppl. 8, 66 (1724)

    Google Scholar 

  31. F Schwabl, Quantum mechanics (Springer, Berlin, 1992)

    Book  MATH  Google Scholar 

  32. A Khare and U Sukhatme, Supersymmetry in quantum mechanics (World Scientific, Singapore, 2001)

    MATH  Google Scholar 

  33. J J Peña, A R Ponce and J Morales, J. Phys.: Conf. Ser. 738, 012095 (2016)

    Google Scholar 

  34. T Breiten, S Dolgov and M Stoll, Numer. Algebra Control. Optim. 11, 407 (2021)

    Article  Google Scholar 

  35. A A Bastami, M R Belić and N Z Petrović, Electron. J. Diff. Equ. 66, 1 (2010)

    Google Scholar 

  36. M A Lohe, J. Math. Phys. 60, 072701 (2019)

    Article  ADS  Google Scholar 

  37. F V Vergés and M D Fragoso, 60th IEEE Conference on Decision and Control (CDC) (2021) pp. 3936–3941

  38. G Darboux, Bull. Sci. Math. 2, 60 (1878); 123 (1878); 151 (1878)

  39. V K Chandrasekar, M Senthilvelan and M Lakshmanan, Proc. R. Soc. London A 461, 2451 (2005)

    ADS  Google Scholar 

  40. M Prelle and M Singer, Trans. Am. Math. Soc. 279, 215 (1983)

    Article  Google Scholar 

  41. L G S Duarte, S E S Duarte and L A C P da Mota, J. Phys. A: Math. Gen. 35, 1001 (2002)

    Article  ADS  Google Scholar 

  42. J F Carinena, J. Phys.: Conf. Ser. 175, 012009 (2009)

  43. C Muriel and M C Nucci, Open Commun. Nonlinear Math. Phys. 1, 41 (2021)

    Article  Google Scholar 

  44. M S Bruzon, M L Gandarias and M Senthilvelan, J. Math. Phys. 53, 023512 (2012)

    Article  ADS  Google Scholar 

  45. C Muriel and J L Romero, Nonlinear Anal. Real World Appl. 16, 191 (2014)

    Article  Google Scholar 

  46. M Senthilvelan, R Mohanasubha and M Lakshmanan, Pramana – J. Phys. 85, 755 (2015)

    Google Scholar 

  47. J F Cariñena, P Guha and M F Rañada, Nonlinearity 22, 2953 (2009)

    Article  ADS  Google Scholar 

  48. S N Pandey, P S Bindu, M Senthilvelan and M Lakshmanan, J. Math. Phys. 50, 102701 (2009)

    Article  ADS  Google Scholar 

  49. A Bhuvaneswari, R A Kraenkel and M Senthilvelan, Nonlinear Anal. Real World Appl. 13, 1102 (2012)

    Article  Google Scholar 

  50. J Chazy, Acta Math. 34, 317 (1911)

    Article  Google Scholar 

  51. P M Mathews and M Lakshmanan, Quart. Appl. Math. 32, 215 (1974)

    Article  Google Scholar 

  52. P M Mathews and M Lakshmanan, Nuovo Cimento A 26, 299 (1975)

    Article  ADS  Google Scholar 

  53. M Lakshmanan and K Eswaran, J. Phys. A: Math. Gen. 8, 1658 (1975)

    Article  ADS  Google Scholar 

  54. P W Higg, J. Phys. A: Math. Gen. 12, 309 (1979)

    Article  ADS  Google Scholar 

  55. H I Leemon, J. Phys. A: Math. Gen. 12, 489 (1979)

    Article  ADS  Google Scholar 

  56. R Mohanasubha and M Senthilvelan, Commun. Nonlinear Sci. Numer. Simul. 43, 111 (2017)

    Article  ADS  Google Scholar 

  57. A Bhuvaneswari, V K Chandrasekar, M Senthilvelan and M Lakshmanan, J. Math. Phys. 53, 073504 (2012)

    Article  ADS  Google Scholar 

  58. R Mohanasubha, V K Chandrasekar, M Senthilvelan and M Lakshmanan, Proc. R. Soc. A 471, 20140720 (2015)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

RMS is funded by the Centre for Computational Modeling, Chennai Institute of Technology, India, vide funding number CIT/CCM/2022/RP-005. VKC thanks DST, New Delhi for computational facilities under the DST-FIST Programme (Grant No. SR/FST/PS-1/2020/135). The work of VKC is also supported by SERB-DST-MATRICS (Grant No. MTR/2018/000676) and DST-CRG project (Grant No. CRG/2020/004353). The work of MS forms part of a research project sponsored by NBHM, Government of India, under the Grant No. 02011/20/2018 NBHM (R.P)/R &DII/15064. The work of ML is supported by a DST-SERB National Science Chair.

Author information

Authors and Affiliations

  1. Centre for Computational Modeling, Chennai Institute of Technology, Chennai , 600 069, India

    R Mohanasubha

  2. Department of Physics, Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA University, Thanjavur, 613 401, India

    V K Chandrasekar

  3. Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli, 620 024, India

    M Senthilvelan & M Lakshmanan

Authors
  1. R Mohanasubha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V K Chandrasekar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M Senthilvelan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M Lakshmanan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V K Chandrasekar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mohanasubha, R., Chandrasekar, V.K., Senthilvelan, M. et al. Finding non-local and contact/dynamical symmetries of Riccati chain. Pramana - J Phys 97, 30 (2023). https://doi.org/10.1007/s12043-022-02496-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-022-02496-8

Keywords

PACS

Search

Navigation