Skip to main content
SpringerLink
Log in

Complex Methods for Lie Symmetry Analysis

  • Chapter
  • First Online:
Symmetries and Applications of Differential Equations

Part of the book series: Nonlinear Physical Science ((NPS))

  • 519 Accesses

Abstract

When Lie developed symmetry analysis, he took the equations to be defined in the complex domain but did not explicitly use the entailed complex analyticity. Making it explicit necessitates the incorporation of the Cauchy–Riemann equations into the original system of equations, which modifies the symmetries of the system. This point was followed up by us, and some of our students, in a series of papers (and theses). It was found that complex methods, when they are applicable, provide more powerful tools for obtaining solutions and integrals of differential equations, even enabling us to find solutions of systems of differential equations that possess no symmetries. In this chapter we review the methods developed and then pose the crucial question that was begged in saying “when they are applicable.” When would they be applicable and why, or how, does the complex method work? We indicate some lines to pursue to try to find the answers, or at least partial answers, to these questions.

Dedicated to the memory of one of the most innovative workers in the field of Symmetry Analysis, after Sophus Lie, Nail Hairullovich Ibragimov, who initiated many new methods for using Lie Analysis to deal with differential equations.

FM is Visiting Professor at UNSW for 2020.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Lie S (1967) Differential equations. Chelsea, New York

    Google Scholar 

  2. Lie S (1883) Klassification und integration von gewönlichen differentialgleichungen zwischen x, y, die eine gruppe von transformationen gestaten. Arch Math

    Google Scholar 

  3. Lie S (1891) Lectures on differential equations with known infinitesimal transformations. Leipzig, Teubner (in German, Lie’s lectures by Sheffers G)

    Google Scholar 

  4. Lie S (1893) Theorie der Transformationsgruppen. III. Teubner, Leipzig

    Google Scholar 

  5. Ibragimov NH (1999) Elementary Lie group analysis and ordinary differential equations. Wiley, Chichester

    Google Scholar 

  6. Stephani H (1996) Differential equations: their solutions using symmetries. Cambridge University Press, Cambridge

    Google Scholar 

  7. Tressé A (1894) Sur les invariants differentiels des groupes continus de transformations. Acta Math 18:1

    Google Scholar 

  8. Chern SS (1937) Sur la geometrie d’une equation differentielle du troiseme orde. CR Acad Sci Paris, 1227; (1940) The geometry of the differential equation \(y^{\prime \prime \prime } = F(x,y,y^{\prime },y^{\prime \prime })\). Sci Rep Nat Tsing Hua Univ. 4, 97

    Google Scholar 

  9. Mahomed FM, Leach PGL (1990) Symmetry Lie algebra of nth. order ordinary differential equations. J Math Anal Appl 151:80

    Google Scholar 

  10. Grebot G (1997) The linearization of third order ODEs; the characterization of third order ordinary differential equations admitting a transitive fibre-preserving point symmetry group. J Math Anal Appl 206:364

    Google Scholar 

  11. Wafo Soh C, Mahomed FM (2001) Linearization criteria for a system of second-order ordinary differential equations. Int J Nonlinear Mech 36:671

    Google Scholar 

  12. Gorringe VM, Leach PGL (1988) Lie point symmetries for systems of \(2\)nd order linear ordinary differential equations. Quaestiones Mathematicae 11:95

    Google Scholar 

  13. Wafo Soh C, Mahomed FM (2000) Symmetry Breaking for a System of Two Linear Second-Order Ordinary Differential Equation. Nonlinear Dyn 22:121

    Google Scholar 

  14. Qadir A (2020) Einstein’s general theory of relativity. Cambridge Scholars Publishers, Cambridge

    Google Scholar 

  15. Aminova AV, Aminov NA-M (2000) Projective geometry of systems of differential equations: general conception. Tensor N S 62:65; (2006) Projective geometry of systems of second-order differential equations. Sbornik: Math 197:951

    Google Scholar 

  16. Feroze T, Mahomed FM, Qadir A (2006) The connection between isometries and symmetries of geodesic equations of the underlying spaces. Nonlinear Dyn 45:65

    Google Scholar 

  17. Mahomed FM, Qadir A (2007) Linearization criteria for a system of second order quadratically semi-linear ordinary differential equations. Nonlinear Dyn 48:417

    Article  MathSciNet  Google Scholar 

  18. Bokhari AH, Qadir A (1985) A prescription for \(n\)-dimensional vierbeins. ZAMP 36:184

    MathSciNet  MATH  Google Scholar 

  19. Mahomed FM, Qadir A (2009) Invariant linearization criteria for systems of cubically semi-linear second order ordinary differential equations. J Nonlin Math Phys 16:1

    MATH  Google Scholar 

  20. Fredericks E, Mahomed FM, Momonia E, Qadir A (2008) Constructing a space from the system of geodesic equations. Comp Phys Commun 179:438

    Article  Google Scholar 

  21. Ali S (2009) Complex symmetry analysis. PhD thesis, NUST CAMP

    Google Scholar 

  22. Ali S, Mahomed FM, Qadir A (2009) Complex Lie symmetries for scalar second-order ordinary differential equations. Nonlinear Anal: Real World App 10:3335

    Article  MathSciNet  Google Scholar 

  23. Safdar M (2003) Solvability of differential equations by complex symmetry analysis. PhD Thesis, NUST CAMP

    Google Scholar 

  24. Safdar M, Qadir A, Ali S (2011) Inequivalence of classes of linearizable systems of cubically semi linear ordinary differential equations obtained by real and complex symmetry analysis. Math Comp Appl 16:923

    MATH  Google Scholar 

  25. Safdar M, Qadir A, Ali S (2011) Linearizability of systems of ordinary differential equations obtained by complex symmetry analysis. Math Problems Eng 2011:171834

    Google Scholar 

  26. Ali S, Safdar M, Qadir A (2014) Linearization from complex Lie point transformations. J Appl Math 2014:793247

    Google Scholar 

  27. Dutt HM, Safdar M (2015) Linearization of two-dimensional complex linearizable system of second order ordinary differential equations. Appl Math Sci 9:2889

    Google Scholar 

  28. Noether E (1918) Invariante Variationsprobleme Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse. 1918:235

    Google Scholar 

  29. Kosmann-Schwarzbach Y (2010) The Noether theorems: invariance and conservation laws in the twentieth century. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, Berlin

    Google Scholar 

  30. Kara AH, Vawda FE, Mahomed FM (1994) Symmetries of first integrals and solutions of differential equations. Lie Groups Appl 1:27

    MathSciNet  MATH  Google Scholar 

  31. Bokhari AH, Al-Dweik AY, Zaman FD, Kara AH, Mahomed FM (2010) Generalization of the double reduction theory. Nonlinear Analysis B 11:3763

    Article  MathSciNet  Google Scholar 

  32. Ali S, Mahomed FM, Qadir A (2008) Complex Lie symmetries for variational problems. J Nonlin Math Phys 25:25

    Article  MathSciNet  Google Scholar 

  33. Zimmerman AH (1987) Generalized integral transformations. Dover Publications, Mineola

    Google Scholar 

  34. Tassaddiq A (2012) Some representations of the extended Fermi-Dirac and Bose-Einstein functions with application. NUST CAMP

    Google Scholar 

  35. Tassaddiq A, Qadir A (2011) Fourier transform and distributional representation of the generalized gamma function with some applications. Appl Math Comp 218:1084

    Article  MathSciNet  Google Scholar 

  36. Bender CM, Boetcher S (1998) Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys Rev Lett 80:5243

    Article  MathSciNet  Google Scholar 

  37. Safdar M, Qadir A, Farooq MU (2019) Comparison of Noether symmetries and first integrals of two-dimensional systems of second order ordinary differential equations by real and complex methods. Symmetry 11:1180

    Article  Google Scholar 

  38. Farooq MU, Ali S, Qadir A (2010) Invariants of two-dimensional systems via complex Lagrangians with applications. Commun Nonlin Sci Num Sim 16:1804

    Article  MathSciNet  Google Scholar 

  39. Kara AH, Mahomed FM (2006) Noether-type symmetries and conservation laws via partial Lagrangians. Nonlinear Dyn 45:367

    Article  MathSciNet  Google Scholar 

  40. Qadir A, Mahomed FM (2015) Higher dimensional systems of differential equations obtainable by iterative use of complex methods. Int J Mod Phys (Conf Ser) 38:1560077. In: Ali S, Mahomed FM, Qadir A (eds)

    Google Scholar 

  41. Leach PGL, Maartens R, Maharaj SD (1992) Self-similar solutions of the generalized Emden-Fowler equation. Int J Non-Linear Mech 27:583

    Article  MathSciNet  Google Scholar 

  42. Gonzalez-Gascon F, Gonzalez-Lopez A (1988) Newtonian systems of differential equations, integrable via quadratures, with trivial group of point symmetry. Phys Lett A 129:153

    Article  MathSciNet  Google Scholar 

  43. Mahomed KS, Momoniat E (2014) Symmetry classification of first integrals for scalar dynamical equations. Int J Non-Linear Mech 59:52

    Article  Google Scholar 

  44. Safdar M, Ali S, Mahomed FM (2011) Linearization of systems of four second order ordinary differential equations. Pramana - J Phys 77:171834

    Google Scholar 

  45. Dutt HM, Safdar M, Qadir A (2019) Linearization criteria for two dimensional systems of third order ordinary differential equations by complex approach. Arabian J Math 8:163

    Article  MathSciNet  Google Scholar 

  46. Dutt HM, Safdar M, Qadir A (2018) Classification of two-dimensional linearizable systems of third order ordinary differential equations by complex methods. In: Aslam MJ, Saifullah K (eds) Proceedings of the \(14^{th}\) regional conference in mathematical physics, pp 278–283, 2015. World Scientific

    Google Scholar 

  47. Dutt HM, Qadir A (2014) Reduction of fourth order ordinary differential equations to second and third Lie linearizable forms. Comm Nonlin Sci Numerical Sim 19:2653

    Article  MathSciNet  Google Scholar 

  48. Dutt HM, Qadir A (2017) Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations. Quaestiones Mathematicae 2017:1

    MATH  Google Scholar 

  49. Dutt HM, Qadir A (2018) Classification of scalar fourth order ordinary differential equations linearizable via generalized Lie-Bäcklund transformations. In: Kac VG, Olver PJ, Winternitz P, Özer T (eds)Proceedings of the symmetries, differential equations and applications III, pp 67–74. Springer Proceedings in Mathematics & Statistics

    Google Scholar 

Download references

Acknowledgements

We are grateful to Rimsha Khalil for pointing out some typographical errors in the chapter. FMM is grateful to the NRF of South Africa for support.

Author information

Authors and Affiliations

  1. Abdus Salam School for Mathematical Sciences, Government College University, Lahore, Pakistan

    Asghar Qadir

  2. DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

    Fazal M. Mahomed

  3. School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW, 2052, Australia

    Fazal M. Mahomed

Authors
  1. Asghar Qadir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fazal M. Mahomed
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mechanical and Mechatronics Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, USA

    Albert C. J. Luo

  2. Department of High Performance Computing, Ufa State Aviation Technical University, Ufa, Russia

    Rafail K. Gazizov

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Higher Education Press

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Qadir, A., Mahomed, F.M. (2021). Complex Methods for Lie Symmetry Analysis. In: Luo, A.C.J., Gazizov, R.K. (eds) Symmetries and Applications of Differential Equations. Nonlinear Physical Science. Springer, Singapore. https://doi.org/10.1007/978-981-16-4683-6_4

Download citation

Publish with us

Policies and ethics

Search

Navigation