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On the Partial Preliminary Group Classification of the Nonlinear Five-Dimensional D’Alembert Equation

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We apply nonconjugate subgroups of the Poіncaré group P(1,4) to perform a partial preliminary group classification of the nonlinear five-dimensional d’Alembert equation. We also present a brief survey of the results obtained.

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References

  1. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Appl. Math., Vol. 4, Philadelphia (1981).

  2. B. M. Barbashov and V. V. Nesterenko, Introduction to the Relativistic String Theory, World Sci., Singapore (1990).

    Book  Google Scholar 

  3. P. L. Bhatnagar, Nonlinear Waves in One-Dimensional Dispersive Systems, Clarendon, Oxford (1979).

    MATH  Google Scholar 

  4. V. A. Dorodnitsyn, I. V. Knyazeva, and S. R. Svirshchevskii, “Group properties of heat conduction equation with a source in the two-dimensional and three-dimensional cases,” Differents. Uravn., 19, No. 7, 1215–1224 (1983).

    MathSciNet  Google Scholar 

  5. S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Consultants Bureau, New York (1984).

    Google Scholar 

  6. V. G. Kadyshevskii, “A new approach to the theory of electromagnetic interactions,” Fiz. Element. Chastits Atom. Yadra, 11, No. 1, 5–39 (1980).

    MathSciNet  Google Scholar 

  7. L. V. Ovsyannikov, “Group properties of the nonlinear heat-conduction equation,” Dokl. Akad. Nauk SSSR, 125, No. 3, 492–495 (1959).

    MathSciNet  MATH  Google Scholar 

  8. R. O. Popovych and I. A. Ehorchenko, “Group classification of generalized eikonal equations,” Ukr. Mat. Zh., 53, No. 11, 1513–1520 (2001); English translation : Ukr. Math. J., 53, No. 11, 1841–1850 (2001).

    Article  Google Scholar 

  9. V. M. Fedorchuk, “Nonsplit subalgebras of the Lie algebra of the generalized Poincaré group P(1,4) ,” Ukr. Mat. Zh., 33, No. 5, 696–700 (1981); English translation : Ukr. Math. J., 33, No. 5, 535–538 (1981).

    Article  MathSciNet  Google Scholar 

  10. V. M. Fedorchuk, “Splitting subalgebras of the Lie algebra of the generalized Poincaré group P(1,4) ,” Ukr. Mat. Zh., 31, No. 6, 717–722 (1979); English translation : Ukr. Math. J., 31, No. 6, 554–558 (1979).

    Article  MathSciNet  Google Scholar 

  11. V. M. Fedorchuk and V. I. Fedorchuk, “On the functional bases of the first-order differential invariants of continuous subgroups of the Poincaré group P(1,4) ,” Mat. Metody Fiz.-Mekh. Polya, 48, No. 4, 51–58 (2005).

    MathSciNet  MATH  Google Scholar 

  12. W. I. Fushchych and A. G. Nikitin, Symmetries of the Equations of Quantum Mechanics, Allerton Press, New York (1994).

    Google Scholar 

  13. S. V. Khabirov, “A property of defining equations for the Lie algebra in the group classification problem for wave equations,” Sib. Mat. Zh., 50, No. 3, 647–668 (2009); English translation: Sib. Math. J., 50, No. 3, 515–532 (2009).

    Article  MathSciNet  Google Scholar 

  14. W. F. Ames, R. J. Lohner, and E., Adams, “Group properties of u tt = [F(u)u x ] x ,” Int. J. Non-Linear Mech., 16, No. 5-6, 439–447 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Andriopoulos, S. Dimas, P. G. L. Leach, and D. Tsoubelis, “On the systematic approach to the classification of differential equations by group theoretical methods,” J. Comput. Appl. Math., 230, No. 1, 224–232 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. M. Fedorchuk and V. I. Fedorchuk, “On first-order differential invariants of the non-conjugate subgroups of the Poincare group P(1,4) ,” in: Differential Geometry and its Applications: Proc. 10th Int. Conf. on DGA 2007, in Honour of Leonhard Euler (Olomouc, Czech Republic, Aug. 27–31, 2007), World Sci. Publ. (2008), pp. 431–444.

  17. W. I. Fushchich and R. M. Cherniha, “The Galilean relativistic principle and nonlinear partial differential equations,” J. Phys. A: Math. Gen., 18, No. 18, 3491–3503 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  18. W. I. Fushchich and N. I. Serov, “The symmetry and some exact solutions of the nonlinear many-dimensional Liouville, d’Alembert and eikonal equations,” J. Phys. A: Math. Gen., 16, 3645 –3658 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  19. W. I. Fushchіch, A. F. Barannіk, L. F. Barannіk, and V. M. Fedorchuk, “Continuous subgroups of the Poіncaré group P(1,4) ,” J. Phys. A: Math. Gen., 18, No. 15, 2893–2899 (1985).

    Article  Google Scholar 

  20. M. L. Gandarias and N. H. Ibragimov, “Equivalence group of a fourth-order evolution equation unifying various non-linear models,” Commun. Nonlinear Sci. Numer. Simul., 13, No. 2, 259–268 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. M. Grundland, J. Harnad, and P. Winternitz, “Symmetry reduction for nonlinear relativistically invariant equations,” J. Math. Phys., 25, No. 4, 791–806 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. M. Grundland, J. A. Tuszyński, and P. Winternitz, “Applications of the three-dimensional “ φ6 ”-model to structural phase transitions,” in: Proc. XV Int. Colloq. on Group Theoretical Methods in Physics (Philadelphia, PA, 1986), World Sci. Publ., Teaneck, NJ (1987), pp. 589–601.

  23. A. M. Grundland, J. A. Tuszyński, and P. Winternitz, “Group theory and solutions of classical field theories with polynomial nonlinearities,” Found. Phys., 23, No. 4, 633–665 (1993).

    Article  MathSciNet  Google Scholar 

  24. H. R. Hernández and P. J. Olver, “Classification of invariant wave equations,” J. Math. Phys., 37, No. 12, 6414–6438 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Juráš, “Some classification results for hyperbolic equations F(x, y,u,u x ,u y ,u xx , u xy ,u yy ) = 0 ,” J. Differ. Equations, 164, No. 2, 296–320 (2000).

    Article  MATH  Google Scholar 

  26. S. Lie, “Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen”, Arch. Math., 6, No. 3, 328–368 (1881).

    MATH  Google Scholar 

  27. S. V. Meleshko and S. Moyo, “On the complete group classification of the reaction–diffusion equation with a delay,” J. Math. Anal. Appl., 338, No. 1, 448–466 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  28. A. G. Nikitin, “Group classification of systems and non-linear reaction–diffusion equations,” Ukr. Mat. Visnyk, 2, No. 2, 149–200 (2005).

    MathSciNet  MATH  Google Scholar 

  29. P. J. Olver, “Differential invariants and invariant differential equations,” in: Modern Group Analysis V: Theory and Appl. in Math. Model: Proc. Int. Conf. on Modern Group Analysis V (Johannesburg, South Africa, Jan. 16–22, 1994), Vol. 1, Issue 1: Lie Groups and Their Applications (1994), pp. 177–192.

  30. E. Pucci and M. C. Salvatori, “Group properties of a class of semilinear hyperbolic equations,” Int. J. Non-Linear Mech., 21, No. 2, 147–155 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Rideau and P. Winternitz, “Evolution equations invariant under two-dimensional space-time Schrödinger group,” J. Math. Phys., 34, No. 2, 558–570 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  32. M. B. Sheftel and A. A. Malykh, “On classification of second-order PDEs possessing partner symmetries,” J. Phys. A: Math. Gen., 42, No. 39, 1–20 (2009).

    Article  MathSciNet  Google Scholar 

  33. C. Sophocleous and R. Tracinà, “Differential invariants for quasi-linear and semi-linear wave-type equations,” Appl. Math. Comput., 202, No. 1, 216–228 (2008).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv, Ukraine

    V. I. Fedorchuk

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  1. V. I. Fedorchuk
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 55, No. 3, pp. 35–43, July–September, 2012.

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Fedorchuk, V.I. On the Partial Preliminary Group Classification of the Nonlinear Five-Dimensional D’Alembert Equation. J Math Sci 194, 166–175 (2013). https://doi.org/10.1007/s10958-013-1517-0

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