91
Views
4
CrossRef citations to date
0
Altmetric
Original Articles

Discretization of Liouville type nonautonomous equations preserving integrals

&
Pages 620-642 | Received 19 Jul 2016, Accepted 25 Sep 2016, Published online: 14 Oct 2016

References

  • V. E. Adler, On a discrete analog of the Tzitzeica equation (arXiv:1103.5139).
  • M. Bruschi, D. Levi, and O. Ragnisco, Discrete version of the nonlinear Schrdinger equation with linearlyx-dependent coefficients, Il Nuovo Cimento A Series 11 53(1) (1979) 21–30. doi: 10.1007/BF02776479
  • N. F. Gareeva and A. V. Zhiber, The second order integrals of the hyperbolic equations and evolutionary equations, in Proceedings of the International Conference ”Algebraic and analytic methods in the theory of the differential equations” (1996) Orel, edited by A.G. Meshkov, 39–42.
  • R. N. Garifullin and R. I. Yamilov, Generalized symmetry classification of discrete equations of a class depending on twelve parameters, Journal of Physics A: Mathematical and Theoretical 45.34 (2012) 345205. doi: 10.1088/1751-8113/45/34/345205
  • J. Gibbons and B. A. Kupershmidt, Time discretizations of lattice integrable systems, Physics Letters A 165.2 (1992) 105–110. doi: 10.1016/0375-9601(92)90084-Y
  • E. Goursat, Recherches sur quelques équations aux dérivés partielles du second ordre, Annales de la faculté des Sciences de l’Université de Toulouse 2e série, 1:1 (1899) 3178.
  • I. T. Habibullin, N. Zheltukhina, and A. Sakieva, Discretization of hyperbolic type Darboux integrable equations preserving integrability, J. Math. Phys. 52 (2011) 093507. doi: 10.1063/1.3628587
  • I. Habibullin, I. N. Zheltukhina and A. Pekcan, On the classification of Darboux integrable chains, J. Math. Phys. 49 (2008) 102702. doi: 10.1063/1.2992950
  • I. Habibullin, N. Zheltukhina and A. Pekcan, Complete list of Darboux integrable chains of the form t1x = tx + d(t,t1), J. Math. Phys. 50 (2009) 102710. doi: 10.1063/1.3251334
  • R. Hirota and K. Kimura, Discretization of the Euler top, Journal of the Physical Society of Japan 69 (2000) 627–630. doi: 10.1143/JPSJ.69.627
  • Nail H. Ibragimov, Transformation groups applied to mathematical physics, Vol. 3 Springer, 2001.
  • O. V. Kaptsov, On the Goursat classification problem, Programming and Computer Software 38 (2) 102–104. doi: 10.1134/S0361768812020041
  • M. E. Lainé, Sur une équation de la forme s = pϕ (x; y; z; q) integrable par la méthode de Darboux, Comptes rendus, 183 (1926) 1254–1256.
  • J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Communications in Mathematical Physics 139.2 (1991) 217–243. doi: 10.1007/BF02352494
  • M. Murata, et al., How to discretize differential systems in a systematic way, Journal of Physics A: Mathematical and Theoretical 43.31 (2010) 315203. doi: 10.1088/1751-8113/43/31/315203
  • Alan C. Newell, Solitons in mathematics and physics, Philadelphia: Society for Industrial and applied Mathematics, 1985.
  • Y. B. Suris, The problem of integrable discretization: Hamiltonian approach, 219 Springer, 2003.
  • E. Vessiot, Sur les équations aux dérivés partielles du second order, F(x; y; z; p; q; r; s;t) = 0, integrable par la méthode de Darboux, J. Math. pure appl. 18 (1939) 1–61.
  • D. Zakharov, A discrete analogue of the modified Novikov-Veselov hierarchy, (arXiv.org.nlin.arXiv:0904.3728v2).
  • K. Zheltukhin and N. Zheltukhina, On existence of an x - integral for a semi-discrete chain of hyperbolic type, Journal of Physics Conference Series 670(1) (2016) 012055.
  • K. Zheltukhin and N. Zheltukhina, Semi-discrete hyperbolic equations admitting five dimensional characteristic x-ring, Journal of Nonlinear Mathematical Physics 23(3) (2016) 351–367. doi: 10.1080/14029251.2016.1199497
  • A. V. Zhiber, R.D. Murtazina, I. T Habibullin and A. B. Shabat, Characteristic Lie rings and nonlinear integrable equation, M.Izhevsk (2012) 3–76 ( in Russian).
  • A. V. Zhiber and V. V. Sokolov, Exactly integrable hyperbolic equations of Liouville type, Russian Mathematical Surveys 56(1) (2001) 61–101. doi: 10.1070/RM2001v056n01ABEH000357

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.