Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 070, 17 pages      arXiv:1005.5288     http://dx.doi.org/10.3842/SIGMA.2010.070

C-Integrability Test for Discrete Equations via Multiple Scale Expansions

Christian Scimiterna and Decio Levi
Dipartimento di Ingegneria Elettronica, Università degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy

Received May 29, 2010, in final form August 20, 2010; Published online August 31, 2010

Abstract
In this paper we are extending the well known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential difference equation. In this case the equation satisfies the A1, A2 and A3 linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable.

Key words: linearizable discrete equations; linearizability theorem; multiple scale expansion; obstructions to linearizability; discrete Burgers.

pdf (326 kb)   ps (205 kb)   tex (20 kb)

References

  1. Agrotis M., Lafortune S., Kevrekidis P.G., On a discrete version of the Korteweg-de Vries equation, Discrete Contin. Dyn. Syst. (2005), suppl., 22-29.
  2. Burgers J.M., A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948), 171-199.
  3. Calogero F., Why are certain nonlinear PDEs both widely applicable and integrable?, in What is Integrability?, Editor V.E. Zakharov, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, 1-62.
  4. Calogero F., Eckhaus W., Necessary conditions for integrability of nonlinear PDEs, Inverse Problems 3 (1987), L27-L32.
    Calogero F., Eckhaus W., Nonlinear evolution equations, rescalings, model PDEs and their integrability. I, Inverse Problems 3 (1987), 229-262.
    Calogero F., Eckhaus W., Nonlinear evolution equations, rescalings, model PDEs and their integrability. II, Inverse Problems 4 (1987), 11-33.
    Calogero F., Degasperis A., Ji X-D., Nonlinear Schrödinger-type equations from multiple scale reduction of PDEs. I. Systematic derivation, J. Math. Phys. 41 (2000), 6399-6443.
    Calogero F., Degasperis A., Ji X-D., Nonlinear Schrödinger-type equations from multiple scale reduction of PDEs. II. Necessary conditions of integrability for real PDEs, J. Math. Phys. 42 (2001), 2635-2652.
    Calogero F., Maccari A., Equations of nonlinear Schrödinger type in 1+1 and 2+1 dimensions obtained from integrable PDEs, in Inverse Problems: an Interdisciplinary Study (Montpellier, 1986), Adv. Electron. Electron Phys., Suppl. 19, Editors C.P. Sabatier, Academic Press, London, 1987, 463-480.
  5. Cole J.D., On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225-236.
  6. Degasperis A., Private communication.
  7. Degasperis A., Holm D.D., Hone A.N.I., A new integrable equation with peakon solutions, Teoret. Mat. Fiz. 133 (2002), 170-183 (English transl.: Theoret. and Math. Phys. 133 (2002), 1463-1474).
  8. Degasperis A., Manakov S.V., Santini P.M., Multiple-scale perturbation beyond the nonlinear Schrödinger equation. I, Phys. D 100 (1997), 187-211.
  9. Degasperis A., Procesi M., Asymptotic integrability, in Symmetry and Perturbation Theory, SPT98 (Rome, 1998), Editors A. Degasperis and G. Gaeta, World Sci. Publ., River Edge, NJ, 1999, 23-37.
    Degasperis A., Multiscale expansion and integrability of dispersive wave equations, in Integrability, Editor A.V. Mikhailov, Springer, Berlin, 2009, 215-244.
  10. Hernandez Heredero R., Levi D., Petrera M., Scimiterna C., Multiscale expansion of the lattice potential KdV equation on functions of an infinite slow-varyness order, J. Phys. A: Math. Theor. 40 (2007), F831-F840, arXiv:0706.1046.
  11. Hernandez Heredero R., Levi D., Petrera M., Scimiterna C., Multiscale expansion on the lattice and integrability of partial difference equations, J. Phys. A: Math. Theor. 41 (2008), 315208, 12 pages, arXiv:0710.5299.
  12. Hernandez Heredero R., Levi D., Petrera M., Scimiterna C., Multiscale expansion and integrability properties of the lattice potential KdV equation, J. Nonlinear Math. Phys. 15 (2008), suppl. 3, 323-333, arXiv:0709.3704.
  13. Hietarinta J., Viallet C., Singularity confinement and chaos in discrete systems, Phys. Rev. Lett. 81 (1998), 325-328, solv-int/9711014.
  14. Hopf E., The partial differential equation ut + uux = uxx, Comm. Pure Appl. Math. 3 (1950), 201-230.
  15. Kodama Y., Mikhailov A.V., Obstacles to asymptotic integrability, in Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl., Vol. 26, Birkhäuser Boston, Boston, MA, 1997, 173-204.
    Hiraoka Y., Kodama Y., Normal form and solitons, in Integrability, Editor A.V. Mikhailov, Springer, Berlin, 2009, 175-214, nlin.SI/0206021.
  16. Leon J., Manna M., Multiscale analysis of discrete nonlinear evolution equations, J. Phys. A: Math. Gen. 32 (1999), 2845-2869, solv-int/9902005.
  17. Levi D., Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV, J. Phys. A: Math. Gen. 38 (2005), 7677-7689, nlin.SI/0505061.
  18. Levi D., Hernandez Heredero R., Multiscale analysis of discrete nonlinear evolution equations: the reduction of the dNLS, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 440-448.
  19. Levi D., Petrera M., Discrete reductive perturbation technique, J. Math. Phys. 47 (2006), 043509, 20 pages, math-ph/0510084.
  20. Levi D., Petrera M., Continuous symmetries of the lattice potential KdV equation, J. Phys. A: Math. Theor. 40 (2007), 4141-4159, math-ph/0701079.
  21. Levi D., Ragnisco O., Bruschi M., Continuous and discrete matrix Burgers' hierarchies, Nuovo Cimento B (11) 74 (1983), 33-51.
  22. Levi D., Scimiterna C., The Kundu-Eckhaus equation and its discretizations, J. Phys. A: Math. Theor. 42 (2009), 465203, 8 pages, arXiv:0904.4844.
  23. Ramani A., Grammaticos B., Tamizhmani K.M., Painlevé analysis and singularity confinement: the ultimate conjecture, J. Phys. A: Math. Gen. 26 (1993), L53-L58.
  24. Santini P.M., The multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test. I, J. Phys. A: Math. Theor. 43 (2010), 045209, 27 pages, arXiv:0908.1492.
  25. Scimiterna C., Multiscale techniques for nonlinear difference equations, Ph.D. Thesis, Roma Tre University, 2009.
  26. Schoombie S.W., A discrete multiscales analysis of a discrete version of the Korteweg-de Vries equation, J. Comp. Phys. 101 (1992), 55-70.
  27. Whitham G.B., Linear and nonlinear waves, Wiley-Interscience, New York, 1974.

Previous article   Next article   Contents of Volume 6 (2010)