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

SIGMA 7 (2011), 079, 24 pages      arXiv:1108.3648
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf-Cole Transformations

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

Received April 15, 2011, in final form August 11, 2011; Published online August 18, 2011

In this paper we propose some linearizability tests of partial difference equations on a quad-graph given by one point, two points and generalized Hopf-Cole transformations. We apply the so obtained tests to a set of nontrivial examples.

Key words: quad-graph equations; linearizability; point transformations; Hopf-Cole transformations.

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  1. Abel N.H., Methode generale pour trouver des fonctions d'une seule quantite variable lorsqu'une propriete de ces fonctions est exprimee par une equation entre deux variables, Mag. Naturvidenskab. 1 (1823), 1-10, reproduced in Ouvres Completes, Vol. I, Christiania, 1881, 1-10.
    Aczel J., Lectures on functional equations and their applications, Mathematics in Science and Engineering, Vol. 19, Academic Press, New York - London, 1966.
  2. Cole J.D., On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225-236.
  3. Hernandez Heredero R., Levi D., Scimiterna C., A discrete linearizability test based on multiscale analysis, J. Phys. A: Math. Theor. 43 (2010), 502002, 14 pages, arXiv:1011.0141.
  4. Hietarinta J., A new two-dimensional lattice model that is 'consistent around a cube', J. Phys. A: Math. Gen. 37 (2004), L67-L73, nlin.SI/0311034.
  5. Hopf E., The partial differential equation ut+uux= uxx, Comm. Pure Appl. Math. 3 (1950), 201-230.
  6. Levi D., Ragnisco O., Bruschi M., Continuous and discrete matrix Burgers hierarchies, Nuovo Cimento B 74 (1983), 33-51.
  7. Levi D., Yamilov R.I., The generalized symmetry method for discrete equations, J. Phys. A: Math. Theor. 42 (2009), 454012, 18 pages, arXiv:0902.4421.
  8. Levi D., Yamilov R.I., Generalized symmetry integrability test for discrete equations on the square lattice, J. Phys. A: Math. Theor. 44 (2011), 145207, 22 pages, arXiv:1011.0070.
  9. Mikhailov A.V., Wang J.P., Xenitidis P., Recursion operators, conservation laws and integrability conditions for difference equations, Theoret. and Math. Phys. 167 (2011), 421-443, arXiv:1004.5346.
  10. Miura R.M., Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9 (1968), 1202-1204.
    Miura R.M., Gardner C.S., Kruskal M.D., Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys. 9 (1968), 1204-1209.
  11. Ramani A., Joshi N., Grammaticos B., Tamizhmani T., Deconstructing an integrable lattice equation, J. Phys. A: Math. Gen. 39 (2006), L145-L149.
  12. Startsev S.Ya., On non-point invertible transformations of difference and differential-difference equations, SIGMA 6 (2010), 092, 14 pages, arXiv:1010.0361.
  13. Uma Maheswari C., Sahadevan R., On the conservation laws for nonlinear partial difference equations, J. Phys. A: Math. Theor. 44 (2011), 275203, 16 pages.

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