Geodesic
Linearizability and a fake Lax pair for a nonlinear nonautonomous quad-graph equation consistent around the cube
Teoretičeskaâ i matematičeskaâ fizika, Tome 189 (2016) no. 1, pp. 69-83 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We discuss the linearization of a nonautonomous nonlinear partial difference equation belonging to the Boll classification of quad-graph equations consistent around the cube. We show that its Lax pair is fake. We present its generalized symmetries, which turn out to be nonautonomous and dependent on an arbitrary function of the dependent variables defined at two lattice points. The se generalized symmetries are differential–difference equations, which admit peculiar Bäcklund transformations in some cases.
Keywords: partial difference equation, $C$-integrability, Bäcklund transformation, fake Lax pair. 
@article{TMF_2016_189_1_a5,
     author = {G. Gubbiotti and D. Levi and Ch. Scimiterna},
     title = {Linearizability and a~fake {Lax} pair for a~nonlinear nonautonomous quad-graph equation consistent around the~cube},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {69--83},
     year = {2016},
     volume = {189},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2016_189_1_a5/}
}
TY  - JOUR
AU  - G. Gubbiotti
AU  - D. Levi
AU  - Ch. Scimiterna
TI  - Linearizability and a fake Lax pair for a nonlinear nonautonomous quad-graph equation consistent around the cube
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2016
SP  - 69
EP  - 83
VL  - 189
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2016_189_1_a5/
LA  - ru
ID  - TMF_2016_189_1_a5
ER  - 
%0 Journal Article
%A G. Gubbiotti
%A D. Levi
%A Ch. Scimiterna
%T Linearizability and a fake Lax pair for a nonlinear nonautonomous quad-graph equation consistent around the cube
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2016
%P 69-83
%V 189
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2016_189_1_a5/
%G ru
%F TMF_2016_189_1_a5
G. Gubbiotti; D. Levi; Ch. Scimiterna. Linearizability and a fake Lax pair for a nonlinear nonautonomous quad-graph equation consistent around the cube. Teoretičeskaâ i matematičeskaâ fizika, Tome 189 (2016) no. 1, pp. 69-83. http://geodesic.mathdoc.fr/item/TMF_2016_189_1_a5/

[1] D. Levi, R. Benguria, Proc. Nat. Acad. Sci. USA, 77:9 (1981), 5025–5027 | DOI | MR

[2] J. J. C. Nimmo, W. K. Schief, Proc. Roy. Soc. London Ser. A, 453:1957 (1997), 255–279 | DOI | MR | Zbl

[3] A. Doliwa, P. M. Santini, Phys. Lett. A, 233:4–6 (1997), 365–372 | DOI | MR | Zbl

[4] A. I. Bobenko, Yu. B. Suris, Int. Math. Res. Not., 2002:11 (2002), 573–611 | DOI | MR | Zbl

[5] T. Bridgman, W. Hereman, G. R. W. Quispel, P. H. van der Kamp, Found. Comput. Math., 13:4 (2013), 517–544 | DOI | MR | Zbl

[6] F. W. Nijhoff, Phys. Lett. A, 297:1–2 (2002), 49–58 | DOI | MR | Zbl

[7] F. W. Nijhoff, A. Walker, Glasg. Math. J., 43A (2001), 109–123 | DOI | MR | Zbl

[8] R. Yamilov, J. Phys. A: Math. Gen., 39:45 (2006), R541–R623 | DOI | MR | Zbl

[9] V. E. Adler, A. I. Bobenko, Yu. B. Suris, Commun. Math. Phys., 233:3 (2003), 513–543 | DOI | MR | Zbl

[10] V. E. Adler, A. I. Bobenko, Yu. B. Suris, Funkts. analiz i ego pril., 43:1 (2009), 3–21 | DOI | DOI | MR | Zbl

[11] R. Boll, J. Nonlinear Math. Phys., 18:3 (2011), 337–365 | DOI | MR | Zbl

[12] R. Boll, J. Nonlinear Math. Phys., 19:4 (2012), 1292001, 3 pp. | MR | Zbl

[13] R. Boll, Classification and Lagrangian structure of $3D$ consistent quad-equations, Ph. D. thesis, Technische Universität, Berlin, 2012 | MR | Zbl

[14] J. Hietarinta, C. Viallet, Nonlinearity, 25:7 (2012), 1955–1966 | DOI | MR | Zbl

[15] P. D. Xenitidis, V. G. Papageorgiou, J. Phys. A: Math. Theor., 42:45 (2009), 454025, 13 pp. | DOI | MR | Zbl

[16] G. Gubbiotti, C. Scimiterna, D. Levi, J. Nonlinear Math. Phys., 23:4 (2016), 507–543 | DOI | MR

[17] S. Tremblay, B. Grammaticos, A. Ramani, Phys. Lett. A, 278:6 (2001), 319–324 | DOI | MR | Zbl

[18] C. Viallet, Algebraic entropy for lattice equations, arXiv: math-ph/0609043

[19] G. Gubbiotti, R. I. Yamilov, Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations I: First integrals, arXiv: 1608.03506

[20] G. Gubbiotti, C. Scimitema, R. I. Yamilov, Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations II: First integrals (in preparation)

[21] J. Hietarinta, J. Phys. A: Math. Gen., 37:6 (2004), L67–L73 | DOI | MR | Zbl

[22] J. Hietarinta, J. Nonlinear Math. Phys., 12, supp. 2 (2005), 223–230 | DOI | MR | Zbl

[23] A. Ramani, N. Joshi, B. Grammaticos, T. Tamizhmani, J. Phys. A: Math. Gen., 39:8 (2006), L145–L149 | DOI | MR | Zbl

[24] J. Atkinson, J. Phys. A: Math. Theor., 42:45 (2009), 454005, 7 pp. | DOI | MR | Zbl

[25] M. Hay, S. Butler, Simple identification of fake Lax pair, arXiv: 1311.2406

[26] F. Calogero, M. C. Nucci, J. Math. Phys., 32:1 (1991), 72–74 | DOI | MR | Zbl

[27] D. Levi, A. Sym, G. Z. Tu, A working algorithm to isolate integrable surfaces in $E^3$, preprint DF-INFN No 761, INFN, Roma, 1990

[28] Y. Q. Li, B. Li, S. Y. Lou, Constraints for evolution equations with some special forms of Lax pairs and distinguishing Lax pairs by available constraints, arXiv: 1008.1375

[29] M. Marvan, Acta Appl. Math., 72:1–2 (2002), 51–65 | DOI | MR | Zbl

[30] M. Marvan, Acta Appl. Math., 109:1 (2010), 239–255 | DOI | MR | Zbl

[31] M. Marvan, Acta Appl. Math., 83:1–2 (2004), 39–68 | DOI | MR | Zbl

[32] S. Yu. Sakovich, True and fake Lax pairs: how to distinguish them, arXiv: nlin.SI/0112027

[33] S. Yu. Sakovich, Acta Appl. Math., 83:1 (2004), 69–83, arXiv: nlin/0212019 | DOI | MR | Zbl