Geodesic
A Completeness Study on Certain $2\times2$ Lax Pairs Including Zero Terms
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We expand the completeness study instigated in [J. Math. Phys. 50 (2009), 103516, 29 pages] which found all $2\times2$ Lax pairs with non-zero, separable terms in each entry of each Lax matrix, along with the most general nonlinear systems that can be associated with them. Here we allow some of the terms within the Lax matrices to be zero. We cover all possible Lax pairs of this type and find a new third order equation that can be reduced to special cases of the non-autonomous lattice KdV and lattice modified KdV equations among others.
Keywords: discrete integrable systems; Lax pairs. 
@article{SIGMA_2011_7_a88,
     author = {Mike C. Hay},
     title = {A {Completeness} {Study} on {Certain} $2\times2$ {Lax} {Pairs} {Including} {Zero} {Terms}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2011},
     volume = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a88/}
}
TY  - JOUR
AU  - Mike C. Hay
TI  - A Completeness Study on Certain $2\times2$ Lax Pairs Including Zero Terms
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2011
VL  - 7
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a88/
LA  - en
ID  - SIGMA_2011_7_a88
ER  - 
%0 Journal Article
%A Mike C. Hay
%T A Completeness Study on Certain $2\times2$ Lax Pairs Including Zero Terms
%J Symmetry, integrability and geometry: methods and applications
%D 2011
%V 7
%U http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a88/
%G en
%F SIGMA_2011_7_a88
Mike C. Hay. A Completeness Study on Certain $2\times2$ Lax Pairs Including Zero Terms. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a88/

[1] Hay M., “A completeness study on discrete $2\times 2$ Lax pairs”, J. Math. Phys., 50 (2009), 103516, 29 pp. | DOI | MR | Zbl

[2] Adler V.E., Bobenko A.I., Suris Yu.B., “Classification of integrable equations on quad-graphs. The consistency approach”, Comm. Math. Phys., 233 (2003), 513–543, arXiv: nlin.SI/0202024 | DOI | MR | Zbl

[3] Grammaticos B., Papageorgiou V., Ramani A., “Discrete dressing transformations and Painlevé equations”, Phys. Lett. A, 235 (1997), 475–479 | DOI | MR

[4] Matsuura N., “Discrete time evolution of planar discrete curves”, DMHF2007: COE Conference on the Development of Mathematics with High Functionality, Book of Abstracts, ed. M.T. Nakao, Faculty of Mathematics, Kyushu University, Fukuoka, 2007, 61–64

[5] Kajiwara K., Ohta Y., “Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation”, J. Phys. Soc. Japan, 77 (2008), 054004, 9 pp., arXiv: 0802.0757 | DOI

[6] Hirota R., “Nonlinear partial difference equations. I. A difference analogue of the Korteweg–de Vries equation”, J. Phys. Soc. Japan, 43 (1977), 1424–1433 | DOI | MR

[7] Grammaticos B., Halburd R.G., Ramani A., Viallet C.-M., “How to detect the integrability of discrete systems”, J. Phys. A: Math. Theor., 42 (2009), 454002, 30 pp. | DOI | MR | Zbl

[8] Nijhoff F.W., Capel H.W., “The discrete Korteweg–de Vries equation”, Acta Appl. Math., 39 (1995), 133–158 | DOI | MR | Zbl

[9] Levi D., Yamilov R.I., “The generalized symmetry method for discrete equations”, J. Phys. A: Math. Theor., 42 (2009), 454012, 18 pp., arXiv: 0902.4421 | DOI | MR | Zbl

[10] Ramani A., Grammaticos B., Satsuma J., Willox R., “On two (not so) new integrable partial difference equations”, J. Phys. A: Math. Theor., 42 (2009), 282002, 6 pp. | DOI | MR | Zbl

[11] Hietarinta J., “A new two-dimensional model that is ‘consistent around a cube’”, J. Phys. A: Math. Gen., 37 (2004), L67–L73, arXiv: nlin.SI/0311034 | DOI | MR | Zbl

[12] Hietarinta J., “Searching for CAC-maps”, J. Nonlinear Math. Phys., 12:2 (2005), 223–230 | DOI | MR | Zbl

[13] Boll R., Classification of 3D consistent quad-equations, arXiv: 1009.4007

[14] Adler V.E., Suris Yu.B., “$\mathrm Q_4$: integrable master equation related to an elliptic curve”, Int. Math. Res. Not., 2004:47 (2004), 2523–2553, arXiv: nlin.SI/0309030 | DOI | MR | Zbl

[15] Ramani A., Joshi N., Grammaticos B., Tamizhmani T., “Deconstructing an integrable lattice equation”, J. Phys. A: Math. Gen., 39 (2006), L145–L149 | DOI | MR | Zbl