Geodesic
A~conjecture concerning nonlocal terms of recursion operators
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 7, pp. 23-33.

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We provide examples to extend a recent conjecture concerning relation between zero curvature representations and nonlocal terms of inverse recursion operators to all recursion operators in dimension two. Namely, we conjecture that nonlocal terms of recursion operators are always related to a suitable zero-curvature representation, not necessarily depending on a parameter or taking values in a semisimple algebra. In particular, the conventional pseudodifferential recursion operators correspond to Abelian Lie algebras. 
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H. Baran; M. Marvan. A~conjecture concerning nonlocal terms of recursion operators. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 7, pp. 23-33. http://geodesic.mathdoc.fr/item/FPM_2006_12_7_a2/

[1] Zakharov V. E., Shabat A. B., “Integrirovanie nelineinykh uravnenii matematicheskoi fiziki metodom obratnoi zadachi rasseyaniya. II”, Funkts. analiz i ego pril., 13:3 (1979), 13–22 | MR | Zbl

[2] Bilge A. H., “On the equivalence of linearization and formal symmetries as integrability tests for evolution equations”, J. Phys. A, 26 (1993), 7511–7519 | DOI | MR | Zbl

[3] Bocharov A. V., Chetverikov V. N., Duzhin S. V., Khor'kova N. G., Krasil'shchik I. S., Samokhin A. V., Torkhov Yu. N., Verbovetsky A. M., Vinogradov A. M., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Translations of Math. Monographs, Amer. Math. Soc., Providence, 1999 ; vol 182 | MR

[4] Foursov M. V., “Classification of certain integrable coupled potential KdV and modified KdV-type equations”, J. Math. Phys., 41 (2000), 6173–6185 | DOI | MR | Zbl

[5] Fulton W., Harris J., Representation Theory. A First Course, Springer, Berlin, 2001 | MR

[6] Gürses M., Karasu A., Sokolov V. V., “On construction of recursion operators from Lax representation”, J. Math. Phys., 40 (1999), 6473–6490 | DOI | MR | Zbl

[7] Guthrie G. A., “Recursion operators and non-local symmetries”, Proc. Roy. Soc. London Ser. A, 446, 1994, 107–114 | MR | Zbl

[8] Humphreys J. E., Introduction to Lie Algebras and Representation Theory, Springer, New York, 1972 | MR | Zbl

[9] Igonin S., Martini R., “Prolongation structure of the Krichever–Novikov equation”, J. Phys. A, 35 (2002), 9801–9810 | DOI | MR | Zbl

[10] Karasu (Kalkanl\i) A., Karasu A., Sakovich S. Yu., “A strange recursion operator for a new integrable system of coupled Korteweg–de Vries equations”, Acta Appl. Math., 83 (2004), 85–94 | DOI | MR | Zbl

[11] Krasil'shchik I. S., Kersten P. H. M., Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations, Kluwer Academic, Dordrecht, 2000 | MR

[12] Krasil'shchik I. S., Kersten P. H. M., “Complete integrability of the coupled KdV–mKdV system”, Lie Groups, Geometric Structures and Differential Equations One Hundred Years After Sophus Lie, Advanced Stud. in Pure Math., 37, eds. T. Morimoto, H. Sato, K. Yamaguchi, Math. Soc. of Japan, Tokyo, 2002, 151–171 | MR

[13] Lax P. D., “Periodic solutions of the KdV equation”, Comm. Pure Appl. Math., 28 (1975), 141–188 | DOI | MR | Zbl

[14] Marvan M., “Another look on recursion operators. Differential Geometry and Applications”, Proc. Conf. Brno, Masaryk University, Brno, 1995, 393–402 | MR

[15] Marvan M., “Some local properties of Bäcklund relations”, Acta Appl. Math., 54 (1998), 1–25 | DOI | MR | Zbl

[16] Marvan M., “Recursion operators for the Einstein equations with symmetries”, Proc. Conf. “Symmetry in Nonlinear Mathematical Physics” (Kyiv, Ukraine, June 23–28, 2003. Part I), 2004; Proc. Inst. Math. NAS Ukraine, vol 50, 179–183 | MR | Zbl

[17] Marvan M., Sergyeyev A., “Recursion operator for the Nizhnik–Veselov–Novikov equation”, J. Phys. A, 36 (2003), L87–L92 | DOI | MR | Zbl

[18] Olver P. J., “Evolution equations possessing infinitely many symmetries”, J. Math. Phys., 18 (1977), 1212–1215 | DOI | MR | Zbl

[19] Sakovich S. Yu., “Cyclic bases of zero-curvature representations: Five illustrations to one concept”, Acta Appl. Math., 83 (2004), 69–83 | DOI | MR | Zbl

[20] Sergyeyev A., Locality of symmetries generated by nonstandard recursion operators: A new application for implectic-symplectic factorization, Preprint Math. Inst. Opava, GA 2/2006 | MR

[21] Sergyeyev A., “A strange recursion operator demystified”, J. Phys. A, 38 (2005), L257–L262 | DOI | MR | Zbl

[22] Sergyeyev A., Demskoi D., The Sasa–Satsuma (complex mKdV II) and the complex sine-Gordon II equation revisited: Recursion operators, nonlocal symmetries and more, arXiv: nlin.SI/0512042

[23] Takhtadzhyan L. A., Faddeev L. D., Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, 1987 | MR