Geodesic
Commuting differential operators in two-dimension
Ufa mathematical journal, Tome 3 (2011) no. 2, pp. 89-95 Cet article a éte moissonné depuis la source Math-Net.Ru

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A generalization to the multi-dimensional case of commutative rings of differential operators is considered. An algorithm for construction of commuting two-dimensional differential operators is formulated for a special kind of operators related to the simple one-dimensional model proposed by Burchnall and Chaundy in 1932. The problem of classifying such commutative pairs is discussed. The suggested algorithm is based on necessary conditions for general commutativity and the reducibility lemma proved in the present paper.
Keywords: commuting ring of differential operators, commuting two-dimensional differential operators. 
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A. B. Shabat; Z. S. Elkanova. Commuting differential operators in two-dimension. Ufa mathematical journal, Tome 3 (2011) no. 2, pp. 89-95. http://geodesic.mathdoc.fr/item/UFA_2011_3_2_a9/

[1] I. Schur, “Uber vertauschbare lineare Differentialausdrucke”, Sitzungsber. Berliner Math. Gen., 4 (1905), 2–8 | Zbl

[2] J. L. Burchnall, T. W. Chaundy, “Commutative ordinary differential operators”, Proc. London Math. Soc. (2), 21 (1923), 420–440 ; Proc. Roy. Soc. London (A), 118 (1928), 557–583 | DOI | MR | Zbl | DOI | Zbl

[3] J. L. Burchnall, T. W. Chaundy, “Commutative ordinary diff. operators, II. The identity $P^n=Q^m$”, Proc. Roy. Soc. London (A), 134 (1932), 471–485 | DOI

[4] Sokolov V. V., “Primery kommutativnykh kolets diff. operatorov”, Funkts. analiz i ego pril., 12:1 (1978), 82–83 | MR | Zbl

[5] J. Hietarinta, “Pure quantum integrability”, Phys. Lett. A, 246 (1998), 97–104 | DOI | MR | Zbl

[6] Shabat A. B., Lektsii po teorii solitonov, KChGU, Karachaevsk, 2008, 60 pp.

[7] Mironov A. E., “O kommutiruyuschikh differentsialnykh operatorakh ranga 2”, Sib. elektr. matem. izvestiya, 6 (2009), 533–536 | MR | Zbl

[8] Shabat A. B., Elkanova Z. S., “O kommutiruyuschikh differentsialnykh operatorakh”, TMF, 162:3 (2010), 334–344 | DOI | MR | Zbl

[9] Gabiev R. A., Shabat A. B., “O differentsialnykh operatorakh kommutiruyuschikh v glavnom”, TMF, 2011 (to appear)