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The computational approach to commuting ordinary differential operators of orders six and nine

Published online by Cambridge University Press:  17 February 2009

Geoff A. Latham
Geoff A. Latham
Affiliation:
Centre for Mathematics and its Applications, ANU, GPO Box 4, Canberra ACT 2601.
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Abstract

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Entirely elementary methods are employed to determine explicit formulae for the coefficients of commuting ordinary differential operators of orders six and nine which correspond to an elliptic curve. These formulae come from solving the nonlinear ordinary differential equations which are equivalent to the commutativity condition. Most solutions turn out to be rational expressions in one or two arbitrary functions and their derivatives. The corresponding Burchnall-Chaundy curves are computed.

Type
Research Article
Information
The ANZIAM Journal , Volume 35 , Issue 4 , April 1994 , pp. 399 - 419
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Baker, H. F., “Note on the foregoing paper ‘Commutative ordinary differential operators’, by J. L. Burchnall and T. W. Chaundy”, Proc. Royal Soc. London A 118 (1928) 584–593.Google Scholar
[2]Burchnall, J. L. and Chaundy, T. W., “Commutative ordinary differential operators”, Proc. London Math. Soc. 21 (1923) 420–440.CrossRefGoogle Scholar
[3]Burchnall, J. L. and Chaundy, T. W., “Commutative ordinary differential operators”, Proc. Royal Soc. London A 118 (1928) 557–583.Google Scholar
[4]Burchnall, J. L. and Chaundy, T. W., “Commutative ordinary differential operators II”, Proc. Royal Soc. London A 134 (1932) 471–485.Google Scholar
[5]Dehornoy, P., “Opérateurs différentiels et courbes elliptiques”, Compositio Math. 43 (1981) 71–99.Google Scholar
[6]Dixmier, J., “Sur les algébres de Weyl”, Bull. Soc. Math. France 96 (1968) 209–242.Google Scholar
[7]Grinevich, P. G., “Rational solutions for the equation of commutation of differential operators”, Funct. Anal. Appl. 16 (1982) 15–19.CrossRefGoogle Scholar
[8]Grünbaum, F. A., “Commuting pairs of linear ordinary differential operators of orders four and six”, Physica 31D (1988) 424–433.Google Scholar
[9]Grünbaum, F. A., “The Kadomtsev–Petviashvili equation: an elementary approach to the ‘rank 2’ solutions of Krichever and Novikov”, Phys. Lett. A 139 (1989) 146–150.CrossRefGoogle Scholar
[10]Krichever, I. M., “Integration of nonlinear equations by methods of algebraic geometry”, Funct. Anal. Appl. 11 (1977) 12–26.CrossRefGoogle Scholar
[11]Krichever, I. M., “Methods of algebraic geometry in the theory of nonlinear equations”, Russian Math. Surveys 32 (1977) 185–213.CrossRefGoogle Scholar
[12]Krichever, I. M., “Commutative rings of ordinary differential operators”, Funct. Anal. Appl. 12 (1978) 175–185.CrossRefGoogle Scholar
[13]Krichever, I. M. and Novikov, S. P., “Holomorphic fiberings and nonlinear equations: finite zone solutions of rank 2”, Soviet Math. Dokl. 20 (1979) 650–654.Google Scholar
[14]Krichever, I. M. and Novikov, S. P., “Holomorphic bundles over algebraic curves and nonlinear equations”, Russian Math. Surveys 35 (1980) 53–79.CrossRefGoogle Scholar
[15]Latham, G. A., “Solutions of the KP equation associated to rank three commuting differential operators over a singular elliptic curve”, Physica 41D (1990) 55–66.Google Scholar
[16]Mokhov, I. O., “Commuting differential operators of rank 3 corresponding to an elliptic curve”, Russian Math. Surveys 39 (1984) 133–134.CrossRefGoogle Scholar
[17]Mokhov, I. O., “Commutative differential operators of rank three and nonlinear equations”, Izvestiya Akad. Nauk SSSR 53 (1989) 1291–1315 (Russian), Math. USSR Izvestiya 35 (1990) 629–655 (English).Google Scholar
[18]Previato, E. and Wilson, G., “Vector bundles over curves and solutions of the KP equations”, Proc. Symp. Pure Math. 49 (1989) 553–569.CrossRefGoogle Scholar
[19]Previato, E. and Wilson, G., “Differential operators and rank two bundles over elliptic curves”, Compositio Math. 81 (1992) 107–119.Google Scholar
[20]Schur, I., “Über vertauschbare linear Differentialausdrüke”, Sitz. Berliner Math. Ges. 4 (1905) 2–8, See also Ges. Abhandlungen I (Springer Verlag, Berlin 1973) 170–176.Google Scholar
[21]Weiss, J., “The Painlevé property for partial differential equations II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative”, J. Math. Phys. 24 (1983) 1405–1413.CrossRefGoogle Scholar
[22]Wilson, G., “Algebraic curves and soliton equations”, in Geometry today (eds. Arbarello, E., Procesi, C. and Strickland, E.), (Birkhaüser, Boston—Basel—Stuttgart, 1985) 303–329.Google Scholar