# nLab Burchnall-Chaundy theory

Burchnall-Chaundy theory refers to ramifications of the theory of commuting differential operators. Under some assumptions commuting operators have to satisfy an algebraic relation (defining therefore an algebraic curve, Burchnall-Chaundy curve). The main origin is 1928 paper of Burchnall and Chaundy.

• J. L. Burchnall, T.W. Chaundy, Commutative ordinary differential operators, Proc. London Math. Soc. 3 (21), 420–440 (1923); Commutative ordinary differential operators Proc. Royal Soc. Lond. Ser. A Math. Phys. Eng. Sci. 118, 557–583 (1928); Commutative ordinary differential operators ii, the identity

• G. Wilson, Commuting flows and conservation laws for Lax equations, Math. Proc. Camb. Phil. Soc. 86 (1979) 131–143 doi

• I. M. Krichever, Integration of non-linear equations by methods of algebraic geometry Funct. Anal, and Its Appl. 11 (1977), 15–31 (Russian), 12–26 (English); Methods of algebraic geometry in the theory of non-linear equations Uspekhi Mat. Nauk 32, 6 (1977) 183–208 = Russian Math. Surveys 32, 6 (1977) 185–213.

• Emma Previato, Multivariable Burchnall–Chaundy theory, Phil. Trans. R. Soc. A (2008) 366, 1155–1177 doi

• Maurice J. Dupré, James F. Glazebrook, Emma Previato, A Banach algebra version of the Sato Grassmannian and commutative rings of differential operators, Acta Appl Math (2006) 92: 241 doi

• James F. Glazebrook, Banach algebras associated to Lax pairs, Rep. Math. Phys. 75:2 (2015) 279-301 doi; A quasi-triangular Hopf structure derived from the Burchnall–Chaundy $C^*$-algebra, Reports on Mathematical Physics 71:1 (2013) 93-111 doi

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