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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