Il a été démontré par Sokolov que la hiérarchie de l'équation de Krichever–Novikov est hamiltonienne pour l'opérateur hamiltonien et possède deux opérateurs de récursion faiblement non locaux de degrés 4 et 6, et . Nous montrons ici que , et sont des opérateurs hamiltoniens compatibles pour lesquels la hiérarchie de l'équation de Krichever–Novikov est hamiltonienne.
It has been proved by Sokolov that Krichever–Novikov equation's hierarchy is hamiltonian for the Hamiltonian operator and possesses two weakly non-local recursion operators of degrees 4 and 6, and . We show here that , and are compatible Hamiltonians operators for which the Krichever–Novikov equation's hierarchy is hamiltonian.
Accepté le :
Publié le :
@article{CRMATH_2017__355_7_744_0, author = {Carpentier, Sylvain}, title = {Compatible {Hamiltonian} operators for the {Krichever{\textendash}Novikov} equation}, journal = {Comptes Rendus. Math\'ematique}, pages = {744--747}, publisher = {Elsevier}, volume = {355}, number = {7}, year = {2017}, doi = {10.1016/j.crma.2017.05.009}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2017.05.009/} }
TY - JOUR AU - Carpentier, Sylvain TI - Compatible Hamiltonian operators for the Krichever–Novikov equation JO - Comptes Rendus. Mathématique PY - 2017 SP - 744 EP - 747 VL - 355 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.05.009/ DO - 10.1016/j.crma.2017.05.009 LA - en ID - CRMATH_2017__355_7_744_0 ER -
%0 Journal Article %A Carpentier, Sylvain %T Compatible Hamiltonian operators for the Krichever–Novikov equation %J Comptes Rendus. Mathématique %D 2017 %P 744-747 %V 355 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.05.009/ %R 10.1016/j.crma.2017.05.009 %G en %F CRMATH_2017__355_7_744_0
Carpentier, Sylvain. Compatible Hamiltonian operators for the Krichever–Novikov equation. Comptes Rendus. Mathématique, Tome 355 (2017) no. 7, pp. 744-747. doi : 10.1016/j.crma.2017.05.009. http://www.numdam.org/articles/10.1016/j.crma.2017.05.009/
[1] A sufficient condition for a rational differential operator to generate an integrable system, Jpn. J. Math., Volume 12 (2017), pp. 33-89
[2] Singular degree of a rational matrix pseudodifferential operator, Int. Math. Res. Not., Volume 2015 (2015) no. 13, pp. 5162-5195
[3] On recursion operators for elliptic models, Nonlinearity, Volume 21 (2008) no. 6, pp. 1253-1264
[4] Non-local Hamiltonian structures and applications to the theory of integrable systems, Jpn. J. Math., Volume 8 (2013) no. 2, pp. 233-347
[5] Higher Hamiltonian structures (the sl2 case), JETP Lett., Volume 58 (1993) no. 8, pp. 658-664
[6] Evolution equations with nontrivial Lie–Bäcklund group, Funkc. Anal. Prilozh., Volume 14 (1980) no. 1, pp. 25-36
[7] Holomorphic bundles over algebraic curves and non-linear equations, Russ. Math. Surv., Volume 35 (1980) no. 6, pp. 53-79
[8] , Lecture Notes in Physics, vol. 120, Springer-Verlag, Berlin (1980), p. 233
[9] On the local systems Hamiltonian in the weakly non-local Poisson brackets, Physica D, Volume 156 (2001), pp. 53-80
[10] Symmetries of differential equations and the problem of integrability (Mikhailov, A.V., ed.), Integrability, Lecture Notes in Physics, vol. 767, Springer, 2008
[11] On the hamiltonian property of the Krichever–Novikov equation, Sov. Math. Dokl., Volume 30 (1984) no. 1
Cité par Sources :