Geodesic
Integrable Hierarchy of the Quantum Benjamin–Ono Equation
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A hierarchy of pairwise commuting Hamiltonians for the quantum periodic Benjamin–Ono equation is constructed by using the Lax matrix. The eigenvectors of these Hamiltonians are Jack symmetric functions of infinitely many variables $x_1,x_2,\ldots$. This construction provides explicit expressions for the Hamiltonians in terms of the power sum symmetric functions $p_n=x_1^n+x_2^n+\cdots$ and is based on our recent results from [Comm. Math. Phys. 324 (2013), 831–849].
Keywords: Jack symmetric functions; quantum Benjamin–Ono equation; collective variables. 
@article{SIGMA_2013_9_a77,
     author = {Maxim Nazarov and Evgeny Sklyanin},
     title = {Integrable {Hierarchy} of the {Quantum} {Benjamin{\textendash}Ono} {Equation}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2013},
     volume = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a77/}
}
TY  - JOUR
AU  - Maxim Nazarov
AU  - Evgeny Sklyanin
TI  - Integrable Hierarchy of the Quantum Benjamin–Ono Equation
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2013
VL  - 9
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a77/
LA  - en
ID  - SIGMA_2013_9_a77
ER  - 
%0 Journal Article
%A Maxim Nazarov
%A Evgeny Sklyanin
%T Integrable Hierarchy of the Quantum Benjamin–Ono Equation
%J Symmetry, integrability and geometry: methods and applications
%D 2013
%V 9
%U http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a77/
%G en
%F SIGMA_2013_9_a77
Maxim Nazarov; Evgeny Sklyanin. Integrable Hierarchy of the Quantum Benjamin–Ono Equation. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a77/

[1] Ablowitz M. J., Fokas A. S., Satsuma J., Segur H., “On the periodic intermediate long wave equation”, J. Phys. A: Math. Gen., 15 (1982), 781–786 | DOI | MR | Zbl

[2] Awata H., Matsuo Y., Odake S., Shiraishi J., “Collective field theory, {C}alogero–{S}utherland model and generalized matrix models”, Phys. Lett. B, 347 (1995), 49–55, arXiv: hep-th/9411053 | DOI | MR | Zbl

[3] Benjamin T. B., “Internal waves of permanent form in fluids of great depth”, J. Fluid Mech., 29 (1967), 559–592 | DOI | Zbl

[4] Cai W., Jing N., “Applications of a {L}aplace–{B}eltrami operator for {J}ack polynomials”, European J. Combin., 33 (2012), 556–571, arXiv: 1101.5544 | DOI | MR | Zbl

[5] Kaup D. J., Lakoba T. I., Matsuno Y., “Complete integrability of the {B}enjamin–{O}no equation by means of action-angle variables”, Phys. Lett. A, 238 (1998), 123–133 | DOI | MR | Zbl

[6] Kaup D. J., Matsuno Y., “The inverse scattering transform for the {B}enjamin–{O}no equation”, Stud. Appl. Math., 101 (1998), 73–98 | DOI | MR | Zbl

[7] Krichever I., Vaninsky K. L., “The periodic and open {T}oda lattice”, Mirror Symmetry ({M}ontreal, {QC}, 2000), v. IV, AMS/IP Stud. Adv. Math., 33, Amer. Math. Soc., Providence, RI, 2002, 139–158, arXiv: hep-th/0010184 | MR | Zbl

[8] Macdonald I. G., Symmetric functions and {H}all polynomials, Oxford Mathematical Monographs, 2nd ed., The Clarendon Press, Oxford University Press, 1995 | MR | Zbl

[9] Nazarov M. L., Sklyanin E. K., “Sekiguchi–{D}ebiard operators at infinity”, Comm. Math. Phys., 324 (2013), 831–849, arXiv: 1212.2781 | DOI | MR | Zbl

[10] Nazarov M. L., Sklyanin E. K., “Macdonald operators at infinity”, J. Algebr. Comb., 2013, arXiv: 1212.2960 | DOI | MR

[11] Okounkov A., Pandharipande R., “Quantum cohomology of the {H}ilbert scheme of points in the plane”, Invent. Math., 179 (2010), 523–557, arXiv: math.AG/0411210 | DOI | MR | Zbl

[12] Ono H., “Algebraic solitary waves in stratified fluids”, J. Phys. Soc. Japan, 39 (1975), 1082–1091 | DOI | MR

[13] Polychronakos A. P., “Waves and solitons in the continuum limit of the {C}alogero–{S}utherland model”, Phys. Rev. Lett., 74 (1995), 5153–5157, arXiv: hep-th/9411054 | DOI | MR | Zbl

[14] Schiffmann O., Vasserot E., “Cherednik algebras, $W$-algebras and the equivariant cohomology of the moduli space of instantons on $\mathbb{A}^2$”, Publ. Math. IHÉS, 118 (2013), 213–342, arXiv: 1202.2756 | DOI | Zbl

[15] Shiraishi J., “A family of integral transformations and basic hypergeometric series”, Comm. Math. Phys., 263 (2006), 439–460 | DOI | MR | Zbl

[16] Sklyanin E. K., “Bispectrality for the quantum open Toda chain”, J. Phys. A: Math. Theor., 46 (2013), 382001, 9 pp., arXiv: 1306.0454 | DOI | Zbl

[17] Stembridge J., Maple packages for symmetric functions, posets, root systems, and finite Coxeter groups, http://www.math.lsa.umich.edu/ ̃ jrs/maple.html#SF

[18] Takasaki K., “Integrable systems whose spectral curves are the graph of a function”, Superintegrability in Classical and Quantum Systems, CRM Proc. Lecture Notes, 37, Amer. Math. Soc., Providence, RI, 2004, arXiv: nlin.SI/0211021 | MR | Zbl

[19] Ujino H., Wadati M., Hikami K., “The quantum {C}alogero–{M}oser model: algebraic structures”, J. Phys. Soc. Japan, 62 (1993), 3035–3043 | DOI | MR | Zbl

[20] Vaninsky K. L., “On explicit parametrisation of spectral curves for {M}oser–{C}alogero particles and its applications”, Int. Math. Res. Not., 1999 (1999), 509–529, arXiv: solv-int/9808018 | DOI | MR | Zbl