Geodesic
On skew-symmetric and general deformations of Lax pseudodifferential operators
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 7, pp. 101-116.

Voir la notice de l'article provenant de la source Math-Net.Ru

A nonlinear deformation is conjectured for the reduction of the third KP flow on the subspace of skew-symmetric operators, and the conjecture is proved for the linearized flow. As a by-product, we find a peculiar (nonquantum) polynomial deformation of the numbers $\left\{\binom{2n+1}{2s+1}\frac{4^{s+1}-1}{s+1}B_{2s+2}\right\}$, where $B_m$'s are the Bernoulli numbers. General open questions and generalizations are also discussed. The conjecture is extended to all the flows, and its linearized version is proved. 
@article{FPM_2006_12_7_a7,
     author = {B. A. Kupershmidt},
     title = {On skew-symmetric and general deformations of {Lax} pseudodifferential operators},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {101--116},
     publisher = {mathdoc},
     volume = {12},
     number = {7},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_7_a7/}
}
TY  - JOUR
AU  - B. A. Kupershmidt
TI  - On skew-symmetric and general deformations of Lax pseudodifferential operators
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2006
SP  - 101
EP  - 116
VL  - 12
IS  - 7
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2006_12_7_a7/
LA  - ru
ID  - FPM_2006_12_7_a7
ER  - 
%0 Journal Article
%A B. A. Kupershmidt
%T On skew-symmetric and general deformations of Lax pseudodifferential operators
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2006
%P 101-116
%V 12
%N 7
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2006_12_7_a7/
%G ru
%F FPM_2006_12_7_a7
B. A. Kupershmidt. On skew-symmetric and general deformations of Lax pseudodifferential operators. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 7, pp. 101-116. http://geodesic.mathdoc.fr/item/FPM_2006_12_7_a7/

[1] Kupershmidt B. A., Manin Yu. I., “Uravneniya dlinnykh voln so svobodnoi poverkhnostyu. II. Gamiltonova struktura i vysshie uravnenii”, Funkts. analiz i ego pril., 12:1 (1978), 25–37 | MR | Zbl

[2] Manin Yu. I., “Algebraicheskie aspekty nelineinykh differentsialnykh uravnenii”, Itogi nauki i tekhn. Ser. Sovr. probl. matematiki, 11, VINITI, M., 1978, 5–152 | MR

[3] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992 | MR

[4] Benney D. J., “Some properties of long nonlinear waves”, Stud. Appl. Math., L11 (1973), 45–50 | Zbl

[5] Kupershmidt B. A., “Deformations of integrable systems”, Proc. Roy. Irish Acad. Sect. A, 83, no. 1, 1983, 45–74 | MR | Zbl

[6] Kupershmidt B. A., “Normal and universal forms in integrable hydrodynamical systems”, Proc. of NASA Ames-Berkeley Conf. on Nonlinear Problems in Optimal Control and Hydrodynamics, eds. L. R. Hunt, C. F. Martin, Math. Sci. Press, 1984, 357–378 | MR

[7] Kupershmidt B. A., “Mathematics of dispersive water waves”, Comm. Math. Phys., 99 (1985), 51–73 | DOI | MR | Zbl

[8] Kupershmidt B. A., KP or mKP. Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems, Amer. Math. Soc., Providence, 2000 | MR | Zbl

[9] Kupershmidt B. A., Stud. Appl. Math., 116 (2006), 415–434 | DOI | MR | Zbl

[10] Kupershmidt B. A., Wilson G., “Modifying Lax equations and the second Hamiltonian structure”, Invent. Math., 62 (1981), 403–436 | DOI | MR

[11] Lebedev D. R., Manin Yu. I., “Conservation laws and Lax representation of Benney's long wave equations”, Phys. Lett. A, 74 (1979), 154–156 | DOI | MR

[12] Sloane N., On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/ñjas/sequences