Geodesic
Quantizing the KdV Equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 129 (2001) no. 2, pp. 333-344 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the quantization procedure for the Gardner–Zakharov–Faddeev and Magri brackets using the fermionic representation for the KdV field. In both cases, the corresponding Hamiltonians are sums of two well-defined operators. Each operator is bilinear and diagonal with respect to either fermion or boson (current) creation/annihilation operators. As a result, the quantization procedure needs no space cutoff and can be performed on the entire axis. In this approach, solitonic states appear in the Hilbert space, and soliton parameters become quantized. We also demonstrate that the dispersionless KdV equation is uniquely and explicitly solvable in the quantum case. 
@article{TMF_2001_129_2_a12,
     author = {A. K. Pogrebkov},
     title = {Quantizing the {KdV} {Equation}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {333--344},
     year = {2001},
     volume = {129},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2001_129_2_a12/}
}
TY  - JOUR
AU  - A. K. Pogrebkov
TI  - Quantizing the KdV Equation
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2001
SP  - 333
EP  - 344
VL  - 129
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2001_129_2_a12/
LA  - ru
ID  - TMF_2001_129_2_a12
ER  - 
%0 Journal Article
%A A. K. Pogrebkov
%T Quantizing the KdV Equation
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2001
%P 333-344
%V 129
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2001_129_2_a12/
%G ru
%F TMF_2001_129_2_a12
A. K. Pogrebkov. Quantizing the KdV Equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 129 (2001) no. 2, pp. 333-344. http://geodesic.mathdoc.fr/item/TMF_2001_129_2_a12/

[1] C. S. Gardner, J. M. Green, M. D. Kruskal, R. M. Miura, Phys. Rev. Lett., 19 (1967), 1095 | DOI | Zbl

[2] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov: Metod obratnoi zadachi, Nauka, M., 1980 | MR

[3] F. Kalodzhero, A. Degasperis, Spektralnye preobrazovaniya i solitony. Metody resheniya i issledovaniya nelineinykh evolyutsionnykh uravnenii, Mir, M., 1985 | MR

[4] C. Gardner, J. Math. Phys., 12 (1971), 1548 | DOI | MR | Zbl

[5] F. Magri, “A geometrical approach to the nonlinear solvable equations”, Nonlinear Evolution Equations and Dynamical Systems, Lect. Notes Phys., 120, eds. M. Boiti, F. Pempinelli, G. Soliani, Springer, Berlin, 1980, 233 | DOI | MR

[6] P. B. Wiegmann, A. Zabrodin, Commun. Math. Phys., 213 (2000), 523 | DOI | MR | Zbl

[7] V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov, Commun. Math. Phys., 177 (1996), 381 | DOI | MR | Zbl

[8] P. Di Francesco, P. Mathieu, D. Sénéchal, Mod. Phys. Lett. A, 7 (1992), 701 | DOI | MR | Zbl

[9] J. Barcelos-Neto, A. Constandache, A. Das, Phys. Lett. A, 268 (2000), 342 | DOI | MR | Zbl

[10] I. Krichever, Commun. Math. Phys., 143 (1992), 415 | DOI | MR | Zbl

[11] A. Vaitman, Problemy v relyativistskoi dinamike kvantovannykh polei, Nauka, M., 1968

[12] S. Coleman, Phys. Rev. D, 11 (1975), 2088 | DOI

[13] S. Mandelstam, Phys. Rev. D, 11 (1975), 3026 | DOI | MR

[14] A. K. Pogrebkov, V. N. Sushko, TMF, 24 (1975), 425 ; 26 (1976), 419 | MR

[15] F. Kolomo, A. G. Izergin, V. E. Korepin, V. Tognetti, TMF, 94 (1993), 19

[16] N. M. Bogolyubov, A. G. Izergin, V. E. Korepin, Korrelyatsionnye funktsii integriruemykh sistem i kvantovyi metod obratnoi zadachi, Nauka, M., 1992 | MR | Zbl

[17] N. A. Slavnov, TMF, 108 (1996), 179 | DOI | MR | Zbl

[18] N. Ilieva, W. Thirring, Eur. Phys. J. C, 6 (1999), 705 | DOI