Geodesic
Investigation of the exact integrability of the multiwave Schrödinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 110 (1997) no. 2, pp. 242-253 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

It is shown that the multiwave nonlinear Schrödinger equation, which describes evolution of several quasi-monochromatic waves with equal group velocities, is not exactly integrable: it does not have infinite sequence of local conservation laws and symmetries. Exact integrability of the system of equations $w_t^i=\alpha_iw_{xx}^i+a_{klm}^iw^kw^lw^m$ with the nondegenerate diagonal matrix at the second-order derivative is studied. 
@article{TMF_1997_110_2_a4,
     author = {S. V. Belyutin},
     title = {Investigation of the exact integrability of the multiwave {Schr\"odinger} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {242--253},
     year = {1997},
     volume = {110},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1997_110_2_a4/}
}
TY  - JOUR
AU  - S. V. Belyutin
TI  - Investigation of the exact integrability of the multiwave Schrödinger equation
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1997
SP  - 242
EP  - 253
VL  - 110
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_1997_110_2_a4/
LA  - ru
ID  - TMF_1997_110_2_a4
ER  - 
%0 Journal Article
%A S. V. Belyutin
%T Investigation of the exact integrability of the multiwave Schrödinger equation
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1997
%P 242-253
%V 110
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_1997_110_2_a4/
%G ru
%F TMF_1997_110_2_a4
S. V. Belyutin. Investigation of the exact integrability of the multiwave Schrödinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 110 (1997) no. 2, pp. 242-253. http://geodesic.mathdoc.fr/item/TMF_1997_110_2_a4/

[1] R. Dodd, Dzh. Eilbek, Dzh. Gibbon, Kh. Morris, Solitony i nelineinye volnovye uravneniya, Mir, M., 1988 | MR

[2] A. Nyuell, Solitony v matematike i fizike, Mir, M., 1989 | MR

[3] H. C. Yuen, B. M. Lake, Adv. Appl. Mech., 22 (1982), 67–229 | DOI | MR | Zbl

[4] A. S. Monin, “O kogerentnykh strukturakh v turbulentnykh techeniyakh”, Etyudy o turbulentnosti, Nauka, M., 1994, 8–16

[5] D. J. Benney, A. C. Newell, J. Math. and Phys., 46 (1967), 133–139 | DOI | MR | Zbl

[6] D. J. Benney, G. J. Roskes, Studies Appl. Math., 48:4 (1969), 377–385 | DOI | Zbl

[7] S. V. Belyutin, Modulyatsionnaya neustoichivost voln i rezonans gruppovykh skorostei v dvukhsloinoi zhidkosti, Diss. ...kand. fiz.-matem. nauk, TGU, Tyumen, 1993

[8] V. E. Zakharov, A. B. Shabat, ZhETF, 61 (1971), 118–134

[9] V. E. Zakharov, S. V. Manakov, TMF, 19:3 (1974), 332–343 | MR | Zbl

[10] S. V. Manakov, ZhETF, 65:2 (1973), 505–516

[11] S. I. Svinolupov, Commun. Math. Phys., 143 (1992), 559–575 | DOI | MR | Zbl

[12] A. V. Mikhailov, A. B. Shabat, V. V. Sokolov, “Simmetriinyi podkhod k klassifikatsii integriruemykh uravnenii”, Integriruemost i kineticheskie uravneniya dlya solitonov, Naukova dumka, Kiev, 1990 | Zbl

[13] A. V. Mikhailov, A. B. Shabat, TMF, 62:2 (1985), 163–185 | MR | Zbl

[14] A. Fordy, P. Kulish, Commun. Math. Phys., 89 (1983), 427–443 | DOI | MR | Zbl

[15] V. E. Zakharov, E. A. Kuznetsov, Physica D, 18 (1986), 455–463 | DOI | MR | Zbl

[16] F. Kalodzhero, “Pochemu nekotorye uravneniya odnovremenno shiroko rasprostraneny i integriruemy?”, Integriruemost i kineticheskie uravneniya dlya solitonov, Naukova dumka, Kiev, 1990