Large-time asymptotic of solutions of the nonlinear Schrodinger equation in $2+1$~dimensions
Izvestiya. Mathematics , Tome 43 (1994) no. 2, pp. 373-384.

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

Large-time asymptotics is found for solutions of the Cauchy problem for the nonlinear Schrodinger equation in $2+1$ dimensions. The leading term of the asymptotic expression decreases like $t^{-1}$ as time increases.
@article{IM2_1994_43_2_a9,
     author = {P. I. Naumkin},
     title = {Large-time asymptotic of solutions of the nonlinear {Schrodinger} equation in $2+1$~dimensions},
     journal = {Izvestiya. Mathematics },
     pages = {373--384},
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1994_43_2_a9/}
}
TY  - JOUR
AU  - P. I. Naumkin
TI  - Large-time asymptotic of solutions of the nonlinear Schrodinger equation in $2+1$~dimensions
JO  - Izvestiya. Mathematics 
PY  - 1994
SP  - 373
EP  - 384
VL  - 43
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1994_43_2_a9/
LA  - en
ID  - IM2_1994_43_2_a9
ER  - 
%0 Journal Article
%A P. I. Naumkin
%T Large-time asymptotic of solutions of the nonlinear Schrodinger equation in $2+1$~dimensions
%J Izvestiya. Mathematics 
%D 1994
%P 373-384
%V 43
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1994_43_2_a9/
%G en
%F IM2_1994_43_2_a9
P. I. Naumkin. Large-time asymptotic of solutions of the nonlinear Schrodinger equation in $2+1$~dimensions. Izvestiya. Mathematics , Tome 43 (1994) no. 2, pp. 373-384. http://geodesic.mathdoc.fr/item/IM2_1994_43_2_a9/

[1] Kelley P. L., “Self-focusing of optical beams”, Phys. Rev. Lett., 15 (1965), 1005–1008 | DOI

[2] Talanov V. I., “O samofokusirovke volnovykh luchei v nelineinykh sredakh”, Pisma v ZhETF, 2:5 (1965), 218–222

[3] Taniuti T., Washimi H., “Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma”, Phys. Rev. Lett., 21 (1968), 209–212 | DOI

[4] Zakharov V. E., “Ustoichivost periodicheskikh voln konechnoi amplitudy na poverkhnosti glubokoi zhidkosti”, PMTF, 1968, no. 6, 86–95

[5] Ablovits M., Sigur X., Solitony i metod obratnoi zadachi, Mir, M., 1987, 479 pp. | MR

[6] Dodd R., Eilbek Dzh., Gibbon Dzh., Morris X., Solitony i nelineinye volnovye uravneniya, Mir, M., 1988, 694 pp. | MR

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

[8] Nyuell A., Solitony v matematike i fizike, Mir, M., 1989, 326 pp. | MR

[9] Takhtadzhyan L. L., Faddeev L. D., Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986, 528 pp. | MR | Zbl

[10] Zakharov V. E., Synakh B. C., “O kharaktere osobennosti pri samofokusirovke”, ZhETF, 68:3 (1975), 940–947

[11] Zakharov V. E., Shabat A. B., “Tochnaya teoriya dvumernoi samofokusirovki i odnomernoi avtomodulyatsii voln v nelineinoi srede”, ZhETF, 61 (1971), 118–134

[12] Zakharov V. E., Manakov S. V., “Asimptoticheskoe povedenie nelineinykh volnovykh sistem, integriruemykh metodom obratnoi zadachi”, ZhETF, 71 (1976), 203–215 | MR

[13] Manakov S. V., “Nelineinaya difraktsiya Fraungofera”, ZhETF, 65 (1973), 1392–1398

[14] Segur H., Ablowitz M. J., “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation, I”, J. Math. Phys., 17 (1976), 710–713 | DOI | MR

[15] Segur H., “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. II”, J. Math. Phys., 17 (1976), 714–716 | DOI | MR

[16] Novokshenov V. Yu., “Asimptotika pri $t\to\infty$ resheniya zadachi Koshi dlya nelineinogo uravneniya Shredingera”, DAN SSSR, 251:4 (1980), 799–801 | MR

[17] Its A. R., “Asimptotika reshenii nelineinogo uravneniya Shredingera i izomonodromnye deformatsii sistem lineinykh differentsialnykh uravnenii”, DAN SSSR, 261:1 (1981), 14–18 | MR | Zbl

[18] Ginibre J., Veto G., “On a class of nonlinear Schrödinger equations”, J. Funct. Anal., 32 (1979), 1–71 | DOI | MR

[19] Koto T., “On nonlinear Schrödinger equations”, Ann. Inst. Henri Poincaré. Physique Théorique, 46 (1987), 113–129 | MR

[20] Pecher H., von Wahl W., “Time dependent nonlinear Schrödinger equations”, Manuscripta Math., 27 (1979), 125–157 | DOI | MR | Zbl

[21] Strauss W. A., “Nonlinear Scattering theory at low energy”, J. Funct. Anal., 41:1 (1981), 110–133 ; 43:3, 281–293 | DOI | MR | Zbl | DOI | Zbl

[22] Glassey R. T., “On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations”, J. Math. Phys., 18 (1977), 1794–1797 | DOI | MR | Zbl

[23] Merle F., “On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrodinger equation with critical exponent and critical mass”, Comm. Pure Appl. Math., 45 (1992), 203–254 | DOI | MR | Zbl

[24] Tsutsumi M., Hayashi N., “Classical solutions of nonlinear Schrödinger equations in higher dimensions”, Math. Z., 177 (1981), 217–237 | DOI | MR

[25] Weinstein M. J., “The nonlinear Schrödinger equation-Singularity formation. Stability and dispersion”, Contemporary Mathematics, 99 (1989), 213–232 | MR | Zbl

[26] Constantin P., Saut J.-C., “Local smoothing properties of Schrödinger equations”, Indiana Univ. Math. J., 38 (1989), 791–810 | DOI | MR | Zbl

[27] Cazenave T., An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos, 22, Rio de Janeiro, 1989

[28] Naumkin P. I., Shishmarev I. A., “Ob asimptotike pri $t\to\infty$ reshenii nelineinykh evolyutsionnykh uravnenii s dissipatsiei”, Matem. zametki, 45:4 (1989), 118–121 | MR | Zbl

[29] Naumkin P. I., Shishmarev I. A., “Asimptotika pri $t\to\infty$ reshenii nelineinogo nelokalnogo uravneniya Shredingera”, Matem. sb., 182:7 (1991), 1024–1042

[30] Fedoryuk M. V., Asimptotika: Integraly i ryady, Nauka, M., 1987, 544 pp. | MR