Centre de diffusion de revues académiques mathématiques

 
 
 
 

Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent | Article suivant
Anna Kazeykina
Solitons and large time behavior of solutions of a multidimensional integrable equation
(Solitons et comportement en grand temps des solutions d’une équation multidimensionnelle intégrable)
Journées équations aux dérivées partielles (2013), Exp. No. 6, 17 p., doi: 10.5802/jedp.102
Article PDF
Class. Math.: 35Q53, 37K40, 37K15, 35P25
Mots clés: équation de Novikov-Veselov, méthode de diffusion inverse, équation de Schrödinger bidimensionnelle, solitons, comportement en grand temps

Résumé - Abstract

L’équation de Novikov-Veselov est un analogue (2+1)-dimensionnel de l’équation classique de Korteweg-de Vries, intégrable via la transformation de diffusion inverse pour l’équation de Schrödinger bidimensionnelle stationnaire. Dans cet exposé on présente quelques résultats récents sur l’existence et l’absence de solitons algébriquement localisés pour l’équation de Novikov-Veselov ainsi que quelques résultats sur le comportement en grand temps des “inverse scattering” solutions de cette équation.

Bibliographie

[1] Boiti M., Leon J.J.-P., Manna M., Pempinelli F. On a spectral transform of a KdV-like equation related to the Schrödinger operator in the plane. Inverse Problems. 3, 25–36 (1987)  MR 875315 |  Zbl 0624.35071
[2] Bogdanov L. V. The Veselov-Novikov equation as a natural generalization of the Korteweg-de Vries equation. Teoret. Mat. Fiz. 70(2), 309-314 (1987), translation in Theoret. and Math. Phys. 70(2), 219-223 (1987)  MR 894472 |  Zbl 0639.35072
[3] de Bouard A., Saut J.-C. Solitary waves of generalized Kadomtsev-Petviashvili equations. Ann. Inst. Henri Poincaré, Analyse Non Linéaire. 14(2), 211-236 (1997) Numdam |  MR 1441393 |  Zbl 0883.35103
[4] Calderón A. P. On an inverse boundary problem. Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasiliera de Matematica, Rio de Janeiro. 61-73 (1980)
[5] Chang J.-H. The Gould-Hopper polynomials in the Novikov-Veselov equation. J. Math. Phys. 52(9), 092703 (2011)  MR 2867810 |  Zbl 1272.35168
[6] Faddeev L.D. Growing solutions of the Schrödinger equation. Dokl. Akad. Nauk SSSR. 165(3), 514-517 (1965), translation in Sov. Phys. Dokl. 10, 1033-1035 (1966)  Zbl 0147.09404
[7] Ferapontov E.V. Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry. Diff. Geom. Appl. 11, 117-128 (1999)  MR 1712135 |  Zbl 0990.53008
[8] Gelfand I.M. Some aspects of functional analysis and algebra. Proceedings of the International Congress of Mathematicians. Amsterdam: Erven P. Noordhoff N.V., Groningen; North-Holland Publishing Co. 1, 253-276 (1954)  MR 95423 |  Zbl 0079.32602
[9] Gohberg I.C., Krein M.G. Introduction to the theory of linear nonselfadjoint operators. Moscow: Nauka (1965)  MR 220070
[10] Grinevich P.G. Rational solitons of the Veselov–Novikov equation are reflectionless potentials at fixed energy. Teoret. Mat. Fiz. 69(2), 307-310 (1986), translation in Theor. Math. Phys. 69, 1170-1172 (1986)  MR 884498 |  Zbl 0617.35121
[11] Grinevich P.G. Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity. Russ. Math. Surv. 55(6), 1015–1083 (2000)  MR 1840357 |  Zbl 1022.81057
[12] Grinevich P.G., Novikov, R.G. Transparent potentials at fixed energy in dimension two. Fixed energy dispersion relations for the fast decaying potentials. Commun. Math. Phys. 174, 409-446 (1995)  MR 1362172 |  Zbl 0843.35090
[13] Grinevich P.G., Novikov S.P. Two-dimensional “inverse scattering problem” for negative energies and generalized-analytic functions. I. Energies below the ground state. Funkts. Anal. Prilozh. 22(1), 23-33 (1988), translation in Funct. Anal. Appl. 22(1), 19-27 (1988)  MR 936696 |  Zbl 0672.35074
[14] Kazeykina A.V. A large time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with non-singular scattering data. Inverse Problems. 28(5), 055017 (2012)  MR 2923202 |  Zbl 1238.35134
[15] Kazeykina A.V. Kazeykina A.V. Absence of conductivity-type solitons for the Novikov-Veselov equation at zero energy. Funct. Anal. Appl., 47(1), 64-66 (2013)  Zbl 1280.35127
[16] Kazeykina A.V. Absence of solitons with sufficient algebraic localization for the Novikov-Veselov equation at nonzero energy. Funct. Anal. Appl., 48(1), 24-35 (2014)
[17] Kazeykina A.V., Novikov R.G. A large time asymptotics for transparent potentials for the Novikov–Veselov equation at positive energy. J. Nonlinear Math. Phys. 18(3), 377-400 (2011)  MR 2846100 |  Zbl 1228.35203
[18] Kazeykina A.V., Novikov R.G. Large time asymptotics for the Grinevich–Zakharov potentials. Bulletin des Sciences Mathématiques. 135, 374-382 (2011)  MR 2799814 |  Zbl 1219.35237
[19] Konopelchenko B., Moro A. Integrable equations in nonlinear geometrical optics. Studies in Applied Mathematics. 113(4), 325-352 (2004)  MR 2094235 |  Zbl 1141.78302
[20] Lassas M., Mueller J.L., Siltanen S., Stahel A. The Novikov-Veselov Equation and the Inverse Scattering Method, Part I: Analysis. Physica D. 241, 1322-1335 (2012)  MR 2947348 |  Zbl 1248.35187
[21] Manakov S.V. The inverse scattering method and two-dimensional evolution equations. Uspekhi Mat. Nauk. 31(5), 245–246 (1976) (in Russian)  MR 467037 |  Zbl 0345.35055
[22] Manakov S.V., Zakharov V.E., Bordag L.A., Its A.R., Matveev V.B. Two–dimensional solitons of the Kadomtsev–Petviashvili equation and their interaction. Physics Letters A. 63(3), 205–206 (1977)
[23] Nachman A.I. Global uniqueness for a two-dimensional inverse boundary value problem. Annals of Mathematics. 143, 71-96 (1995)  MR 1370758 |  Zbl 0857.35135
[24] Novikov R.G. The inverse scattering problem on a fixed energy level for the two–dimensional Schrödinger operator. Journal of Funct. Anal. 103, 409-463 (1992)  MR 1151554 |  Zbl 0762.35077
[25] Novikov R.G. Absence of exponentially localized solitons for the Novikov–Veselov equation at positive energy. Physics Letters A. 375, 1233-1235 (2011)  MR 2770407 |  Zbl 1242.35196
[26] Novikov S.P., Veselov A.P. Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formula and evolutions equations. Dokl. Akad. Nauk SSSR. 279, 20–24 (1984), translation in Sov. Math. Dokl. 30, 588-591 (1984)  MR 769198 |  Zbl 0613.35020
[27] Novikov S.P., Veselov A.P. Finite-zone, two-dimensional Schrödinger operators. Potential operators. Dokl. Akad. Nauk SSSR. 279, 784–788 (1984), translation in Sov. Math. Dokl. 30, 705–708 (1984)  MR 771574 |  Zbl 0602.35024
[28] Perry P.A. Miura maps and inverse scattering for the Novikov-Veselov equation. Analysis & PDE, to appear. arXiv: 1201.2385v2 (2012)  MR 3218811
[29] Tsai T.-Y. The Schrödinger operator in the plane. Inverse Problems. 9, 763-787 (1993)  MR 1251205 |  Zbl 0797.35140
[30] Vekua I.N. Generalized analytic functions. Oxford: Pergamon Press (1962)  MR 150320 |  Zbl 0100.07603
Copyright Cellule MathDoc 2018 | Crédit | Plan du site