Solitons and large time behavior of solutions of a multidimensional integrable equation

Kazeykina, Anna

Journées équations aux dérivées partielles (2013), article no. 6, 17 p.

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.

Novikov-Veselov equation is a (2+1)-dimensional analog of the classic Korteweg-de Vries equation integrable via the inverse scattering translform for the 2-dimensional stationary Schrödinger equation. In this talk we present some recent results on existence and absence of algebraically localized solitons for the Novikov-Veselov equation as well as some results on the large time behavior of the “inverse scattering solutions” for this equation.

Classification : 35Q53, 37K40, 37K15, 35P25
Keywords: Novikov-Veselov equation, inverse scattering method, two-dimensional Schrödinger equation, solitons, large time behavior
Mot clés : équation de Novikov-Veselov, méthode de diffusion inverse, équation de Schrödinger bidimensionnelle, solitons, comportement en grand temps

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[7] Ferapontov E.V. Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry. Diff. Geom. Appl. 11, 117-128 (1999) | MR | Zbl

[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 | Zbl

[9] Gohberg I.C., Krein M.G. Introduction to the theory of linear nonselfadjoint operators. Moscow: Nauka (1965) | MR

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

[18] Kazeykina A.V., Novikov R.G. Large time asymptotics for the Grinevich–Zakharov potentials. Bulletin des Sciences Mathématiques. 135, 374-382 (2011) | MR | Zbl

[19] Konopelchenko B., Moro A. Integrable equations in nonlinear geometrical optics. Studies in Applied Mathematics. 113(4), 325-352 (2004) | MR | Zbl

[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 | Zbl

[21] Manakov S.V. The inverse scattering method and two-dimensional evolution equations. Uspekhi Mat. Nauk. 31(5), 245–246 (1976) (in Russian) | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[28] Perry P.A. Miura maps and inverse scattering for the Novikov-Veselov equation. Analysis & PDE, to appear. arXiv: 1201.2385v2 (2012) | MR

[29] Tsai T.-Y. The Schrödinger operator in the plane. Inverse Problems. 9, 763-787 (1993) | MR | Zbl

[30] Vekua I.N. Generalized analytic functions. Oxford: Pergamon Press (1962) | MR | Zbl