Geodesic
Spectral Theory of the Nonstationary Schr\"odinger Equation with a~Bidimensionally Perturbed One-Dimensional Potential
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear dynamics, Tome 251 (2005), pp. 10-53.

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

We derive and describe in detail the extension of the inverse scattering transform method to the case of linear spectral problems with potentials that do not decay in some space directions. Our presentation is based on the extended resolvent approach. As a basic example, we consider the nonstationary Schrödinger equation with a potential that is a perturbation of a generic one-dimensional potential by means of a decaying function of two variables. We give the corresponding modifications of the Jost solutions and the spectral data and derive their properties and characterization equations. 
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     author = {M. Boiti and F. Pempinelli and A. K. Pogrebkov and B. Prinari},
     title = {Spectral {Theory} of the {Nonstationary} {Schr\"odinger} {Equation} with {a~Bidimensionally} {Perturbed} {One-Dimensional} {Potential}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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M. Boiti; F. Pempinelli; A. K. Pogrebkov; B. Prinari. Spectral Theory of the Nonstationary Schr\"odinger Equation with a~Bidimensionally Perturbed One-Dimensional Potential. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear dynamics, Tome 251 (2005), pp. 10-53. http://geodesic.mathdoc.fr/item/TM_2005_251_a2/

[1] Kadomtsev B. B., Petviashvili V. I., “Ob ustoichivosti uedinennykh voln v slabo dispergiruyuschikh sredakh”, DAN SSSR, 192 (1970), 753–756 | Zbl

[2] Zakharov V. E., Shabat A. B., “Skhema integrirovaniya nelineinykh uravnenii matematicheskoi fiziki metodom obratnoi zadachi rasseyaniya, I”, Funkts. analiz i ego pril., 8:3 (1974), 43–53 | MR | Zbl

[3] Dryuma V. S., “Ob analiticheskom reshenii dvumernogo uravneniya Kortevega–de Vriza (KDV)”, Pisma v ZhETF, 19:2 (1974), 753–755

[4] Zakharov V. E., Manakov S. V., “Soliton theory”, Sov. Sci. Rev. A.: Phys. Rev., 1 (1979), 133–190 | MR

[5] Manakov S. V., “The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev–Petviashvili equation”, Physica D., 3 (1981), 420–427 | DOI | MR

[6] Fokas A. S., Ablowitz M. J., “On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev–Petviashvili I equation”, Stud. Appl. Math., 69 (1983), 211–228 | MR | Zbl

[7] Xin Zhou, “Inverse scattering transform for the time dependent Schrödinger equation with applications to the KP I equation”, Commun. Math. Phys., 128 (1990), 551–564 | DOI | MR | Zbl

[8] Fokas A. S., Sung L. Y., “The Cauchy problem for the Kadomtsev–Petviashvili I equation without the zero mass constraint”, Math. Proc. Cambridge Phil. Soc., 125 (1999), 113–138 | DOI | MR | Zbl

[9] Boiti M., Pempinelli F., Pogrebkov A. K., Polivanov M. C., “Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schrödinger problem and KP I equation”, TMF, 93:2 (1992), 181–210 | MR | Zbl

[10] Boiti M., Pempinelli F., Pogrebkov A. K., Polivanov M. C., “Resolvent approach for the non-stationary Schrödinger equation”, Inverse Probl., 8 (1992), 331–364 | DOI | MR | Zbl

[11] Boiti M., Pempinelli F., Pogrebkov A. K., “Solutions of the KP I equation with smooth initial data”, Inverse Probl., 10 (1994), 505–519 | DOI | MR | Zbl

[12] Boiti M., Pempinelli F., Pogrebkov A. K., “Properties of solutions of the Kadomtsev–Petviashvili I equation”, J. Math. Phys., 35 (1994), 4683–4718 | DOI | MR | Zbl

[13] Boiti M., Pempinelli F., Pogrebkov A. K., “Some new methods and results in the theory of $(2+1)$-dimensional integrable equations”, TMF, 99:2 (1994), 185–200 | MR | Zbl

[14] Boiti M., Pempinelli F., Pogrebkov A. K., “Spectral theory of solitons on a generic background for the KP I equation”, Nonlinear physics. Theory and experiment, eds. E. Alfinito, M. Boiti, L. Martina, F. Pempinelli, World Sci., Singapore, 1996, 37–52 | MR | Zbl

[15] Boiti M., Pempinelli F., Pogrebkov A. K., “Solving the Kadomtsev–Petviashvili equation with initial data not vanishing at large distances”, Inverse Probl., 13 (1997), L7–L10 | DOI | MR | Zbl

[16] Boiti M., Pempinelli F., Pogrebkov A. K., Prinari B., “K teorii obratnoi zadachi rasseyaniya dlya dvumernykh neubyvayuschikh potentsialov”, TMF, 116:1 (1998), 3–53 | MR | Zbl

[17] Boiti M., Pempinelli F., Pogrebkov A. K., Prinari B., “Inverse scattering theory of the heat equation for the perturbed 1-soliton potential”, J. Math. Phys., 43 (2002), 1044–1062 | DOI | MR | Zbl

[18] Boiti M., Pempinelli F., Pogrebkov A. K., Prinari B., “Extended resolvent and inverse scattering with an application to KP I”, J. Math. Phys., 44 (2003), 3309–3340 | DOI | MR | Zbl

[19] Fokas A. S., Pogrebkov A. K., “Inverse scattering transform for the KPÍ equation on the background of one-line soliton”, Nonlinearity, 16 (2003), 771–783 | DOI | MR | Zbl

[20] Boiti M., Léon J., Pempinelli F., “Spectral transform and orthogonality relations for the Kadomtsev–Petviashvili equation”, Phys. Lett. A., 141 (1989), 96–100 | DOI | MR

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

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