Abstract:
The inverse scattering method is considered for the nonstationary Schrödinger equation with the potential $u(x_{1},x_{2})$ nondecaying in a finite number of directions in the $x$ plane. The general resolvent approach, which is particularly convenient for this problem, is tested using a potential that is the Bäcklund transformation of an arbitrary decaying potential and that describes a soliton superimposed on an arbitrary background. In this example, the resolvent, Jost solutions, and spectral data are explicitly constructed, and their properties are analyzed. The characterization equations satisfied by the spectral data are derived, and the unique solution of the inverse problem is obtained. The asymptotic potential behavior at large distances is also studied in detail. The obtained resolvent is used in a dressing procedure to show that with more general nondecaying potentials, the Jost solutions may have an additional cut in the spectral-parameter complex domain. The necessary and sufficient condition for the absence of this additional cut is formulated.
Citation:
M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Towards an inverse scattering theory for two-dimensional nondecaying potentials”, TMF, 116:1 (1998), 3–53; Theoret. and Math. Phys., 116:1 (1998), 741–781
\Bibitem{BoiPemPog98}
\by M.~Boiti, F.~Pempinelli, A.~K.~Pogrebkov, B.~Prinari
\paper Towards an inverse scattering theory for two-dimensional nondecaying potentials
\jour TMF
\yr 1998
\vol 116
\issue 1
\pages 3--53
\mathnet{http://mi.mathnet.ru/tmf888}
\crossref{https://doi.org/10.4213/tmf888}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1700690}
\zmath{https://zbmath.org/?q=an:0951.35118}
\transl
\jour Theoret. and Math. Phys.
\yr 1998
\vol 116
\issue 1
\pages 741--781
\crossref{https://doi.org/10.1007/BF02557122}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000076425700001}
Linking options:
https://www.mathnet.ru/eng/tmf888
https://doi.org/10.4213/tmf888
https://www.mathnet.ru/eng/tmf/v116/i1/p3
This publication is cited in the following 16 articles:
M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Building an extended resolvent of the heat operator via twisting transformations”, Theoret. and Math. Phys., 159:3 (2009), 721–733
Boiti, M, “Scattering transform for nonstationary Schrodinger equation with bidimensionally perturbed N-soliton potential”, Journal of Mathematical Physics, 47:12 (2006), 123510
Boiti, M, “On the extended resolvent of the nonstationary Schrodinger operator for a Darboux transformed potential”, Journal of Physics A-Mathematical and General, 39:8 (2006), 1877
M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Spectral Theory of the Nonstationary Schrödinger Equation with a Bidimensionally Perturbed One-Dimensional Potential”, Proc. Steklov Inst. Math., 251 (2005), 6–48
M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Spectral Theory of the Nonstationary Schrodinger Equation with a Two-Dimensionally Perturbed Arbitrary One-Dimensional Potential”, Theoret. and Math. Phys., 144:2 (2005), 1100–1116
O. M. Kiselev, “Asymptotics of solutions of higher-dimensional integrable equations and their perturbations”, Journal of Mathematical Sciences, 138:6 (2006), 6067–6230
Boiti, M, “Extended resolvent and inverse scattering with an application to KPI”, Journal of Mathematical Physics, 44:8 (2003), 3309
Li-Yeng Sung, Scattering, 2002, 1717
Boiti, M, “Towards an inverse scattering theory for non-decaying potentials of the heat equation”, Inverse Problems, 17:4 (2001), 937
Boiti, M, “Inverse scattering transform for the perturbed 1-soliton potential of the heat equation”, Physics Letters A, 285:5–6 (2001), 307
Prinari, B, “On some nondecaying potentials and related Jost solutions for the heat conduction equation”, Inverse Problems, 16:3 (2000), 589
Fokas, AS, “On the integrability of linear and nonlinear partial differential equations”, Journal of Mathematical Physics, 41:6 (2000), 4188
Boiti M., Pempinelli F., Prinari B., Pogrebkov A.K., “Some nondecaying potentials for the nonstationary Schrodinger equation”, Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years After Needs '79, 2000, 33–41
Boiti M., Pempinelli F., Prinari B., Pogrebkov A.K., “Some nondecaying potentials for the heat conduction equation”, Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years After Needs '79, 2000, 42–50
M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Bäcklund and Darboux Transformations for the Nonstationary Schrödinger Equation”, Proc. Steklov Inst. Math., 226 (1999), 42–62
Boiti M., Pempinelli F., Prinari B., Pogrebkov A.K., “N-wave soliton solution on a generic background for KPI equation”, International Seminar Day on Diffraction, Proceedings, 1999, 167–175
Citation in format AMSBIB
Contact us: