M. Boiti, F. Pempinelli, A. K. Pogrebkov, M. K. Polivanov, “Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schrödinger problem and KPI equation”, TMF, 93:2 (1992), 181–210; Theoret. and Math. Phys., 93:2 (1992), 1200

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Teoreticheskaya i Matematicheskaya Fizika, 1992, Volume 93, Number 2, Pages 181–210 (Mi tmf1522)

This article is cited in 23 scientific papers (total in 23 papers)

Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schrödinger problem and KPI equation

M. Boiti^a, F. Pempinelli^a, A. K. Pogrebkov^b, M. K. Polivanov

^a Lecce University
^b Steklov Mathematical Institute, Russian Academy of Sciences

Full-text PDF (2331 kB) Citations (23)

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Abstract: The resolvent operator of the Linear Problem is determined as full Green function continued in the complex domain in two variables. An analog of the known Hilbert Identity is derived. We demonstrate the role of this identity in the study of two-dimensional scattering. Considering the Nonstationary Schrödinger Equation as an example we show that all types of solutions of the Linear Problem as well as Spectral Data known in the literature are given as specific values of this unique function — resolvent. New form of Inverse Problem is formulated.

Received: 28.09.1992

English version:
Theoretical and Mathematical Physics, 1992, Volume 93, Issue 2, Pages 1200–1224
DOI: https://doi.org/10.1007/BF01083519

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: M. Boiti, F. Pempinelli, A. K. Pogrebkov, M. K. Polivanov, “Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schrödinger problem and KPI equation”, TMF, 93:2 (1992), 181–210; Theoret. and Math. Phys., 93:2 (1992), 1200–1224

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/tmf1522

https://www.mathnet.ru/eng/tmf/v93/i2/p181

This publication is cited in the following 23 articles:

A. K. Pogrebkov, “Commutator identities on associative algebras, the non-Abelian Hirota difference equation and its reductions”, Theoret. and Math. Phys., 187:3 (2016), 823–834
A. K. Pogrebkov, “Hirota difference equation: Inverse scattering transform, Darboux transformation, and solitons”, Theoret. and Math. Phys., 181:3 (2014), 1585–1598
St. Petersburg Math. J., 22:3 (2011), 473–483
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
A. K. Pogrebkov, “Commutator identities on associative algebras and the integrability of nonlinear evolution equations”, Theoret. and Math. Phys., 154:3 (2008), 405–417
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 Schrodinger Equation with a Two-Dimensionally Perturbed Arbitrary One-Dimensional Potential”, Theoret. and Math. Phys., 144:2 (2005), 1100–1116
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
Boiti, M, “Extended resolvent and inverse scattering with an application to KPI”, Journal of Mathematical Physics, 44:8 (2003), 3309
A S Fokas, A K Pogrebkov, “Inverse scattering transform for the KPI equation on the background of a one-line soliton*”, Nonlinearity, 16:2 (2003), 771
Boiti, M, “Towards an inverse scattering theory for non-decaying potentials of the heat equation”, Inverse Problems, 17:4 (2001), 937
Prinari, B, “On some nondecaying potentials and related Jost solutions for the heat conduction equation”, Inverse Problems, 16:3 (2000), 589
Pelinovsky, DE, “Eigenfunctions and eigenvalues for a scalar Riemann–Hilbert problem associated to inverse scattering”, Communications in Mathematical Physics, 208:3 (2000), 713
A. K. Pogrebkov, M. C. Prati, “An Ablowitz–Ladik system with a discrete potential: I. Extended resolvent”, Theoret. and Math. Phys., 119:1 (1999), 407–419
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
M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Towards an inverse scattering theory for two-dimensional nondecaying potentials”, Theoret. and Math. Phys., 116:1 (1998), 741–781
M Boiti, F Pempinelli, A Pogrebkov, “Solving the Kadomtsev - Petviashvili equation with initial data not vanishing at large distances”, Inverse Problems, 13:3 (1997), L7
A. K. Pogrebkov, T. I. Garagash, “On a solution of the Cauchy problem for the Boiti–Leon–Pempinelli equation”, Theoret. and Math. Phys., 109:2 (1996), 1369–1378
A. K. Pogrebkov, M. C. Prati, “Resolvent approach to the ablowitz-ladik linear system”, Nuovo Cim B, 111:12 (1996), 1495

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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