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 TMF, 1994, Volume 99, Number 2, Pages 185–200 (Mi tmf1577)

Some new methods and results in the theory of ($2+1$)-dimensional integrable equations

M. Boitia, F. Pempinellia, A. K. Pogrebkovb

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

Abstract: The general resolvent scheme for solving nonlinear integrable evolution equations is formulated. Special attention is paid for the problem of nontrivial dressing and corresponding transformation of spectral data. Kadomtsev–Petviashvili equation is considered as the standard example of integrable models in $2+1$ dimensions. Properties of the solution $u(t,x,y)$ of the Kadomtsev–Petviashvili I equation as well as corresponding Jost solutions and spectral data with given initial data $u(0,x,y)$ belonging to the Schwartz space are presented.

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Theoretical and Mathematical Physics, 1994, 99:2, 511–522

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Citation: M. Boiti, F. Pempinelli, A. K. Pogrebkov, “Some new methods and results in the theory of ($2+1$)-dimensional integrable equations”, TMF, 99:2 (1994), 185–200; Theoret. and Math. Phys., 99:2 (1994), 511–522

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\Bibitem{BoiPemPog94} \by M.~Boiti, F.~Pempinelli, A.~K.~Pogrebkov \paper Some new methods and results in the theory of ($2+1$)-dimensional integrable equations \jour TMF \yr 1994 \vol 99 \issue 2 \pages 185--200 \mathnet{http://mi.mathnet.ru/tmf1577} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1308779} \zmath{https://zbmath.org/?q=an:0850.35094} \transl \jour Theoret. and Math. Phys. \yr 1994 \vol 99 \issue 2 \pages 511--522 \crossref{https://doi.org/10.1007/BF01016132} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994PV07100004} 

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This publication is cited in the following articles:
1. Theoret. and Math. Phys., 99:2 (1994), 583–587
2. T. I. Garagash, A. K. Pogrebkov, “Scattering problem for the differential operator $\partial_x\partial_y+1+a(x,y)\partial_y+ b(x,y)$”, Theoret. and Math. Phys., 102:2 (1995), 117–132
3. 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
4. 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
5. 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
6. Prinari, B, “On some nondecaying potentials and related Jost solutions for the heat conduction equation”, Inverse Problems, 16:3 (2000), 589
7. Boiti, M, “Towards an inverse scattering theory for non-decaying potentials of the heat equation”, Inverse Problems, 17:4 (2001), 937
8. Boiti, M, “Extended resolvent and inverse scattering with an application to KPI”, Journal of Mathematical Physics, 44:8 (2003), 3309
9. 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
10. 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
11. Boiti, M, “Scattering transform for nonstationary Schrodinger equation with bidimensionally perturbed N-soliton potential”, Journal of Mathematical Physics, 47:12 (2006), 123510
12. 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
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