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.
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
\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://mathscinet.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{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1994PV07100004}
Linking options:
https://www.mathnet.ru/eng/tmf1577
https://www.mathnet.ru/eng/tmf/v99/i2/p185
This publication is cited in the following 17 articles:
Ion Bica, Randy Wanye K, “Modeling rogue waves with the Kadomtsev–Petviashvili (KP) equation”, Rocky Mountain J. Math., 48:5 (2018)
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
Boiti, M, “Extended resolvent and inverse scattering with an application to KPI”, Journal of Mathematical Physics, 44:8 (2003), 3309
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
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
T. I. Garagash, A. K. Pogrebkov, “Scattering problem for the differential operator ∂x∂y+1+a(x,y)∂y+b(x,y)”, Theoret. and Math. Phys., 102:2 (1995), 117–132
M. Boiti, F. Pempinelli, A. Pogrebkov, “Dressing of a two-dimensional nontrivial potential”, Physica D: Nonlinear Phenomena, 87:1-4 (1995), 123