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On the Inverse Scattering of the Time‐Dependent Schrödinger Equation and the Associated Kadomtsev‐Petviashvili (I) Equation
Author(s) -
Fokas A. S.,
Ablowitz M. J.
Publication year - 1983
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1983693211
Subject(s) - inverse scattering transform , inverse scattering problem , mathematics , mathematical analysis , scattering , sign (mathematics) , korteweg–de vries equation , initial value problem , kadomtsev–petviashvili equation , riemann–hilbert problem , infinity , algebraic number , inverse problem , inverse , algebraic equation , scattering theory , mathematical physics , characteristic equation , partial differential equation , physics , nonlinear system , quantum mechanics , boundary value problem , geometry
The Kadomtsev‐Petviashvili equation, a two‐spatial‐dimensional analogue of the Korteweg‐deVries equation, arises in physical situations in two different forms depending on a certain sign appearing in the evolution equation. Here we investigate one of the two cases. The initial‐value problem, associated with initial data decaying sufficiently rapidly at infinity, is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation of a nonlocal Riemann‐Hilbert problem in terms of scattering data expressible in closed form in terms of given initial data. The lump solutions, algebraically decaying solitons, are given a definite spectral characterization. Pure lump solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on x, y, t . Many of the above results are also relevant to the problem of inverse scattering for the so‐called time‐dependent Schrödinger equation.