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Inverse Spectral Transform for the Nonlinear Evolution Equation Generating the Davey‐Stewartson and Ishimori Equations
Author(s) -
Konopelchenko B. G.,
Matkarimov B. T.
Publication year - 1990
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/sapm1990824319
Subject(s) - mathematics , inverse scattering transform , eigenfunction , mathematical analysis , nonlinear system , integrable system , korteweg–de vries equation , exponential function , differential equation , spectral theory of ordinary differential equations , inverse problem , inverse scattering problem , eigenvalues and eigenvectors , physics , quasinormal operator , quantum mechanics , banach space , finite rank operator
A 2 + 1‐dimensional nonlinear differential equation integrable by the inverse‐spectral‐transform method with the quartet operator representation is proposed. This GL(2, C )‐valued chiral‐field‐type equation is the generating (prototype) equation for the Davey‐Stewartson and Ishimori equations. It coincides with the nonlinear equation for the Davey‐Stewartson eigenfunction ψ DS . The initial‐value problem for this equation is solved by the techniques for the and the nonlocal Riemann‐Hilbert problem. The classes of exact solutions with the functional parameters and exponential‐rational solutions are constructed by the method. The static lump solution in the case α = i and the exponentially localized solution at α = i are found. Other similar examples of nonlinear integrable equations in 2 + 1 and 1 + 1 dimensions are discussed.