On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane. (English) Zbl 0557.35110
Summary: The inverse problem associated with a rather general system of n first- order equations in the plane is linearized. When the system is hyperbolic, this is achieved by utilizing a Riemann-Hilbert problem; similarly, a ”\({\bar \partial}''\) (DBAR) problem is used when the system is elliptic. The above result can be employed to linearize the initial value problem associated with a variety of physically significant equations in \(2+1\), i.e., two spatial and one temporal dimensions. Concrete results are given for the n-wave interaction in \(2+1\) and for various forms of the Davey-Stewartson equations. Lump solutions (solitons in \(2+1)\) of the latter equation are given a definitive spectral characterization and are obtained through a linear system of algebraic equations.
MSC:
| 35Q99 | Partial differential equations of mathematical physics and other areas of application |
| 35R30 | Inverse problems for PDEs |
| 35P25 | Scattering theory for PDEs |
| 35F20 | Nonlinear first-order PDEs |
Keywords:
inverse scattering transform; inverse problem; system of n first-order equations; n-wave interaction; Davey-Stewartson equations; Lump solutionsReferences:
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