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International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 58, Pages 3117-3128
http://dx.doi.org/10.1155/S0161171204312408

On a direct construction of inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension

1Department of Physics and Astronomy, University of Maryland, College Park 20742, MD, USA
2Department of Mathematical Sciences, George Mason University, Fairfax 22030, VA, USA

Received 30 December 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a method to construct inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension. The temporal component is the adjoint of the linearized equation and the spatial component is a partial differential equation with respect to the spatial variables. Although this idea has been known for the one-spatial dimension for some time, it is the first time that this method is presented for the case of the higher-spatial dimension. We present this method in detail for the Veselov-Novikov equation and the Kadomtsev-Petviashvili equation.