Some Analytical Aspects of the Nonlinear Fourier Transform

Abstract

The inverse scattering transform method for solving nonlinear integrable partial di˙erential equations is a nonlinear analogue of the Fourier transform method for solving suitable initial-value problems for linear partial di˙erential equations. Therefore, the scattering transform is often called the nonlinear Fourier transform. The nonlinear Fourier transform F and its inverse G are analytically computable only for some very special arguments. Therefore, it makes sense to look for perturbational approximations of these transforms. In the paper, we propose an iterative method for constructing arbitrarily good approximations of G for an arbitrary argument. We discuss analytical properties which guarantee that the iterative formula for G converges. We also provide an explicit convergent power series for the calculation of F in powers of the spectral parameter. We expect that this formula will be useful in the study of certain analytical properties of F described by the Paley-Wiener type of theorems.