An Interdisciplinary Journal

* 2015, Vol.18, No.3, pp.381-391*

The inverse scattering transform method for solving nonlinear integrable partial differential
equations is a nonlinear analogue of the Fourier transform method for solving suitable initial-
value problems for linear partial differential 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.

*Key words: *
nonlinear Fourier transform, inverse scattering method, integrable nonlinear differential equations

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Last updated: *October 07, 2015*