  In this paper we present some open problems pertaining to the approximation
theory involved in the solution of the important class of Nonlinear Partial
Differential Equations (NPDEs) of integrable type. For this class of NPDEs, any
Initial Value Problem (IVP) can be theoretically solved by the Inverse
Scattering Transform (IST) technique whose main steps involve the solution of
Volterra equations with structured kernels on unbounded domains, the solution
of Fredholm integral equations and the identification of coefficients and
parameters of monomial-exponential sums. The aim of this paper is twofold:
propose a method for solving the above mentioned problems under particular
hypothesis and arouse interest in these problems in order to develop an
effective method which works under more general assumptions.
