Eigenfunctions and eigenvalues for a scalar Riemann-Hilbert problem associated to inverse scattering

A complete set of eigenfunctions is introduced within the Riemann-Hilbert f ormalism for spectral problems associated to some solvable nonlinear evolut ion equations. In particular, we consider the time-independent and time-dep endent Schrodinger problems which are related to the KdV and KPI equations possessing solitons and lumps, respectively. Non-standard scalar products, orthogonality and completeness relations are derived for these problems. Th e complete set of eigenfunctions is used for perturbation theory and bifurc ation analysis of eigenvalues supported by the potentials under perturbatio ns. We classify two different types of bifurcations of new eigenvalues and analyze their characteristic features. One type corresponds to thresholdles s generation of solitons in the KdV equation, while the other predicts a th reshold for generation of lumps in the KPI equation.