Riemann–Hilbert approach for a Schrödinger-type equation with nonzero boundary conditions

In this paper, we focus on investigating a nonlinear Schrödinger-type equation with nonzero boundary at infinity. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter [Formula: see text] into a single-valued parameter [Formula: see text]. Starting from the Lax pair of the Schrödinger-type equation, we derive its Jost solutions with nonzero boundary conditions, and further analyze the asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. An associated matrix Riemann–Hilbert (RH) problem associated with the problem of nonzero boundary conditions is subsequently presented, and a formulae of [Formula: see text]-soliton solutions for the Schrödinger-type equation by solving the matrix RH problem. As an application of the [Formula: see text]-soliton formulae, we present two kinds of one-soliton solutions and three kinds of two-soliton solutions according to different distributions of spectral parameters, and dynamical features of those solutions are also further analyzed.

The Riemann–Hilbert approach to focusing Kundu–Eckhaus equation with non-zero boundary conditions

In this paper, we investigate the focusing Kundu–Eckhaus equation with non-zero boundary conditions. An appropriate two-sheeted Riemann surface is introduced to map the spectral parameter [Formula: see text] into a single-valued parameter [Formula: see text]. Starting from the Lax pair of Kundu–Eckhaus equation, two kinds of Jost solutions are constructed. Further, their asymptotic, analyticity, symmetries as well as spectral matrix are analyzed in detail. It is shown that the solution of the Kundu–Eckhaus equation with non-zero boundary conditions can be characterized with a matrix Riemann–Hilbert problem. Then a formula of [Formula: see text]-soliton solutions is derived by solving the Riemann–Hilbert problem. As applications of the [Formula: see text]-soliton formula, the first-order explicit soliton solutions with different dynamical features are obtained and analyzed.

Effective Interactions: Resonances and Off-Shell Characteristics

The influence of resonances on the analytical properties and off-shell characteristics of effective interactions has been investigated. This requires, among others, the knowledge of the Jost function in regions of physical interest on the complex kplane when the potentials are given in a tabular form. The latter are encountered in inverse scattering and supersymmetric transformations. To investigate the effects of resonances on the analytical properties of the potential, we employed the Marchenko inverse scattering method to construct, phase and bound state equivalent local potentials but with different resonance spectra. It is shown that the inclusion of resonances changes the shape, strength, and range of the potential which in turn would modify the bound and scattering wave functions in the interior region. This could have important consequences in calculations of transition amplitudes in nuclear reactions, which strongly depend on the behaviour of the wave functions at short distances. Finally, an exact method to obtain the Jost solutions and the Jost functions for a repulsive singular potential is presented. The effectiveness of the method is demonstrated using the Lennard-Jones (12,6) potential.

Bound states in the continuum and time evolution of the generalized eigenfunctions

We study the Jost solutions for the scattering problem of a von Neumann-Wigner type potential, constructed by means of a two times iterated and completely degenerated Darboux transformation. We show that for a particular energy the unnormalizedJost solutions coalesce to give rise to a Jordan cycle of rank two. Performing a pole decomposition of the normalized Jost solutions we find the generalized eigenfunctions: one is a normalizable function corresponding to the bound state in the continuum and the other is a bounded, non-normalizable function. We obtain the time evolution of these functions as pseudo-unitary, characteristic of a pseudo-Hermitian system.