Narrow Results

Darboux transformations in one independent variable have found numerous applications in various field of mathematics and physics. In this paper, we introduce a novel application of these transformations in two dimensions to decouple systems of Schrodinger equations. We derive explicit representation for three classes of such systems which can be decoupled by such transformations. We also show that there is an elegant relationship between these transformations and analytic complex matrix functions.

Algebraic Bargmann and Darboux transformations for equations of a more general form than the Schrödinger ones with an additional functional dependence h(r) in the right-hand side of equations are constructed. The suggested generalized transformations turn into the Bargmann and Darboux transformations for both fixed and variable values of energy and an angular momentum.

This paper studies the Hirota–Maxwell–Bloch (H–MB) system and its nonlocal form. Based on the Darboux Transformations (DTs), for H–MB system, we present general double breathers, what is more, we take appropriate modulation frequency and position parameters to investigate the generative mechanism of rogue wave sequences and different periodic breather sequences. For nonlocal Hirota–Maxwell–Bloch (NH–MB) system, we discuss symmetry preserving and broken soliton solutions under zero background. Besides, we present nine combinations of dark and antidark soliton solutions under continuous waves background when PT-symmetry is broken.

The coupled nonlinear Schrödinger (CNLS) system is often used to describe physical problems such as wave propagation in a birefringent optical fiber. Under investigation in this paper is the nonlocal ${𝒫}{𝒯}$-symmetric reverse-space type of CNLS system, through constructing two types of Darboux transformation (DT), we will derive a new series of nonlocal analytic solutions: (1) Single-periodic and double-periodic solutions; (2) Four different combinations about dark and anti-dark solitons, as well as the bright and dark breathers on a periodic background. Moreover, the fundamental properties and dynamical behaviors of those solutions will be discussed.