Dark-bright Soliton Solutions Research Articles

Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions

The inverse scattering transform for the vector defocusing nonlinear Schrödinger (NLS) equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system.

The solutions of the NLS equations with self-consistent sources

We construct the generalized Darboux transformation with arbitrary functions in time $t$ for the AKNS equation with self-consistent sources (AKNSESCS) which, in contrast with the Darboux transformation for the AKNS equation, provides a non-auto-B\"{a}cklund transformation between two AKNSESCSs with different degrees of sources. The formula for N-times repeated generalized Darboux transformation is proposed. By reduction the generalized Darboux transformation with arbitrary functions in time $t$ for the Nonlinear Schr\"{o}dinger equation with self-consistent sources (NLSESCS) is obtained and enables us to find the dark soliton, bright soliton and positon solutions for NLS$^{+}$ESCS and NLS$^{-}$ESCS. The properties of these solution are analyzed.

Analytic method for solving the nonlinear Schrodinger equation describing pulse propagation in dispersive optic fibres

The authors give a method for obtaining new exact solutions of the nonlinear Schrodinger equation describing pulse propagation in optical fibres for both the anomalous and the normal dispersion regime. The method is based on the construction of a certain complete integrable finite-dimensional dynamical system whose solution determines the exact solutions of the nonlinear Schrodinger equation. By using the phase diagrams associated with the corresponding nonlinear differential equations they classify all the obtained solutions into one of the following categories: bright or dark solitary waves, bright or dark soliton solutions, rational (algebraic) bright or dark solitons, regular or singular periodic waves and stationary solutions. They give a set of particular solutions which describe the periodic wave patterns that are generated by the temporal self-phase modulation instability, the periodic evolution of bright solitons on a continuous wave background and the collision of two dark waves with equal amplitudes.