Novel Particular Solutions, Breathers, and Rogue Waves for an Integrable Nonlocal Derivative Nonlinear Schrödinger Equation

A determinant representation of the n -fold Darboux transformation for the integrable nonlocal derivative nonlinear Schödinger (DNLS) equation is presented. Using the proposed Darboux transformation, we construct some particular solutions from zero seed, which have not been reported so far for locally integrable systems. We also obtain explicit breathers from a nonzero seed with constant amplitude, deduce the corresponding extended Taylor expansion, and obtain several first-order rogue wave solutions. Our results reveal several interesting phenomena which differ from those emerging from the classical DNLS equation.

Shelf solutions and dispersive shocks in a discrete NLS equation: Effects of nonlocality

We study shelf-like breathers and dispersive shock phenomena in a discrete nonlinear Schrödinger (DNLS) equation with a nonlocal nonlinearity. The system models laser light propagation in waveguide arrays made from a nematic liquid crystal substratum. Shelf-like breathers are studied in the regime of small linear intersite coupling, and we report some new theoretical existence and stability results. We also study numerically the evolution from nearby dam-break and more general jump initial conditions for stronger linear intersite coupling. In the defocusing case, we see rarefaction and shock wave profiles, superposed with oscillations. Some of the hyperbolic features of the observed profiles are described approximately by continuous NLS hydrodynamics. Nonlocality is seen to lead to some smoothing of the rapid oscillations seen in the local DNLS.

Chaos and Intermittency in the DNLS Equation Describing the Parallel Alfvén Wave Propagation

When the Hall effect is included in the magnetohydrodynamics equations (Hall-MHD model) the wave propagation modes become coupled, but for propagation parallel to the ambient magnetic field the Alfvén mode decouples from the magnetosonic ones, resulting in circularly polarized waves that are described by the derivative nonlinear Schrödinger (DNLS) equation. In this paper, the DNLS equation is numerically solved using spectral methods for the spatial derivatives and a fourth order Runge-Kutta scheme for time integration. Firstly, the nondiffusive DNLS equation is considered to test the validity of the method by verifying the analytical condition of modulational stability. Later, diffusive and excitatory effects are incorporated to compare the numerical results with those obtained by a three-wave truncation model. The results show that different types of attractors can exist depending on the diffusion level: for relatively large damping, there are fixed points for which the truncation model is a good approximation; for low damping, chaotic solutions appear and the three-wave truncation model fails due to the emergence of new nonnegligible modes.