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.

Solitons, Rogues and Interaction Behaviors of Third-Order Nonlinear Schrodinger Equation

In this paper, we investigate the third-order nonlinear Schr\"{o}dinger equation which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The multi-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detail calculations.PACS numbers: 02.30.Ik, 05.45.Yv.

Strong and weak interactions of rational vector rogue waves and solitons to any n -component nonlinear Schrödinger system with higher-order effects

The higher-order effects play an important role in the wave propagations of ultrashort (e.g. subpicosecond or femtosecond) light pulses in optical fibres. In this paper, we investigate any n -component fourth-order nonlinear Schrödinger ( n -FONLS) system with non-zero backgrounds containing the n -Hirota equation and the n -Lakshmanan–Porsezian–Daniel equation. Based on the loop group theory, we find the multi-parameter family of novel rational vector rogue waves (RVRWs) of the n -FONLS equation starting from the plane-wave solutions. Moreover, we exhibit the weak and strong interactions of some representative RVRW structures. In particular, we also find that the W-shaped rational vector dark and bright solitons of the n -FONLS equation as the second- and fourth-order dispersion coefficients satisfy some relation. Furthermore, we find the higher-order RVRWs of the n -FONLS equation. These obtained rational solutions will be useful in the study of RVRW phenomena of multi-component nonlinear wave models in nonlinear optics, deep ocean and Bose–Einstein condensates.