In this paper, the nonautonomous Lenells-Fokas (LF) model is investigated. The modulational instability analysis of the solutions with variable coefficients in the presence of a small perturbation is studied. Higher-order soliton, breather, earthwormon, and rogue wave solutions of the nonautonomous LF model are derived via the n-fold variable-coefficient Darboux transformation. The solitons and earthwormons display the elastic collisions. It is found that the nonautonomous LF model admits the higher-order periodic rogue waves, composite rogue waves (rogue wave pair), and oscillating rogue waves, whose dynamics can be controlled by the inhomogeneous nonlinear parameters. Based on the second-order rogue wave, a diamond structure consisting of four first-order rogue waves is observed. In addition, the semirational solutions (the mixed rational-exponential solutions) of the nonautonomous LF model are obtained, which can be used to describe the interactions between the rogue waves and breathers. Our results could be helpful for the design of experiments in the optical fiber communications.

Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells-Fokas equation in inhomogeneous fibers

We employ a nonrecursive Darboux transformation formalism for obtaining a hierarchy of rogue wave solutions to the focusing vector nonlinear Schrodinger equations (Manakov system). The exact explicit rogue wave solutions up to the second order are presented. Typical rogue wave patterns such as Peregrine-type, triple, quadruple, and sextuple vector rogue waves, either bright–dark or bright–bright in their respective components, are put forward. Despite the diversity, there exists a universal compossibility that different rogue wave states could coexist for the same background parameters. It is also shown that the higher-order rogue wave hierarchy can indeed be thought of as a nonlinear superposition of a fixed well prescribed number of fundamental rogue waves. These results may help understand the protean rogue wave manifestations in areas ranging from hydrodynamics to nonlinear optics.

https://iopscience.iop.org/article/10.1088/1751-8113/48/21/215202/pdf

Vector rogue waves in the Manakov system: diversity and compossibility

Generalized Darboux transformation and localized waves in coupled Hirota equations

In this paper, we construct a generalized Darboux transformation to the coupled Hirota equations with high-order nonlinear effects like the third dispersion, self-steepening and inelastic Raman scattering terms. As application, an Nth-order localized wave solution on the plane backgrounds with the same spectral parameter is derived through the direct iterative rule. In particular, some semi-rational, multi-parametric localized wave solutions are obtained: (1) Vector generalization of the first- and the second-order rogue wave solution; (2) Interactional solutions between a dark-bright soliton and a rogue wave, two dark-bright solitons and a second-order rogue wave; (3) Interactional solutions between a breather and a rogue wave, two breathers and a second-order rogue wave. The results further reveal the striking dynamic structures of localized waves in complex coupled systems.

In this paper, a vector generalization of the Fokas–Lenells system, which describes for nonlinear pulse propagation in optical fibers by retaining terms up to the next leading asymptotic order, is investigated. Higher-order soliton, breather, and rogue wave solutions of the coupled Fokas–Lenells system are derived via the n -fold Darboux transformation. Meanwhile the dynamic characteristics of those solitary wave solutions have been discussed.

Solitons, breathers and rogue waves for the coupled Fokas–Lenells system via Darboux transformation

General higher-order rogue waves of a vector nonlinear Schrodinger equation (Manakov system) are derived using a Darboux-dressing transformation with an asymptotic expansion method. The $N$th-order semirational solutions containing 3N free parameters are expressed in separation-of-variables form. These solutions exhibit rogue waves on a multisoliton background. They demonstrate that the structure of rogue waves in this two-component system is richer than that in a one-component system. Our results would be of much importance in understanding and predicting rogue wave phenomena arising in nonlinear and complex systems, including optics, fluid dynamics, Bose--Einstein condensates, and finance.

Dynamics of Rogue Waves on a Multisoliton Background in a Vector Nonlinear Schrödinger Equation

Intricate dynamics of rogue waves governed by the Sasa–Satsuma equation

High order rational solitons and their dynamics of the 3-wave resonant interaction equation

General higher order rogue waves of a vector nonlinear Schrodinger equation (Manakov system) are derived using a Darboux-dressing transformation with an asymptotic expansion method. The Nth order semi-rational solutions containing 3N free parameters are expressed in separation of variables form. These solutions exhibit rogue waves on a multisoliton background. They demonstrate that the structure of rogue waves in this two-component system is richer than that in a one-component system. The study of our results would be of much importance in understanding and predicting rogue wave phenomena arising in nonlinear and complex systems, including optics, fluid dynamics, Bose-Einstein condensates and finance and so on.

Gui Mu

Papers

Dynamics of Rogue Waves on a Multisoliton Background in a Vector Nonlinear Schrödinger Equation

Intricate dynamics of rogue waves governed by the Sasa–Satsuma equation

High order rational solitons and their dynamics of the 3-wave resonant interaction equation

Dynamics of rogue waves on a multi-soliton background in a vector nonlinear Schrodinger equation

General high-order rogue waves in the Hirota equation