Rogue waves in the three-level defocusing coupled Maxwell–Bloch equations

The coupled Maxwell–Bloch (CMB) system is a fundamental model describing the propagation of ultrashort laser pulses in a resonant medium with coherent three-level atomic transitions. In this paper, we consider an integrable generalization of the CMB equations with the defocusing case. The CMB hierarchy is derived with the aid of a 3 × 3 matrix eigenvalue problem and the Lenard recursion equation, from which the defocusing CMB model is proposed as a special reduction of the general CMB equations. The n -fold Darboux transformation as well as the multiparametric n th-order rogue wave solution of the defocusing CMB equations are put forward in terms of Schur polynomials. As an application, the explicit rogue wave solutions from first to second order are presented. Apart from the traditional dark rogue wave, bright rogue wave and four-petalled rogue wave, some novel rogue wave structures such as the dark four-peaked rogue wave and the double-ridged rogue wave are found. Moreover, the second-order rogue wave triplets which contain a fixed number of these rogue waves are shown.

New types of chirped soliton solutions for the Fokas–Lenells equation

Purpose The purpose of this paper is to present a reliable treatment of the Fokas–Lenells equation, an integrable generalization of the nonlinear Schrödinger equation. The authors use a special complex envelope traveling-wave solution to carry out the analysis. The study confirms the accuracy and efficiency of the used method. Design/methodology/approach The proposed technique, namely, the trial equation method, as presented in this work has been shown to be very efficient for solving nonlinear equations with spatio-temporal dispersion. Findings A class of chirped soliton-like solutions including bright, dark and kink solitons is derived. The associated chirp, including linear and nonlinear contributions, is also determined for each of these optical pulses. Parametric conditions for the existence of chirped soliton solutions are presented. Research limitations/implications The paper presents a new efficient algorithm for handling an integrable generalization of the nonlinear Schrödinger equation. Practical/implications The authors present a useful algorithm to handle nonlinear equations with spatial-temporal dispersion. The method is an effective method with promising results. Social/implications This is a newly examined model. A useful method is presented to offer a reliable treatment. Originality/value The paper presents a new efficient algorithm for handling an integrable generalization of the nonlinear Schrödinger equation.

The Hamiltonian structure and quasi-periodic solutions for the generalized associated Camassa–Holm equation

In this paper, we study a integrable generalization of the associated Camassa–Holm equation. This equation is shown to be completely integrable and possesses bi-Hamiltonian structure. The bilinear equation and its solition solution are presented through the Bell polynomial technique. Meanwhile, the quasi-periodic wave solution is obtained by Riemann theta-function, and we show the asymptotic properties of the quasi-periodic wave solutions.

Discrete Spectrum of 2 + 1-Dimensional Nonlinear Schrödinger Equation and Dynamics of Lumps

We consider a natural integrable generalization of nonlinear Schrödinger equation to2+1dimensions. By studying the associated spectral operator we discover a rich discrete spectrum associated with regular rationally decaying solutions, the lumps, which display interesting nontrivial dynamics and scattering. Particular interest is placed in the dynamical evolution of the associated pulses. For all cases under study we find that the relevant dynamics corresponds to acentral configurationof a certainN-body problem.