Yu-Shan Xue

Asymmetric Rogue Waves, Breather-to-Soliton Conversion, and Nonlinear Wave Interactions in the Hirota–Maxwell–Bloch System

We study the nonlinear localized waves on constant backgrounds of the Hirota–Maxwell–Bloch (HMB) system arising from the erbium doped fibers. We derive the asymmetric breather, rogue wave (RW) and semirational solutions of the HMB system. We show that the breather and RW solutions can be converted into various soliton solutions. Under different conditions of parameters, we calculate the locus of the eigenvalues on the complex plane which converts the breathers or RWs into solitons. Based on the second-order solutions, we investigate the interactions among different types of nonlinear waves including the breathers, RWs and solitons.We study the nonlinear localized waves on constant backgrounds of the Hirota–Maxwell–Bloch (HMB) system arising from the erbium doped fibers. We derive the asymmetric breather, rogue wave (RW) and semirational solutions of the HMB system. We show that the breather and RW solutions can be converted into various soliton solutions. Under different conditions of parameters, we calculate the locus of the eigenvalues on the complex plane which converts the breathers or RWs into solitons. Based on the second-order solutions, we investigate the interactions among different types of nonlinear waves including the breathers, RWs and solitons.

Conservation laws and solitons for the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics

Under investigation in this paper are the coupled cubic–quintic nonlinear Schrodinger equations describing the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations is obtained via the Ablowitz–Kaup–Newell–Segur scheme and the corresponding Darboux transformation is constructed. One-, two- and three-soliton solutions are presented and an infinite number of conservation laws are also derived. The features of solitons are graphically discussed: (i) head-on and overtaking elastic collisions of the two solitons; (ii) periodic attraction and repulsion of the bounded states of two solitons; (iii) energy-exchanging collisions of the three solitons.

Soliton collision in a general coupled nonlinear Schrödinger system via symbolic computation

A general coupled nonlinear Schrodinger system with the self-phase modulation, cross-phase modulation and four-wave mixing terms is investigated. The system is still integrable with the variable coefficients. Through the Hirota bilinear method, one- and two-soliton solutions are derived via symbolic computation. With the asymptotic analysis, it is found that the two-soliton solutions admit the inelastic and elastic collisions depending on the choice of solitonic parameters. A new inelastic collision phenomenon occurring in this system is that both the amplitudes of two components of each soliton get suppressed or enhanced after the collision, which might provide us with a different approach of signal amplification.

Soliton-like solutions of the coupled Hirota–Maxwell–Bloch system in optical fibers with symbolic computation

With the aid of symbolic computation, the coupled Hirota–Maxwell–Bloch system is investigated with third-order dispersion and higher-order nonlinear effects, which govern the nonlinear pulse propagation in an erbium-doped optical fiber medium. In addition, the Lax pair for the system is explicitly constructed and the soliton-like solutions are derived using the Darboux transformation, which makes it possible to generate the multi-soliton solutions in a recursive manner. Through the graphical analysis of some obtained analytic one- and two-soliton-like solutions, stable propagation and collision between two solitons are discussed. Furthermore, the conservation laws for the system are presented.

Direct analysis of the bright-soliton collisions in the focusing vector nonlinear Schrödinger equation

The multi-component Wronskian representation of the bright N-soliton solution to the focusing vector nonlinear Schrodinger equation for certain optical fibers allows a direct algebraic method to study the collision behavior of N vector solitons with m components. In this letter, this method is used to analyze the two- and three-soliton collisions via symbolic computation. The phase-shift formulae induced by the vector-soliton collisions are given explicitly, and the parametric conditions are revealed for the amplitude preservation of all the soliton components in the collision. In addition, the generalized linear fractional transformations are also derived to directly describe the state change for each component of colliding solitons.

Gauge transformation between the first-order nonisospectral and isospectral Heisenberg hierarchies

Abstract By means of the Ablowitz–Kaup–Newell–Segur system with symbolic computation, the gauge transformation between the first-order nonisospectral and isospectral Heisenberg spectral problems is obtained. Correspondingly, the equivalence relation between the first-order nonisospectral and isospectral Heisenberg hierarchies is given. In addition, further applications and potential research directions of our method are discussed.

Integrable nonlinear differential-difference hierarchy and Darboux transformation

In this paper, with a free function embedded into a discrete zero-curvature equation, an integrable nonlinear differential-difference hierarchy is derived via the extension of original hierarchy with symbolic computation. Based on the Lax pair, infinitely many conservation laws and Darboux transformations are constructed for the first nonlinear differential-difference equations in the hierarchy. As an application of the Darboux transformation, some explicit solutions of those sample equations are given.

INTEGRABLE ASPECTS AND SOLITON-LIKE SOLUTIONS OF AN INHOMOGENEOUS COUPLED HIROTA–MAXWELL–BLOCH SYSTEM IN OPTICAL FIBERS WITH SYMBOLIC COMPUTATION

For describing wave propagation in an inhomogeneous erbium-doped nonlinear fiber with higher-order dispersion and self-steepening, an inhomogeneous coupled Hirota–Maxwell–Bloch system is considered with the aid of symbolic computation. Through Painleve singularity structure analysis, the integrable condition of such a system is analyzed. Via the Painleve-integrable condition, the Lax pair is explicitly constructed based on the Ablowitz–Kaup–Newell–Segur scheme. Furthermore, the analytic soliton-like solutions are obtained via the Darboux transformation which makes it exercisable to generate the multi-soliton solutions in a recursive manner. Through the graphical analysis of some obtained analytic one- and two-soliton-like solutions, our concerns are mainly on the envelope soliton excitation, the propagation features of optical solitons and their interaction behaviors in actual fiber communication. Finally, the conservation laws for the system are also presented.

SOLITONS AND LOCALIZED EXCITATIONS FOR THE (2+1)-DIMENSIONAL DISPERSIVE LONG WAVE SYSTEM VIA SYMBOLIC COMPUTATION

For describing some nonlinear localized excitations, the (2+1)-dimensional dispersive long wave (DLW) system is investigated with symbolic computation in this paper. Based on two different dependent variable transformations obtained through the truncated Painleve expansion, the (2+1)-dimensional DLW system can be bilinearized or linearized. Through the Hirota bilinear method, the analytic one-, two-, three-, and N-soliton solutions are derived. On the other hand, by means of the variable separation approach, localized excitations, such as the resonant dromion, resonant solitoff, lump and compacton excitations, are obtained. Figures are plotted to illustrate the structures of those solutions.