Ya Sun

Solitons and chaos of the Klein-Gordon-Zakharov system in a high-frequency plasma

In this paper, we study the Klein-Gordon-Zakharov (KGZ) system, which describes the interaction between the Langmuir wave and ion sound wave in a high-frequency plasma. By means of the Hirota method and symbolic computation, bright and mixed-type soliton solutions are obtained. For the one soliton, amplitude of E is positively related to β2, and that of n is inversely related to β2, while they are both positively related to α, where E refers to the high-frequency part of the electrostatic potential of the electric field raised by the electrons, and n represents the deviation of ion density from its equilibrium, β2 and α are the plasma frequency and ion sound speed, respectively. Head-on interactions between the two bright solitons and two mixed-type ones are respectively displayed. With β2 increasing, the head-on interaction is transformed into an overtaking one. Bright bound-state solitons are investigated, and the interaction period decreases with α increasing. Furthermore, with the external forces Γ1(t...

Akhmediev breathers, Kuznetsov–Ma solitons and rogue waves in a dispersion varying optical fiber

Dispersion varying fibres have applications in optical pulse compression techniques. We investigate Akhmediev breathers, Kuznetsov–Ma (KM) solitons and optical rogue waves in a dispersion varying optical fibre based on a variable-coefficient nonlinear Schrodinger equation. Analytical solutions in the forms of Akhmediev breathers, KM solitons and rogue waves up to the second order of that equation are obtained via the generalised Darboux transformation and integrable constraint. The properties of Akhmediev breathers, KM solitons and rogue waves in a dispersion varying optical fibre, e.g. dispersion decreasing fibre (DDF) or a periodically distributed system (PDS), are discussed: in a DDF we observe the compression behaviours of KM solitons and rogue waves on a monotonically increasing background. The amplitude of each peak of the KM soliton increases, while the width of each peak of the KM soliton gradually decreases along the propagation distance; in a PDS, the amplitude of each peak of the KM soliton varies periodically along the propagation distance on a periodic background. Different from the KM soliton, the Akhmediev breather and rogue waves repeat their behaviours along the propagation distance without the compression.

Soliton collisions for a generalized variable-coefficient coupled Hirota–Maxwell–Bloch system for an erbium-doped optical fiber

In this paper, we construct soliton solutions for a generalized variable-coefficient coupled Hirota–Maxwell–Bloch system, which can describe the ultrashort optical pulse propagation in a nonlinear, dispersive fiber doped with two-level resonant atoms. Under certain transformations and constraints, one- and two-soliton solutions are obtained via the Hirota method and symbolic computation, and soliton collisions are graphically presented and analyzed. One soliton is shown to maintain its amplitude and shape during the propagation. Soliton collision is elastic, while bright two-peak solitons and dark two-peak solitons are also observed. We discuss the influence of the coefficients for the group velocity, group-velocity dispersion (GVD), self-phase modulation, distribution of the dopant, and Stark shift on the soliton propagation and collision features, with those coefficients are set as some constants and functions, respectively. We find the group velocity and self-phase modulation can change the solitons’ amplitudes and widths, and the solitons become curved when the GVD and distribution of the dopant are chosen as some functions. When the Stark shift is chosen as a certain constant, the two peaks of bright two-peak solitons and dark two-peak solitons are not parallel. In addition, we observe the periodic collision of the two solitons.

Symbolic-computation study of bright solitons in the optical waveguides and Bose-Einstein condensates

We investigate solitons in optical waveguides and Bose–Einstein condensates (BECs) governed by a (3+1)-dimensional Gross–Pitaevskii system, which describes the propagation of electromagnetic waves in the optical waveguides and ground-state wave functions of the BECs. We use the symbolic computation and Hirota method to derive analytic bright one- and two-soliton solutions under certain conditions. Soliton amplitude/width amplification and the influence of time-modulated dispersion on the bright-soliton shape are studied via graphic analysis. Through the analysis of bright solitons in optical waveguides and BECs, we find that both the amplitude and the width of bright solitons can become larger during propagation with certain choices of time-modulated dispersion, and that the shape of the bright soliton can also be affected by the time-modulated dispersion; when the time-modulated dispersion is different, we can obtain bright parabolic-like and periodic-type solitons.

B?cklund transformation and N-soliton solutions for a (2+1)-dimensional nonlinear evolution equation in nonlinear water waves

Under investigation in this paper is a (2+1)-dimensional nonlinear evolution equation generated via the Jaulent?Miodek hierarchy for nonlinear water waves. With the aid of binary Bell polynomials and symbolic computation, bilinear forms and a B?cklund transformations are derived. -soliton solutions are obtained through the Hirota method. Soliton propagation is discussed analytically. The bell-shaped soliton, anti-bell-shaped soliton and shock wave can be seen with some parameters selected. Soliton interactions are analyzed graphically: four kinds of elastic interactions are presented: two parallel bell-shaped solitons, two parallel anti-bell-shaped solitons, three parallel bell-shaped solitons and three parallel anti-bell-shaped solitons. We see that (1) the solitons maintain their original amplitudes, widths and directions except for some phase shifts after each interaction, and (2) the smaller the soliton amplitude is, the faster the soliton travels.

Solitons for the (2+1)-dimensional nonlinear Schrödinger-Maxwell-Bloch equations in an erbium-doped fibre

Under investigation in this paper is a set of the -dimensional nonlinear Schrödinger–Maxwell–Bloch (NLS–MB) equations, which describes the optical pulse propagation in an erbium-doped fibre. Employing the Hirota method and symbolic computation, we obtain the one- two- and N-soliton solutions. We prove that the interactions between the two solitons are elastic through the asymptotic analysis on the soliton solutions. Figures are plotted to show the one soliton and the interaction between the two solitons. We find that the decrease of the frequency shift from the resonance can make the angle between the propagation directions and the amplitudes of the two MB solitons bigger, implying that the change of the amplitudes of the two MB solitons is closely related to the propagation directions of the two NLS solitons.

Bright solitons and their interactions of the (3 + 1)-dimensional coupled nonlinear Schrödinger system for an optical fiber

Under investigation in this paper is the (3 + 1)-dimensional coupled nonlinear Schrodinger system for an optical fiber with birefringence. With the Hirota method, bilinear forms of the system are derived via an auxiliary function, and the bright one- and two-soliton solutions are constructed. Based on those soliton solutions, soliton propagation and interaction are investigated analytically and graphically. Non-singular cases of the bright one-soliton solutions are presented, from which the single-peak and two-peak solitons can arise, respectively. Through the analysis on the bright two-soliton solutions, the elastic and inelastic interactions are investigated. Three kinds of the elastic interactions are presented, between the two one-peak solitons, a one-peak soliton and a two-peak soliton, and the two two-peak solitons.

Bright soliton solutions and collisions for a ( 3 + 1 ) -dimensional coupled nonlinear Schrödinger system in optical-fiber communication

A ( 3 + 1 ) -dimensional coupled nonlinear Schrodinger system is investigated, which describes the pulses in an optical fiber and transverse effects in a nonlinear optical system. With the aid of symbolic computation and Hirota method, bright one- and two-soliton solutions of the system are derived. On the basis of the soliton solutions, we will graphically discuss the head-on collisions which include the elastic and inelastic collisions between the two bright solitons. After an elastic collision, two colliding solitons keep their shapes, amplitudes and velocities unchanged except for some phase shifts, while inelastically, the intensity of one soliton is enhanced, while the other gets suppressed.

Soliton solutions and chaotic motions for the ( 2 + 1 ) -dimensional Zakharov equations in a laser-induced plasma

The ( 2 + 1 ) -dimensional Zakharov equations arising from the propagation of a laser beam in a plasma are studied in this paper. Analytic soliton solutions are obtained by means of the symbolic computation, based on which we find that | E | is inversely related to ω p e , but positively related to m i and c s , while n is inversely related to ω p e and ω L , but positively related to n 0 , with E as the envelope of the high-frequency electric field, n as the plasma density, while ω p e , ω L , n 0 , m i and c s as the plasma electronic frequency, frequency of the laser beam, mean density of the plasma, mass of an ion and ion-sound velocity in the plasma, respectively. Head-on interaction is found to be transformed into an overtaking one with ω p e increasing or n 0 decreasing. Also, period of the bound-state interaction decreases with ω L decreasing. Considering the driving forces in the laser-induced plasma, we explore the associated chaotic motions as well as the effects of ω L , ω p e , k L , n 0 , m i , c s , ω F 1 and ω F 2 , where k L is the wave number of the laser beam, ω F 1 and ω F 1 represent the frequencies of driving forces, respectively. It is found that the chaotic motions can be weakened with ω p e , c s and ω F 1 increasing, or with n 0 , m i and ω F 2 decreasing, and the periodic motion can occur when ω F 1 reaches the critical value 2 π , while the chaotic motions are independent of ω L and k L .

Multi-soliton Collisions and Bäcklund Transformations for the (2+1)-dimensional Modified Nizhnik–Novikov–Vesselov Equations

Abstract The Korteweg–de Vries (KdV)-type equations can describe the shallow water waves, stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics, while the (2+1)-dimensional Nizhnik–Novikov–Vesselov equations are the isotropic extensions of KdV-type equations. In this paper, we investigate the (2+1)-dimensional modified Nizhnik–Novikov–Vesselov equations. By virtue of the binary Bell polynomials, bilinear forms, multi-soliton solutions and Bäcklund transformations are derived. Effects of some parameters on the solitons and monotonic function are graphically illustrated. We can observe the coalescence of the two solitons in their collision region, where their shapes change after the collision.