Xi-Yang Xie

Solitons and rouge waves for a generalized ( 3 + 1 )-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics

Evolution of the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in fluid mechanics in three spatial dimensions can be described by a generalized ( 3 + 1 )-dimensional variable-coefficient Kadomtsev-Petviashvili equation, which is studied in this paper with symbolic computation. Via the truncated Painleve expansion, an auto-Backlund transformation is derived, based on which, under certain variable-coefficient constraints, one-soliton, two-soliton, homoclinic breather-wave and rouge-wave solutions are respectively obtained via the Hirota method. Graphic analysis shows that the soliton propagates with the varying soliton direction. Change of the value of any one of g ( t ) , m ( t ) , n ( t ) , h ( t ) , q ( t ) and l ( t ) in the equation can cause the change of the soliton shape, while the soliton amplitude cannot be affected by that change, where g ( t ) represents the dispersion, m ( t ) and n ( t ) respectively stand for the disturbed wave velocities along the y and z directions, h ( t ) , q ( t ) and l ( t ) are the perturbed effects, y and z are the scaled spatial coordinates, and t is the temporal coordinate. Soliton direction and type of the interaction between the two solitons can vary with the change of the value of g ( t ) , while they cannot be affected by m ( t ) , n ( t ) , h ( t ) , q ( t ) and l ( t ) . Homoclinic breather wave and rouge wave are respectively displayed, where the rouge wave comes from the extreme behaviour of the homoclinic breather wave.

Analytic study on a ( 2 + 1 )-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetism

In this paper, a ( 2 + 1 )-dimensional nonlinear Schrodinger equation for a ( 2 + 1 )-dimensional Heisenberg ferromagnetic spin chain with the bilinear and anisotropic interactions is investigated. Via the Hirota method and symbolic computation, bilinear forms and multi-soliton solutions are derived. The one, two and three solitons are analyzed graphically and we find the amplitudes and widths of the two and three solitons keep invariant after each interaction. The bell-shape one soliton as well as parallel, crossed two and three solitons are respectively observed. Through the asymptotic analysis, expressions which denote the two solitons before and after the interactions are obtained and interactions between the two solitons are proved to be elastic.

Vector semirational rogue waves and modulation instability for the coupled higher-order nonlinear Schrödinger equations in the birefringent optical fibers

We report the existence and properties of vector breather and semirational rogue-wave solutions for the coupled higher-order nonlinear Schrödinger equations, which describe the propagation of ultrashort optical pulses in birefringent optical fibers. Analytic vector breather and semirational rogue-wave solutions are obtained with Darboux dressing transformation. We observe that the superposition of the dark and bright contributions in each of the two wave components can give rise to complicated breather and semirational rogue-wave dynamics. We show that the bright-dark type vector solitons (or breather-like vector solitons) with nonconstant speed interplay with Akhmediev breathers, Kuznetsov-Ma solitons, and rogue waves. By adjusting parameters, we note that the rogue wave and bright-dark soliton merge, generating the boomeron-type bright-dark solitons. We prove that the rogue wave can be excited in the baseband modulation instability regime. These results may provide evidence of the collision between the mixed ultrashort soliton and rogue wave.

Nonautonomous Matter-Wave Solitons in a Bose-Einstein Condensate with an External Potential

Nonautonomous matter-wave solitons in a Bose–Einstein condensate with an external potential are reported. Via the non-isospectral Ablowitz–Kaup–Newell–Segur system, the Gross–Pitaevskii equation is found to have the double Wronskian solutions. The verification of such solutions is finished through some double Wronskian identities. With the zero-potential Lax pair, the double Wronskian solutions give the nonautonomous N-soliton solutions that contain 2N parameters. For characterizing the asymptotic behavior of the bright two-soliton solutions, the explicit expressions of asymptotic solitons are given. Effects of the linear and harmonic potentials on the bound states between two matter-wave solitons are discussed.

Vector bright soliton behaviors of the coupled higher-order nonlinear Schrödinger system in the birefringent or two-mode fiber

Studied in this paper are the vector bright solitons of the coupled higher-order nonlinear Schrödinger system, which describes the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber. With the help of auxiliary functions, we obtain the bilinear forms and construct the vector bright one- and two-soliton solutions via the Hirota method and symbolic computation. Two types of vector solitons are derived. Single-hump, double-hump, and flat-top solitons are displayed. Elastic and inelastic interactions between the Type-I solitons, between the Type-II solitons, and between the two combined types of the solitons are revealed, respectively. Especially, from the interaction between a Type-I soliton and a Type-II soliton, we see that the Type-II soliton exhibits the oscillation periodically before such an interaction and becomes the double-hump soliton after the interaction, which is different from the previously reported.

Solitary wave and multi-front wave collisions for the Bogoyavlenskii–Kadomtsev–Petviashili equation in physics, biology and electrical networks

In this paper, we investigate a Bogoyavlenskii–Kadomtsev–Petviashili equation, which can be used to describe the propagation of nonlinear waves in physics, biology and electrical networks. We find that the equation is Painleve integrable. With symbolic computation, Hirota bilinear forms, solitary waves and multi-front waves are derived. Elastic collisions between/among the two and three solitary waves are graphically discussed, where the waves maintain their shapes, amplitudes and velocities after the collision only with some phase shifts. Inelastic collisions among the multi-front waves are discussed, where the front waves coalesce into one larger front wave in their collision region.

Prolongation Structure of a Generalised Inhomogeneous Gardner Equation in Plasmas and Fluids

Abstract In this article, the prolongation structure technique is applied to a generalised inhomogeneous Gardner equation, which can be used to describe certain physical situations, such as the stratified shear flows in ocean and atmosphere, ion acoustic waves in plasmas with a negative ion, interfacial solitary waves over slowly varying topographies, and wave motion in a non-linear elastic structural element with large deflection. The Lax pairs, which are derived via the prolongation structure, are more general than the Lax pairs published before. Under the Painlevé conditions, the linear-damping coefficient equals to zero, the quadratic non-linear coefficient is proportional to the dispersive coefficient c(t), the cubic non-linear coefficient is proportional to c(t), leaving no constraints on c(t) and the dissipative coefficient d(t). We establish the prolongation structure through constructing the exterior differential system. We introduce two methods to obtain the Lax pairs: (a) based on the prolongation structure, the Lax pairs are obtained, and (b) via the Lie algebra, we can derive the Pfaffian forms and Lax pairs when certain parameters are chosen. We set d(t) as a constant to discuss the influence of c(t) on the Pfaffian forms and Lax pairs, and to discuss the influence of d(t) on the Pfaffian forms and Lax pairs, we set c(t) as another constant. Then, we get different prolongation structure, Pfaffian forms and Lax pairs.

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.

Solitons, Bäcklund transformation and Lax pair for a (2+1)-dimensional Davey-Stewartson system on surface waves of finite depth

Abstract Under investigation in this paper is a (2+1)-dimensional Davey-Stewartson system, which describes the transformation of a wave-packet on water of finite depth. By virtue of the bell polynomials, bilinear form, Bäcklund transformation and Lax pair are got. One- and two-soliton solutions are obtained via the symbolic computation and Hirota method. Velocity and amplitude of the one-soliton solutions are relevant with the wave number. Graphical analysis indicates that soliton shapes keep unchanged and maintain their original directions and amplitudes during the propagation. Elastic overtaking and head-on interactions between the two solitons are described.

Solitonic and chaotic behaviors for the nonlinear dust-acoustic waves in a magnetized dusty plasma

A model for the nonlinear dust-ion-acoustic waves in a two-ion-temperature, magnetized dusty plasma is studied in this paper. Via the symbolic computation, one-, two- and N-soliton solutions are obtained. It is found that when μeμi<2Ti2Te2+TiTe, the soliton amplitude is positively related to μe, μi, Ti, Zd, and B0, but inversely related to Te and md, with Te, Ti, μe, and μi as the temperature of an electron, temperature of a positive ion, normalized initial density of electrons, and normalized initial density of positive ions, respectively, Zd, B0, and md as the charge number of a dust particle, strength of the static magnetic field, and mass of a dust particle, respectively. It is also found that the two solitons are always parallel during the propagation on the x − y, x − t, and y − t planes, where x, y, and z are the scaled spacial coordinates, and t is the retarded time. Upon the introduction of the driving force Γ(t), both the developed and weak chaotic motions as well as the effect of Γ(t) are explo...