Yun-Po Wang

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.

Conservation laws and dark-soliton solutions of an integrable higher-order nonlinear Schrödinger equation for a density-modulated quantum condensate

In this paper, an integrable higher-order nonlinear Schrodinger equation for a density-modulated quantum condensate is investigated. Based on the Ablowitz–Kaup–Newell–Segur system, an infinite number of conservation laws are obtained. Introducing an auxiliary function, we derive the bilinear forms and construct the dark-soliton solutions with the help of the Hirota method and symbolic computation. Dark one, two, and three solitons are analyzed graphically. Via asymptotic analysis, interactions between the two dark solitons are proved to be elastic. We see that the coefficients in the equation only affect the soliton velocity. We analyze the linear stability of the plane wave solutions in the presence of a small perturbation.

Solitons, Bäcklund transformations, Lax pair and conservation laws for the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients

The transition phenomenon of few-cycle-pulse optical solitons from a pure modified Korteweg–de Vries (mKdV) to a pure sine-Gordon regime can be described by the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients. Based on the Bell polynomials, Hirota method and symbolic computation, bilinear forms and soliton solutions for this equation are obtained. Backlund transformations (BTs) in both the binary Bell polynomial and bilinear forms are obtained. By virtue of the BTs and Ablowitz–Kaup–Newell–Segur system, Lax pair and infinitely many conservation laws for this equation are derived as well.

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.

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.

Conservation Laws and Mixed-Type Vector Solitons for the 3-Coupled Variable-Coefficient Nonlinear Schrödinger Equations in Inhomogeneous Multicomponent Optical Fibre

Abstract In this article, the propagation and collision of vector solitons are investigated from the 3-coupled variable-coefficient nonlinear Schrödinger equations, which describe the amplification or attenuation of the picosecond pulses in the inhomogeneous multicomponent optical fibre with different frequencies or polarizations. On the basis of the Lax pair, infinitely-many conservation laws are obtained. Under an integrability constraint among the variable coefficients for the group velocity dispersion (GVD), nonlinearity and fibre gain/loss, and two mixed-type (2-bright-1-dark and 1-bright-2-dark) vector one- and two-soliton solutions are derived via the Hirota method and symbolic computation. Influence of the variable coefficients for the GVD and nonlinearity on the vector soliton amplitudes and velocities is analysed. Through the asymptotic and graphic analysis, bound states and elastic and inelastic collisions between the vector two solitons are investigated: Not only the elastic but also inelastic collision between the 2-bright-1-dark vector two solitons can occur, whereas the collision between the 1-bright-2-dark vector two solitons is always elastic; for the bound states, the GVD and nonlinearity affect their types; with the GVD and nonlinearity being the constants, collision period decreases as the GVD increases but is independent of the nonlinearity.