Xing Lü

Dynamics of Alfvén solitons in inhomogeneous plasmas

To provide an analytical scheme for the dynamical behavior of nonlinear Alfven waves in inhomogeneous plasmas, this paper investigates a generalized variable-coefficient derivative nonlinear Schrodinger equation. In the sense of admitting the Lax pair and infinitely many conservation laws, the integrability of this equation is established under certain coefficient constraint which suggests which inhomogeneities support stable Alfven solitons. The Hirota method is adopted to construct the one- and multi-Alfven-soliton solutions. The inhomogeneous soliton features are also discussed through analyzing some important physical quantities. A sample model is treated with our results, and graphical illustration presents two energy-radiating Alfven soliton structures.

Integrability aspects with optical solitons of a generalized variable-coefficient N-coupled higher order nonlinear Schrödinger system from inhomogeneous optical fibers

For describing the long-distance communication and manufacturing problems of N fields propagation in inhomogeneous optical fibers, we consider a generalized variable-coefficient N-coupled nonlinear Schrodinger system with higher order effects such as the third-order dispersion, self-steepening and self-frequency shift. Using the Painleve singularity structure analysis, we obtain two cases for this system to admit the Painleve property. Then for case (1) we derive the optical dark solitons via solving the Hirota bilinear equations; and based on the obtained (2N+1)×(2N+1) Lax pair, we construct the Darboux transformation to obtain the optical bright solitons (including the multisoliton profiles) for case (2). Finally, the features of optical solitons (both dark and bright ones) in inhomogeneous optical fibers are analyzed and graphically discussed.

Lax pair, Bäcklund transformation and multi-soliton solutions for the Boussinesq–Burgers equations from shallow water waves

Abstract Under investigation in this paper is the set of the Boussinesq–Burgers (BB) equations, which can be used to describe the propagation of shallow water waves. Based on the binary Bell polynomials, Hirota method and symbolic computation, the bilinear form and soliton solutions for the BB equations are derived. Backlund transformations (BTs) in both the binary-Bell-polynomial and bilinear forms are obtained. Through the BT in the binary-Bell-polynomial form, a type of solutions and Lax pair for the BB equations are presented as well. Propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis. Shock wave and bell-shape solitons are respectively obtained for the horizontal velocity field u and height v of the water surface. In both the head-on and overtaking collisions, the shock waves for the u profile change their shapes, which denotes that the collisions for the u profile are inelastic. However, the collisions for the v profile are proved to be elastic through the asymptotic analysis. Our results might have some potential applications for the harbor and coastal design.

MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painleve expansion and Hirota bilinear method. Firstly, based on the truncation of the Painleve series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painleve expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

Darboux transformation and soliton solutions for the generalized coupled variable-coefficient nonlinear Schrödinger-Maxwell-Bloch system with symbolic computation

In an inhomogeneous nonlinear light guide doped with two-level resonant atoms, the generalized coupled variable-coefficient nonlinear Schrödinger-Maxwell-Bloch system can be used to describe the propagation of optical solitons. In this paper, the Lax pair and conservation laws of that model are derived via symbolic computation. Furthermore, based on the Lax pair obtained, the Darboux transformation is constructed and soliton solutions are presented. Figures are plotted to reveal the following dynamic features of the solitons: (1) Periodic mutual attractions and repulsions of four types of bound solitons: of two one-peak bright solitons; of two one-peak dark solitons; of two two-peak bright solitons and of two two-peak dark solitons; (2) Two types of elastic interactions of solitons: of two bright solitons and of two dark solitons; (3) Two types of parallel propagations of parabolic solitons: of two bright solitons and of two dark solitons. Those results might be useful in the study of optical solitons in some inhomogeneous nonlinear light guides.

Integrability study on a generalized (2+1)-dimensional variable-coefficient Gardner model with symbolic computation

Gardner model describes certain nonlinear elastic structures, ion-acoustic waves in plasmas, and shear flows in ocean and atmosphere. In this paper, by virtue of the computerized symbolic computation, the integrability of a generalized (2+1)-dimensional variable-coefficient Gardner model is investigated. Painlevé integrability conditions are derived among the coefficient functions, which reduce all the coefficient functions to be proportional only to γ(t), the coefficient of the cubic nonlinear term u(2)u(x). Then, an independent transformation of the variable t transforms the reduced γ(t)-dependent equation into a constant-coefficient integrable one. Painlevé test shows that this is the only case when our original generalized (2+1)-dimensional variable-coefficient Gardner model is integrable.

Analytic study on soliton-effect pulse compression in dispersion-shifted fibers with symbolic computation

The soliton-effect pulse compression of ultrashort solitons in a dispersion-shifted fiber (DSF) is investigated based on solving the higher-order nonlinear Schrödinger equation with the effects of third-order dispersion (TOD), self-steepening (SS) and stimulated Raman scattering (SRS). By using Hirotas bilinear method with a set of parametric conditions, the analytic one-, two- and three-soliton solutions of this model are obtained. According to those solutions, the higher-order soliton is shown to be compressed in the DSF for the pulse with width in the range of a few picoseconds or less. An appealing feature of the soliton-effect pulse compression is that, in contrast to the second-order soliton compression due to the combined effects of negative TOD and SRS, the third-order soliton can significantly enhance the soliton compression in the DSF with small values of the group-velocity dispersion (GVD) at the operating wavelength.

Soliton Solution, Bäcklund Transformation, and Conservation Laws for the Sasa-Satsuma Equation in the Optical Fiber Communications

Under investigation in this paper, with symbolic computation, is the Sasa-Satsuma (SS) equation which can describe the propagation of ultra short pulses in optical fiber communications. By virtue of the Ablowitz-Kaup-Newell-Segur procedure, the Lax pair for the SS equation is directly established. Based on such a Lax pair, a Bäcklund transformation is constructed, through which the explicit onesoliton solution is derived.Meanwhile, an infinite number of conservation laws is provided to indicate the integrability of the SS equation in the Liouville sense. To further understand the stability of the one-soliton solution, we employ the split-step Fourier method to simulate the propagation of the soliton pulses under the finite initial perturbations. In addition, the interaction of two adjacent pulses with different separation distances is investigated through numerical simulation. Analytic and numerical results discussed in this paper are expected to be applied to the description of the optical pulse propagation.

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.

Symbolic Computation Study of a Generalized Variable-Coefficient Two-Dimensional Korteweg-de Vries Model with Various External-Force Terms from Shallow Water Waves, Plasma Physics, and Fluid Dynamics

Abstract The variable-coefficient two-dimensional Korteweg-de Vries (KdV) model is of considerable significance in describing many physical situations such as in canonical and cylindrical cases, and in the propagation of surface waves in large channels of varying width and depth with nonvanishing vorticity. Under investigation hereby is a generalized variable-coefficient two-dimensional KdV model with various external-force terms. With the extended bilinear method, this model is transformed into a variable-coefficient bilinear form, and then a Bäcklund transformation is constructed in bilinear form. Via symbolic computation, the associated inverse scattering scheme is simultaneously derived on the basis of the aforementioned bilinear Bäcklund transformation. Certain constraints on coefficient functions are also analyzed and finally some possible cases of the external-force terms are discussed