Dark Bound Solitons and Soliton Chains for the Higher-Order Nonlinear Schrödinger Equation

Dark bound solitons and soliton chains without interactions are investigated for the higher-order nonlinear Schrödinger (HNLS) equation, which can model the propagation of the femtosecond optical pulse under some physical situations in nonlinear fiber optics. Via the modulation of parameters for the analytic solutions, different types of dark bound solitons and soliton chains can be derived for the HNLS equation. In addition, stabilities of those structures are checked through numerical simulations. Our discussions are expected to be helpful in interpreting those new structures, and applied to the long-distance transmission of the femtosecond pulses in optical fibers.     

Multi-hump bright solitons in a Schrödinger–mKdV system

We consider the problem of energy transport in a Davydov model along an anharmonic crystal medium obeying quartic longitudinal interactions corresponding to rigid interacting particles. The Zabusky and Kruskal unidirectional continuum limit of the original discrete equations reduces, in the long wave approximation, to a coupled system between the linear Schrödinger (LS) equation and the modified Korteweg–de Vries (mKdV) equation. Single- and two-hump bright soliton solutions for this LS–mKdV system are predicted to exist by variational means and numerically confirmed. The one-hump bright solitons are found to be the anharmonic supersonic analogue of the Davydov's solitons while the two-hump (in both components) bright solitons are found to be a novel type of soliton consisting of a two-soliton solution of mKdV trapped by the wave function associated to the LS equation. This two-hump soliton solution, as a two component solution, represents a new class of polaron solution to be contrasted with the two-soliton interaction phenomena from soliton theory, as revealed by a variational approach and direct numerical results for the two-soliton solution.     

Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schrödinger equation

We make a quantitative study on the soliton interactions in the nonlinear Schrödinger equation (NLSE) and its variable-coefficient (vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities (especially the soliton accelerations and interaction forces); whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles, particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.     

Cascade compression induced by nonlinear barriers in propagation of optical solitons

We investigate the nonlinear tunneling of optical solitons through both dispersion and nonlinear barriers by employing the exact solution of the generalized nonlinear Schrödinger equation with variable coefficients. The extensive numerical simulations show that the optical solitons can be efficiently compressed when they pass through adequate engineered nonlinear barriers. A cascade compression system in a dispersion decreasing fiber with nonlinear barriers on an exponential background is proposed and the cascade compression of optical pulses is further investigated in detail. Finally, the stability to various initial perturbations of the cascade compressed optical soliton and the interaction between two neighboring compressed solitons were investigated too.     

非均匀光纤中暗孤子传输特性研究

由于变系数非线性Schrödinger方程的增益、色散和非线性项都是变化的, 根据方程这一特点可以研究光脉冲在非均匀光纤中的传输特性. 本文利用Hirota方法, 得到非线性Schrödinger方程的解析暗孤子解. 然后根据暗孤子解对暗孤子的传输特性进行讨论, 并且分析各个物理参量对暗孤子传输的影响. 经研究发现, 通过调节光纤的损耗、色散和非线性效应都能有效的控制暗孤子的传输, 从而提高非均匀光纤中的光脉冲传输质量. 此外, 本文还得到了所求解方程的解析双暗孤子解, 最后对两个暗孤子相互作用进行了探讨. 本文得到的结论有利于研究非均匀光纤中的孤子控制技术.     

Dynamics of large-amplitude solitons

The interaction and generation of solitons in nonlinear integrable systems which allow the existence of a soliton of limiting amplitude are considered. The integrable system considered is the Gardner equation, which includes the Korteweg-de Vries equation (for quadratic nonlinearity) and the modified Korteweg-de Vries equation (for cubic nonlinearity) as special cases. A two-soliton solution of the Gardner equation is derived, and a criterion, which distinguishes between different scenarios for the interaction of two solitons, is determined. The evolution of an initial pulsed disturbance is considered. It is shown, in particular, that solitons of opposite polarity appear during such evolution on the crest of a limiting soliton. Zh. éksp. Teor. Fiz. 116, 318–335 (July 1999)     

Exact chirped gray soliton solutions of the nonlinear Schrödinger equation with variable coefficients

In this paper, we consider the nonlinear Schrödinger equation with variable coefficients, and by using direct transformation of variables and functions, the explicit chirped gray one- and two-soliton solutions are presented. Based on the exact solutions, we in detail analyze the propagation characteristics of the chirped gray soliton, including the stability against either the deviation from integrable condition or the initial perturbation, and interaction between the chirped gray solitons. The results show that the gray soliton can be compressed by choosing the appropriate initial chirp, and the chirped gray pulses can stably propagate along optical fibers remaining the character of solitons.     

Coherent interactions of multi bright spatial solitons in biased photorefractive crystals

The coherent interaction scenarios of two solitons, three solitons, and four solitons are presented. For two-soliton interactions, energy transfer and fusion between solitons are dependent on the relative phase of the interaction solitons, and for multi-soliton interactions, energy transfer will occur in all three relative phase conditions. The magnitude and direction of energy transfer can be controlled respectively by adjusting the interval and the relative phase of solitons.     

布拉格光栅中微扰光声孤子

研究了布拉格光栅中光波和声波的相互作用. 利用多重尺度方法,将光声耦合方程约化为非线性薛定谔方程,并给出了单光声孤子和双光声孤子近似解析解. 进一步讨论了光声孤子抑制光速的机理和双光声孤子的相互作用性质. 关键词： 布拉格光栅 光声耦合 多重尺度方法 光声孤子     

Novel localized wave solutions of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

Based on a special transformation that we introduce, the N-soliton solution of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation is constructed. By applying the long wave limit and restricting certain conjugation conditions to the related solitons, some novel localized wave solutions are obtained, which contain higher-order breathers and lumps as well as their interactions. In particular, by choosing appropriate parameters involved in the N-solitons, two interaction solutions mixed by a bell-shaped soliton and one breather or by a bell-shaped soliton and one lump are constructed from the 3-soliton solution. Five solutions including two breathers, two lumps, and interaction solutions between one breather and two bell-shaped solitons, one breather and one lump, or one lump and two bell-shaped solitons are constructed from the 4-soliton solution. Five interaction solutions mixed by one breather/lump and three bell-shaped solitons, two breathers/lumps and a bell-shaped soliton, as well as mixing with one lump, one breather and a bell-shaped soliton are constructed from the 5-soliton solution. To study the behaviors that the obtained interaction solutions may have, we present some illustrative numerical simulations, which demonstrate that the choice of the parameters has a great impacts on the types of the solutions and their propagation properties. The method proposed can be effectively used to construct localized interaction solutions of many nonlinear evolution equations. The results obtained may help related experts to understand and study the interaction phenomena of nonlinear localized waves during propagations.