Kerr类非线性介质周期结构中的慢Bragg孤子

在耦合模理论的基础上,给出了一维无限大Kerr类非线性介质周期结构中的孤波解,并且指出,孤波的振幅依赖于入射频率以及脉宽两个参量.同时也证明,在布拉格共振极限条件下,孤波解可以简化成所谓的“隙孤子”解或是“慢布拉格孤子”解 关键词： 孤波 慢布拉格孤子 隙孤子 耦合模理论     

Polariton gap solitary waves in semiconductor microcavities

We study nonlinear dynamics of optical pulse propagation in the spectral region inside the polariton gap. It is shown that the Kerr nonlinearity can lead to formation of solitary waves in this region of frequencies.     

修正cKdV方程组的孤立波结构及其稳定性

利用函数展开法求解修正耦合KdV(Coupled KdV,cKdV)方程组,得到几类孤立波解,包括扭结型-钟型、双扭结型、双钟型以及双扭结-双钟型结构的单孤子解.在不同的极限情况下,这些解分别退化为修正cKdV方程的扭结状或钟状孤波解.对孤立波的稳定性进行了数值研究,结果表明:修正cKdV方程既存在稳定的孤立波解,也存在不稳定的孤立波解.     

含高阶非线性效应的薛定谔方程的精确解研究

利用孤子理论,研究了含三次和五次非线性项的非线性薛定谔方程,在参数取不同值时得到了方程的新型亮孤子解、新型暗孤子解和新的三角函数周期解。     

Bessel solitary waves in strongly nonlocal nonlinear media with distributed parameters

Bessel solitary wave solutions to a two-dimensional strongly nonlocal nonlinear Schrödinger equation with distributed coefficients are obtained. Bessel solitary wave solutions have unique characteristics compared with Gaussian solitary wave solutions, Laguerre-Gaussian solitary wave solutions, and Hermite-Gaussian solitary wave solutions. The generalized two-dimensional nonlocal nonlinear Schrödinger equation with distributed coefficients is investigated for the first time to our knowledge.     

From bell-shaped solitary wave to W/M-shaped solitary wave solutions in an integrable nonlinear wave equation

The bifurcation theory of dynamical systems is applied to an integrable nonlinear wave equation. As a result, it is pointed out that the solitary waves of this equation evolve from bell-shaped solitary waves to W/M-shaped solitary waves when wave speed passes certain critical wave speed. Under different parameter conditions, all exact explicit parametric representations of solitary wave solutions are obtained.     

Rogue waves in exciton–polariton condensates under homogeneous pumping

We predict the emergence of rogue wave solutions in one-dimensional exciton–polariton condensates under homogeneous pumping. We model the condensate dynamics in a microwire using the dissipative Gross–Pitaevskii equation for the polariton field, with considers attractive nonlinearity, coupled to the rate equation of the excitonic reservoir density. With the help of the direct ansatz method and similarity transformation, deformed first order rogue wave solutions are constructed and its dynamics analyzed. We show that the deformed rogue wave has a curved background controlled by the pump power and the strength of the nonlinear interaction of polaritons. Moreover, the maximal population of the polaritons appears where high energy of rogue wave is concentrated.     

Bifurcations and Dynamics of Traveling Wave Solutions to a Fujimoto-Watanabe Equation

In this paper, we study the bifurcations and dynamics of traveling wave solutions to a Fujimoto-Watanabe equation by using the method of dynamical systems. We obtain all possible bifurcations of phase portraits of the system in different regions of the parametric space. Then we show the sufficient conditions to guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions, compactions and kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions and periodic wave solutions are implicitly given,while the expressions of kink-like and antikink-like wave solutions are explicitly shown. The dynamics of these new traveling wave solutions will greatly enrich the previews results and further help us understand the physical structures and analyze the propagation of the nonlinear wave.     

一类非线性演化方程的新多级准确解

在Lamé方程和新的Lamé函数的基础上，应用小扰动方法和Jacobi椭圆函数展开法求解一类非线性演化方程（如mKdV方程，非线性Klein-Gordon方程Ⅱ等），获得多种新的多级准确解 .这些多级准确解对应着不同形式的周期波解.这些解在极限条件下可以退化为多种形式的孤 立波解，如带状孤立子、钟形孤立子等. 关键词： Jacobi椭圆函数 Lam函数 多级准确解 非线性演化方程 扰动方法     

Extended Hyperbola Function Method and Its Application to Nonlinear Equations

An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be found by this new method, which include kink-shaped soliton solutions, bell-shaped soliton solutions and new solitary wave.The new method can be applied to other nonlinear equations in mathematical physics.     

非线性演化方程孤立波的同伦分析法求解

利用同伦分析法求解了Burgers方程，得到了其扭结形孤立波的近似解析解，该解非常接近于相应的精确解.结果表明，同伦分析法可用来求解非线性演化方程的孤立波解.同时，也对所用方法进行了一定扩展，得到了Kadomtsev-Petviashvili(KP)方程的钟形孤立子解.经过扩展后的方法能够更方便地用于求解更多非线性演化方程的高精度近似解析解. 关键词： Burgers方程 同伦分析法 KP方程 孤立波解     

Solitary waves and rogue waves in a plasma with nonthermal electrons featuring Tsallis distribution

In this Letter, we discuss the electron acoustic (EA) waves in plasmas, which consist of nonthermal hot electrons featuring the Tsallis distribution, and obtain the corresponding governing equation, that is, a nonlinear Schrödinger (NLS) equation. By means of Modulation Instability (MI) analysis of the EA waves, it is found that both electron acoustic solitary wave and rogue wave can exist in such plasmas. Basing on the Darboux transformation method, we derive the analytical expressions of nonlinear solutions of NLS equations, such as single/double solitary wave solutions and single/double rogue wave solutions. The existential regions and amplitude of solitary wave solutions and the rogue wave solutions are influenced by the nonextensive parameter q and nonthermal parameter α. Moreover, the interaction of solitary wave and how to postpone the excitation of rogue wave are also studied.     

Dispersive Solitary Wave Solutions of Strain Wave Dynamical Model and Its Stability

In the materials of micro-structured, the propagation of wave modeling should take into account the scale of various microstructures. The different kinds solitary wave solutions of strain wave dynamical model are derived via utilizing exp\((-\phi(\xi))\)-expansion and extended simple equation methods. This dynamical equation plays a key role in engineering and mathematical physics. Solutions obtained in this work include periodic solitary waves, Kink and anti-Kink solitary waves, bell-shaped solutions, solitons, and rational solutions. These exact solutions help researchers for knowing the physical phenomena of this wave equation. The stability of this dynamical model is examined via standard linear stability analysis, which authenticate that the model is stable and their solutions are exact. Graphs are depicted for knowing the movements of some solutions. The results show that the current methods, by the assist of symbolic calculation, give an effectual and direct mathematical tools for resolving the nonlinear problems in applied sciences.     

Soliton switching in nonlinear couplers

We present a theoretical overview of soliton switching phenomena in two-mode nonlinear couplers. By complementing numerical studies with perturbative or exact solitary wave solutions, one finds that nonlinear Schrödinger or sine-Gordon solitons tend to maintain their identity in the coupled systems. Moreover, the coupling itself may originate novel vector solitary waves, such as gap solitons in periodic media. The switching dynamics in the presence of dissipative perturbations such as linear gain or intrapulse Raman scattering is also discussed.     

mBBM方程的双扭结孤立波及其动力学稳定性

采用双曲函数展开法得到Modified Benjamin-Bona-Mahony（mBBM）方程的一类扭结-反扭结状的双扭结孤立波解,在不同的极限情况下,此孤立波分别退化为mBBM方程的扭结状和钟状孤立波解.对双扭结型单孤子的结构特征进行分析,构造有限差分格式对其动力学稳定性进行数值研究.有限差分格式为两层隐式格式,在线性化意义下无条件稳定.数值结果表明mBBM方程的双扭结型单孤子在不同类型的扰动下均具有很强的稳定性.对双孤立波的碰撞进行数值模拟,发现既存在弹性碰撞也存在非弹性碰撞.     

Existence and Dynamics of Bounded Traveling Wave Solutions to Getmanou Equation

In this paper, we study the existence and dynamics of bounded traveling wave solutions to Getmanou equations by using the qualitative theory of differential equations and the bifurcation method of dynamical systems. We show that the corresponding traveling wave system is a singular planar dynamical system with two singular straight lines, and obtain the bifurcations of phase portraits of the system under different parameters conditions. Through phase portraits, we show the existence and dynamics of several types of bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, compactons, kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions are given. Additionally, we confirm abundant dynamical behaviors of the traveling wave s olutions to the equation, which are summarized as follows: i) We confirm that two types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system. ii) We confirm that two types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, and the homoclinic orbit of associated system, which is tangent to the singular line at the singular point of associated system.     

耦合KdV方程的双峰孤立子及其稳定性

利用扩展双曲函数法求解了耦合KdV方程,得到了6类精确解,其中一类为具有双峰状结构的单孤子解.在不同的极限情况下,该解分别退化为耦合KdV方程的扭结状或钟状孤波解.文中对双峰孤立波的稳定性进行了数值研究,结果表明:耦合KdV方程的双峰孤立波在长波小振幅扰动和小振幅钟型孤立波扰动下,均稳定. 关键词： 耦合KdV方程 双峰孤立子 稳定性     

Bifurcation and Solitary Waves of the Combined KdV and mKdV Equation

Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigated systematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcation parameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence of breather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutions are given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials is utilized to give simpler forms for the pulse-like solitary wave solutions. In view of the references, our results are the most complete and the theoretical methods are the simplest hitherto.     

Complex Wave Excitations in Generalized Broer-Kaup System

Starting from an improved projective method and a linear variable separation approach, new families of variable separation solutions （including solltary wave solutlons, periodic wave solutions and rational function solutions） with arbitrary functions [or the （2＋ 1）-dimensional general/zed Broer-Kaup （GBK） system are derived. Usually, in terms of solitary wave solutions and/or rational function solutions, one can find abundant important localized excitations. However, based on the derived periodic wave solution in this paper, we reveal some complex wave excitations in the （2＋1）-dimensional GBK system, which describe solitons moving on a periodic wave background. Some interesting evolutional properties for these solitary waves propagating on the periodic wave bactground are also briefly discussed.     

组合KdV方程的双扭结孤立波及其稳定性研究

利用扩展的双曲函数法得到了combined KdV-mKdV (cKdV)方程的几类精确解,其中一类为具有扭结—反扭结状结构的双扭结单孤子解.在不同的极限情况下,该解分别退化为cKdV方程的扭结状或钟状孤波解.理论分析表明,cKdV方程既有传播型孤立波解,也有非传播型孤立波解.文中对双扭结型孤立波解的稳定性进行了数值研究,结果表明,cKdV方程既存在稳定的双扭结型孤立波,也存在不稳定的双扭结型孤立波. 关键词： cKdV方程 双扭结单孤子 稳定性