孤立波激发港口振荡的数值研究

采用Boussinesq数值波浪模型模拟了在孤立波作用下复杂形状港内水体的响应。孤立波在进入港口后会引起港内水体的振荡并被反射,港内波面扰动是一个随时间变化的瞬变波动过程。通过基于连续小波变换的时频分析结果并与现有的理论值进行比较发现,孤立波引起的振荡其主要能量主要集中在港池第一振荡模态上,这为估计复杂形状港口的自振频率提供了一个可行的方法。     

基于简单WENO-RKDG的溃坝问题数值模拟

对溃坝问题水流间断面的高精度、高分辨率数值模拟是水动力学的重要内容。简单加权本质无振荡(WENO)限制器由"问题单元"及其相邻单元的解重构"问题单元"的解,从而抑制数值解的非物理振荡,能够很好地模拟间断问题。本文详细介绍了简单WENO限制器的基本原理和过程。将简单WENO限制器-Runge-Kutta间断Galerkin方法应用于二维浅水控制方程的求解中,对二维矩形明渠中大坝瞬间全溃、局部溃塌所致的水流运动进行了数值模拟,并将数值计算结果与理论分析进行了比较。计算结果表明,方法能够清晰地捕捉到溃坝全过程中的间断,没有非物理的振荡现象发生,简单WENO限制器-RKDG方法能够很好地模拟溃坝波的演进过程。     

孤立波与多个潜水障碍物相互作用的数值研究

本文以完整二维Navier-Stokes方程为控制方程,采用VOF界面跟踪技术和差分方法,数值计算孤立波与多个淹没水下物体相互作用的问题。本文对潜水物体的高度接近水深时,孤立波通过水下孤立直立方柱、两个间距较大的水下直立方柱和两个间距较小的水下直立方柱等三种情况分别进行了计算,给出了波形随时间的演化图,可以看到反射波、前传波和跟随振荡型小波列的生成。对孤立波通过水下孤立直立方柱情形的计算结果,与实验结果和势流理论结果进行了比较。     

求解对流占优问题的无网格自适应稳定化方案

应用标准的无网格方法求解对流占优问题时会出现非物理的数值伪振荡现象,采用MF-SUPG、MFGLS、MFSGS等稳定化方法可以有效地消除数值伪振荡.因此本文基于无网格径向点插值法提出了一种自适应布点方案,并分别与MFSUPG、MFGLS、MFSGS方法相结合.数值模拟表明:当扩散系数较小时,三种稳定化方法均可以有效地消除对流占优问题大部分区域的数值伪振荡,但稳定化后其解在边界处仍有振荡存在,而结合自适应方案后的三种稳定化方法均可以彻底地消除数值伪振荡,且具有计算精度高、稳定性好、算法实施简单、前后处理方便.     

孤立波作用下细长港响应的数值研究

采用Boussinesq数值波浪模型,模拟了不同波高的孤立波分别对常水深和变水深细长港作用时港内的响应.对数值模型的结果进行小波分析和频谱分析并与现有的理论值比较.结果表明,孤立波传入一侧为开敞水域的细长港时,港内激发振荡的能量主要集中在细长港前三个自振模态上,港口的响应频率与理论固有频率非常接近,这为估算细长港池的固有频率提供了一种可行性方法.     

移动载荷作用下浮冰的非线性动力学响应

本文考虑非线性、惯性和阻尼的影响, 研究了任意深度二维理想流体顶部浮冰的振动. 对相关的拟微分算子进行展开并将非线性项保留至三阶后, 完全非线性问题被简化为仅与自由面上的变量相关的三阶截断模型. 为了验证简化模型的准确性, 重点关注了自由孤立波解. 在不考虑阻尼的情况下, 采用多重尺度方法推导了三阶非线性薛定谔方程(NLS), 利用该方程预测了任意水深下原始欧拉方程中自由波包型孤立波解的存在性及三阶截断模型的准确性. 相比于Dinvay等所提出的二阶模型, 三阶截断模型的优势在于其对应的三阶NLS具有准确的非线性项系数, 能够在最小相速度附近更好地模拟冰层的动力学响应. 进一步地对自由孤立波解进行数值计算, 数值结果表明三阶截断模型在分岔曲线和孤立波波形上均与完全欧拉方程吻合良好, 准确性高于二阶截断模型. 基于三阶截断模型, 探究了匀速局域化载荷作用下的浮冰非线性动力学响应并将时间依赖解与实验测量数据进行比较, 数值计算结果与实验记录吻合良好.      

一维溃坝洪水波的高精度数值模拟

将ENO(Essentially Non-Oscillatory)格式和Runge-Kutta时间离散的思想应用于一维Saint-Venant方程组的求解,数值模拟溃坝洪水,得出了水位和流速的沿程分布。经与理论解比较可见,数值解在间断波附近没有出现数值振荡,水位和流速大小均符合较好,表明ENO格式是一类新的高精度无振荡差分格式,采用ENO格式所建立的高分辨率模型能够很好地模拟溃坝波的演进过程。     

基于水平集方法的毛管上升特征研究

毛管上升现象与许多行业密切相关,系统地对此现象进行研究具有重大意义。与传统理论研究方法不同,本文使用N-S方程耦合水平集方法模拟毛管气液上升行为。通过与简化条件的解析解对比,验证了模拟方法的可靠性。此外,详细地研究了毛管振荡现象,并分析了影响毛管振荡行为的主要因素。结果表明,水平集方法能够精确地表征毛管振荡现象,与数值解相比具有更高的精度。毛管长度的增加能够减弱液柱振荡,主要归结于非湿相气体的粘滞力作用;湿相密度和湿相粘度同样对毛管振荡现象影响显著。湿相密度越大,惯性力越大,促进了毛管振荡;而湿相粘度变大,会增大粘滞力作用,因此减弱了毛管振荡现象。毛管振荡是由多种影响因素共同控制的,流体的惯性力是造成毛管振荡的主要原因,而粘滞力是减弱毛管振荡行为的主要因素,使液柱振荡逐渐衰减,并稳定至平衡高度。     

位移浅水内孤立波

研究两层浅水系统中的内孤立波,该系统由两层常密度不可压缩无黏性水组成。利用Lagrange坐标和Hamilton原理,推导了两层浅水系统的位移浅水内波方程,并进一步导出了两层浅水系统的位移内孤立波解。数值实验表明,位移内孤立波与经典的KdV内孤立波吻合很好,说明Lagrange坐标和Hamilton方法适用于内波分析,可以为构造内波分析的保辛方法提供一种途径。     

冲击接触问题增广 Lagrangian 双共轭梯度法

为避免动力接触问题罚函数法由于满足表面接触条件所带来的数值解的振荡,及常规Ｌａｇｒａｎｇｉｎａ乘子法与显式积分算法不相容的缺陷,本文发展了一个既与显式积分算法相容,又可自然地用于隐式自法的增广Ｌａｇｒａｎｇｉａｎ双共轭梯度算法。增广Ｌａｇｒａｎｇｉａｎ双共轭梯度算法既可精确地满足接触约束条件,又可避免数值解的振荡;在改善数值迭代的收敛性的同时又提高了计算效率。     

Nonautonomous bright soliton solutions on continuous wave and cnoidal wave backgrounds in blood vessels

We discuss the dissipative nonlinear Schrödinger equation (NLSE) with a variable coefficient in blood vessels via a NLSE-based constructive method and obtain exact nonautonomous soliton solutions including bright soliton solutions on continuous wave (cw) and cnoidal wave (cnw) backgrounds. Moreover, the dynamical behaviors of these soliton solutions are studied. The impact of the cw background on the separating and interactive soliton behaviors is investigated. These behaviors of the soliton can be modulated by adjusting the amplitude of background wave. The propagation behaviors of solitons on the cnw background in different dispersion systems are also studied. These results are potentially useful for future experiments in various blood vessels.     

Dark and gray solitons of ($$$$)-dimensional nonlocal nonlinear media with periodic response function

By using the standard symmetry reduction method, some exact analytical solutions including gray solitons and gray soliton lattice solutions are derived for the (\(2+1\))-dimensional nonlinear optical media with periodic nonlocal response. Furthermore, dark/gray soliton solutions and dark soliton lattice solutions are found by means of hyperbolic function expansion method and elliptic function expansion method for the nonlocal nonlinear system, respectively. It is found that two critical points exist for soliton solutions, and the switching dynamics of solitons may be described by the critical points.     

Soliton solutions and Bäcklund transformations of a (2 + 1)-dimensional nonlinear evolution equation via the Jaulent–Miodek hierarchy

In this paper, a (2 + 1)-dimensional nonlinear evolution equation generated via the Jaulent–Miodek hierarchy is investigated. Based on the Bell polynomials and Hirota method, bilinear forms and Bäcklund transformations are derived. One- and two-soliton solutions are constructed via symbolic computation. Soliton solutions are obtained through the Bäcklund transformations. We can get three types by choosing different parameters: the kink, bell-shape, and anti-bell-shape solitons. Propagation of the one soliton and elastic interactions between the two solitons are discussed graphically. After the interaction of the two bell-shape or anti-bell-shape solitons, solitonic shapes and amplitudes keep invariant except for some phase shifts, while after the interaction of the kink soliton and anti-bell-shape soliton, the anti-bell-shape soliton turns into a bell-shape one, and the kink soliton keeps its shape, with their amplitudes unchanged.     

Types of coefficient constraints of coupled nonlinear Schrödinger equations for elastic and inelastic interactions between spatial solitons with symbolic computation

With symbolic computation and Hirota method, analytic two-soliton solutions for the coupled nonlinear Schrödinger (CNLS) equations, which describe the propagation of spatial solitons in an AlGaAs slab waveguide, are derived. Two types of coefficient constraints of the CNLS equations to distinguish the elastic and inelastic interactions between spatial solitons are obtained for the first time in this paper. Asymptotic analysis is made to investigate the spatial soliton interactions. The inelastic interactions are studied under the obtained coefficient constraints of the CNLS equations. The influences of parameters for the obtained soliton solutions are discussed. All-optical switching and soliton amplification are studied based on the dynamic properties of inelastic interactions between spatial solitons. Numerical simulations are in good agreement with the analytic results. The presented results have applications in the design of birefringence-managed switching architecture.     

Auto-Darboux Transformation and Exact Solutions of the Brusselator Reaction Diffusion Model

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Nonlocal symmetry and similarity reductions for a $$\varvec{}$$-dimensional Korteweg–de Vries equation

Based on the Lax pair, the nonlocal symmetries to \((2+1)\)-dimensional Korteweg–de Vries equation are investigated, which are also constructed by the truncated Painlevé expansion method. Through introducing some internal spectrum parameters, infinitely many nonlocal symmetries are given. By choosing four suitable auxiliary variables, nonlocal symmetries are localized to a closed prolonged system. Via solving the initial-value problems, the finite symmetry transformations are obtained to generate new solutions. Moreover, rich explicit interaction solutions are presented by similarity reductions. In particular, bright soliton, dark soliton, bell-typed soliton and soliton interacting with elliptic solutions are found. Through computer numerical simulation, the dynamical phenomena of these interaction solutions are displayed in graphical way, which show meaningful structures.     

Propagation properties of optical soliton in an erbium-doped tapered parabolic index nonlinear fiber: soliton control

In this paper, with the aid of symbolic computation, we investigate the generalized nonlinear Schrödinger Maxwell–Bloch equation, which describes the propagation of the optical soliton through an inhomogeneous two-level dielectric tapered fiber medium. By virtue of the Darboux transformation method, two-soliton solutions are generated based on the constructed Lax pair and figures are plotted to illustrate the properties of the obtained solutions. Moreover, through manipulating the dispersion and nonlinearity profiles, various soliton control systems are investigated which is promising for potential applications in the design of soliton compressor, soliton amplification and high-speed optical devices in ultralarge capacity transmission systems. This means that we are able to control the soliton types with suitably selected values of the parameters. Additionally more soliton control techniques are proposed and investigated. We expect that the above analysis could be observed in future experiments.     

The concave or convex peaked and smooth soliton solutions of Camassa-Holm equation

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Nonlinear surface waves on a ferromagnetic fluid of variable depth

A nonlinear ray method is used to study surface waves on a ferromagnetic fluid of variable depth subject to a horizontal magnetic field, and an equation of the KdV type with variable coefficients is derived. An approximate solution of the equation representing a three-dimensional soliton with varying amplitude and phase is constructed and numerical results are presented.     

A new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function

This paper proposes a new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function. Firstly, a Wronskian condition involving a free function is introduced for the equation. Secondly, by solving the Wronskian condition, some exact solutions are presented. Thirdly, the dynamical behaviors are analyzed by choosing specific functions in the Wronskian condition. In addition, some exact solutions are graphically illustrated by using Mathematica symbolic computations. The dynamical behaviors include stationary y-breather, line-soliton resonance, line-soliton-like phenomenon, parabola–soliton interaction, cubic–parabola–soliton resonance, kink behavior, and singular waves. These results not only illustrate the merits of the proposed method in deriving new exact solutions but also novel dynamical behaviors in the (3 + 1)-dimensional nonlinear evolution equation.