一类高阶非线性波方程的子方程与精确行波解

结合子方程和动力系统分析的方法研究了一类五阶非线性波方程的精确行波解.得到了这类方程所蕴含的子方程, 并利用子方程在不同参数条件下的精确解, 给出了研究这类高阶非线性波方程行波解的方法, 并以Sawada Kotera方程为例, 给出了该方程的两组精确谷状孤波解和两组光滑周期波解.该研究方法适用于形如对应行波系统可以约化为只含有偶数阶导数、一阶导数平方和未知函数的多项式形式的高阶非线性波方程行波解的研究.     

一维弱噪声随机Burgers方程的奇摄动解

讨论了一类有界区域上具有有色噪声干扰的随机Burgers方程奇摄动解,其波动率服从弱噪声Ornstein-Uhlenbeck(O-U)过程.由波运动的转移概率密度函数满足的后向Kolmogorov方程,得到随机Burgers的期望所满足的后向Kolmogorov方程.由于期望满足的后向Kolmogorov方程的初边值问题条件涉及到一类确定性Burgers方程的解,因此该问题实际上是Burgers方程和Kolmogorov方程的联立形式.首先,应用奇摄动方法,对一类确定性Burgers方程进行了正则渐近展开,由Schauder估计、Ascoli-Arzela定理证明了非线性抛物方程渐近解的有界性与存在性,由Lax-Milgram定理证明了线性抛物方程渐近解的有界性与存在性,得到波速率的形式渐近解.其次,由奇摄动理论,对期望满足的方程进行了奇摄动渐近展开和边界层矫正,由二阶线性偏微分方程理论,得到边界层函数渐近解存在且有界.应用极值原理、De-Giorgi迭代技术分别证明了波速率和波期望渐近解的余项有界,得到渐近解的一致有效性.     

一类伴有边界摄动的非线性奇摄动四阶微分方程三点边值问题

讨论了-类伴有边界摄动的非线性奇摄动四阶微分方程三点边值问题.在适当的条件下,利用摄动理论和微分不等式技巧证明了解的存在性,给出了其解及其导数的任意n阶-致有效渐近展开式.     

分数阶微分不等式及其在分数阶奇摄动初值问题中的应用（英文）

利用分数阶微分方程与微分不等式之间的关系,得到了分数阶微分不等式的相关理论.基于此理论研究了分数阶微分方程的奇摄动初值问题,证明了其解的存在性.同时通过恰当不等式的解,估计了方程的精确解,进而得到分数阶奇摄动初值问题解的存在性及其渐进行为的一般结论..     

带有弱奇性核的多项分数阶非线性随机微分方程解的适定性

本文主要研究一类带有多项分数阶Caputo导数的非线性随机微分方程初值问题的解的适定性.具体地,首先把多项分数阶随机微分方程等价地转化为随机Volterra积分方程;然后,给出了该随机积分方程的Euler-Maruyama (EM)格式;最后,借助于该EM格式,证明了多项分数阶随机微分方程的解的适定性.     

分数阶微分不等式及其在分数阶奇摄动初值问题中的应用

利用分数阶微分方程与微分不等式之间的关系,得到了分数阶微分不等式的相关理论.基于此理论研究了分数阶微分方程的奇摄动初值问题,证明了其解的存在性.同时通过恰当不等式的解,估计了方程的精确解,进而得到分数阶奇摄动初值问题解的存在性及其渐进行为的一般结论..     

非线性超抛物型方程广义渐近解的研究

讨论了一类超抛物型方程的非线性奇摄动问题.首先引入了相应问题的比较定理,然后利用奇摄动方法构造了问题的形式渐近解,最后利用比较定理,证明了问题广义解的存在性及其渐近性态.     

两参数非线性发展方程的奇摄动尖层解

本文研究了一类具有非线性发展方程奇摄动问题.引入伸长变量和多重尺度,构造了初始边值问题外部解和尖层、边界层和初始层校正项,得到了问题形式解.利用不动点定理,证明了问题的解的一致有效性.推广了对两参数的奇摄动问题的研究结果.     

带奇性右端项的一类线性双曲型方程的摄动

本文讨论了在二维或三维正则区域中一类具有奇性右端项的二阶双曲型方程的初一边值问题的摄动.摄动算子是一个四阶椭圆算子,它线性地依赖于小参数ε.文中考察了摄动问题广义解的存在性及其极限性态,证明了当ε趋于零时,摄动问题的解在一定意义下收敛于原问题的解.     

两参数非线性发展方程的奇摄动尖层解（英文）

本文研究了一类具有非线性发展方程奇摄动问题.引入伸长变量和多重尺度,构造了初始边值问题外部解和尖层、边界层和初始层校正项,得到了问题形式解.利用不动点定理,证明了问题的解的一致有效性.推广了对两参数的奇摄动问题的研究结果.     

Nonlinear Schr?dinger Approximation for the Electron Euler-Poisson Equation

The nonlinear Schr?dinger(NLS for short) equation plays an important role in describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet. In this paper, the authors study the NLS approximation by providing rigorous error estimates in Sobolev spaces for the electron Euler-Poisson equation,an important model to describe Langmuir waves in a plasma. They derive an approximate wave packet-like solution to the evolution equations by the multiscale a...     

非线性Schr\"{o}dinger 方程解的整体存在和爆破的门槛结果

研究了非线性 Schr\"{o}dinger 方程的柯西问题. 通过引进位势井及其外部集合, 得到了解的整体存在性和爆破的门槛结果.     

Variational resolution for some general classes of nonlinear evolutions. Part II

Using our results in [11], we provide existence theorems for general classes of nonlinear evolutions. Then we give examples of applications of our results to parabolic, hyperbolic, Schr¨odinger, Navier-Stokes and other time-dependent systems of equations.     

Various exact solutions of nonlinear Schrödinger equation with two nonlinear terms

Four kinds of exact solutions to nonlinear Schrödinger equation with two higher order nonlinear terms are obtained by a subsidiary ordinary differential equation method (sub-equation method for short). They are the bell type solitary waves, the kink type solitary waves, the algebraic solitary waves and the sinusoidal waves.     

Can Parity‐Time‐Symmetric Potentials Support Families of Non‐Parity‐Time‐Symmetric Solitons?

For the one‐dimensional nonlinear Schrödinger equations with parity‐time (PT) symmetric potentials, it is shown that when a real symmetric potential is perturbed by weak PT‐symmetric perturbations, continuous families of asymmetric solitary waves in the real potential are destroyed. It is also shown that in the same model with a general PT‐symmetric potential, symmetry breaking of PT‐symmetric solitary waves does not occur. Based on these findings, it is conjectured that one‐dimensional PT‐symmetric potentials cannot support continuous families of non‐PT‐symmetric solitary waves.     

Classification of Solitary Wave Bifurcations in Generalized Nonlinear Schrödinger Equations

Bifurcations of solitary waves are classified for the generalized nonlinear Schrödinger equations with arbitrary nonlinearities and external potentials in arbitrary spatial dimensions. Analytical conditions are derived for three major types of solitary wave bifurcations, namely, saddle‐node, pitchfork, and transcritical bifurcations. Shapes of power diagrams near these bifurcations are also obtained. It is shown that for pitchfork and transcritical bifurcations, their power diagrams look differently from their familiar solution‐bifurcation diagrams. Numerical examples for these three types of bifurcations are given as well. Of these numerical examples, one shows a transcritical bifurcation, which is the first report of transcritical bifurcations in the generalized nonlinear Schrödinger equations. Another shows a power loop phenomenon which contains several saddle‐node bifurcations, and a third example shows double pitchfork bifurcations. These numerical examples are in good agreement with the analytical results.     

A novel numerical approach to simulating nonlinear Schrödinger equations with varying coefficients

We describe a novel numerical approach to simulations of nonlinear Schrödinger equations with varying coefficients, based on the discovery of a new and intrinsic conservation law for varying coefficient nonlinear Schrödinger equations. The approach is shown to preserve some crucial classical conservations, such as the spatial ergodicity, and utilized in numerical simulations of periodically and quasi-periodically solitary waves for nonlinear Schrödinger equations with periodic or quasi-periodic coefficients. Some numerical experiments are presented to illustrate the conservative property.     

The exact solutions and the relevant constraint conditions for two nonlinear Schrödinger equations with variable coefficients

Two nonlinear Schrödinger equations with variable coefficients are researched, and the various exact solutions (including the bright and dark solitary waves) of the nonlinear Schrödinger equations are obtained with the aid of a subsidiary elliptic-like equation (sub-ODEs for short), at the same time, the constraint conditions which the coefficients of the nonlinear Schrödinger equations with variable coefficients satisfy are presented. The exact solutions and the constraint conditions are helpful in the application of the nonlinear Schrödinger equations with variable coefficients studied in this paper.     

Explicit and exact travelling wave solutions for the generalized derivative Schrödinger equation

In this paper, a new auxiliary equation expansion method and its algorithm is proposed by studying a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. Being concise and straightforward, the method is applied to the generalized derivative Schrödinger equation. As a result, some new exact travelling wave solutions are obtained which include bright and dark solitary wave solutions, triangular periodic wave solutions and singular solutions. This algorithm can also be applied to other nonlinear wave equations in mathematical physics.     

Analytical construction of peaked solutions for the nonlinear evolution of an electromagnetic pulse propagating through a plasma

Abstract

We obtain analytical solutions, by way of the homotopy analysis method, to a nonlinear wave equation describing the nonlinear evolution of a vector potential of an electromagnetic pulse propagating in an arbitrary pair plasma with temperature asymmetry. As the method is analytical, we are able to construct peaked structures which propagate through the pair plasma, analogous to peakon solutions. These solutions are obtained through a novel matching of inner and outer homotopy solutions. In order to ensure that our analytical results are valid over the whole real line, we also discuss the convergence of the analytical results to the true solution, through minimization of the residual errors resulting from an approximate analytical solution. These results demonstrate the existence of peaked pulses propagating through a pair plasma. The algebraic decay rate of the pulses are determined analytically, as well. The method discussed here can be applied to approximate solutions to similar nonlinear partial differential equations of nonlinear Schr¨odinger type.