一类非线性发展方程整体解的存在性、渐近性与解的爆破

研究一类非线性发展方程初边值问题整体弱解的存在性，渐近性和解的爆破问题，证明在关于非线性项的不同条件下，上述初边值问题分别在大初值和小初始能量的情况下存在整体弱解，并且讨论了弱解的渐近性。还证明：在相反的条件下，上述弱解在有限时刻爆破，并且给出了一个实例。     

凸区域上椭圆方程弱解的边界唯一延拓性和B_p权特性

本文研究非光滑凸区域上的散度型二阶椭圆方程 ｉ（ａｉｊ（Ｘ）　ｊｕ（Ｘ））＝０的非零弱解的近边无穷次消失性和双倍性质，刻划弱解梯度在区域边界上的Ｂｐ权特性，建立弱解和弱解的梯度在凸区域边界的任意开子集上不可能同时消失的边界唯一延拓性定理．     

一类非线性拋物变分不等式解的存在性

本文研究了一类基于非线性抛物算子的变分不等式问题.首先,通过拓展偏微分方程的弱解理论定义了变分不等式的弱解.其次,利用惩罚函数并结合连续方法,证明了变分不等式存在弱解.     

倒向随机微分方程弱解（英）

引入倒向随机微分方程弱解的概念，应用Girsanov变换，建立了两类倒向随机微分方程（0．1）和（0．2）弱解存在的等价性，由此得到倒向 随机微分方程弱解存在的几个充分条件。     

一类三阶梯度流方程弱解的稳定性[英文]

本文研究R~3上的一类三阶梯度流方程弱解的稳定性问题.我们分别证明弱解的一个全局稳定性结果和一个渐近稳定性结果.所得结果改进了已有文献中的一些关于三阶梯度流方程弱解的稳定性结果.     

三维Keller-Segel-Navier-Stokes方程弱解的整体存在性

本文证明具有logistic源的一个3维Keller-Segel-Navier-Stokes方程弱解的整体存在性,并研究了弱解的长时间行为.     

二阶正定型偏微分方程非齐次定解问题的能量方法

本文给出解决二阶正定型偏微分方程非齐次定解问题适定性的 Hilbert 空间框架.由于直接证明这类方程非齐次定解问题的弱解的存在性遇到困难,本文提出 P_-拟弱解的概念,先证明 P_-拟弱解的存在性,然后在适当的定解条件下证明 P_-拟弱解就是 P_-弱解,再证明Sarason 弱强解的一致性.在典型定解问题的适定性的基础上进而可得一类定解问题 Lu=f,的适定性.这里 A 可以是非局部的和非线性的.本文以多元混合型的 Basemana方程为例将此框架具体化.     

二阶正定型偏微分方程非齐次定解问题的能量方法

本文给出解决二阶正定型偏微分方程非齐次定解问题适定性的Hilbert空间框架,由于直接证明这类方程非齐次定解问题的弱解的存在性遇到困难,本文提出P_-拟弱解的概念,先证明P_-拟弱解的存在性,然后在适当的定解条件下证明P_-拟弱解就是P_-弱解,再证明Sarason弱强解的一致性,在典型定解问题的适定性的基础上进而可得一类定解问题Lu=f,的适定性,这里A可以是非局部的和非线性的,本文以多元混合型的Busemann方程为例将此框架具体化。     

Boussinesq方程解的部分正则性

本文考虑Boussinesq方程一类合适弱解的部分正则性.我们先运用广义能量不等式和奇异积分理论得到一些无维量的估计;再通过合适弱解满足的等式,运用迭代技巧,推导出温度场的小性估计;最后由尺度分析(scaling arguments)得到了一类合适弱解的部分正则性.     

具有非标准增长条件的p(x)-Laplace方程弱解的唯一性

本文研究具有非标准增长条件的p(x) -Laplace方程,在给出弱解的先验估计的基础上,得到了弱解的唯一性.     

A series of abundant exact travelling wave solutions for a modified generalized Vakhnenko equation using auxiliary equation method

In this paper, a series of abundant exact travelling wave solutions is established for a modified generalized Vakhnenko equation by using auxiliary equation method. These solutions can be expressed by Jacobi elliptic function. When Jacobi elliptic functions modulus m→1 or 0, the travelling wave solutions degenerate to four types of solutions, namely, the soliton solutions, the hyperbolic function solutions, the trigonometric function solutions, constant solutions.     

New exact solutions to the (2 + 1)-dimensional Ito equation: Extended homoclinic test technique

Exact soliton solutions to the (2 + 1)-dimensional Ito equation are studied based on the idea of extended homoclinic test and bilinear method. Some explicit solutions, such as triangle function solutions, soliton solutions, doubly-periodic wave solutions and periodic solitary wave solutions, are obtained. It shows that the (2 + 1)-dimensional Ito equation has richer solutions. Besides, the elastic interactions of the solutions and their corresponding physical meaning are discussed.     

Classification of optical wave solutions to the nonlinearly dispersive Schrödinger equation

Different kinds of optical wave solutions to the nonlinearly dispersive Schrödinger equation are given according to different parameters’ regions. Those solutions include looped wave solutions, cusped wave solutions, peaked wave solutions, compacted wave solutions. The looped and cusped forms have not been reported in the literature regarding to the study of the nonlinear Schrödinger equation. We also study the limiting behavior of all periodic solutions as the parameters trend to some special values.     

Solutions of the generalized Lennard-Jones system

In this paper, we study solution structures of the following generalized Lennard-Jones system in R~n,x=(-α/|x|~(α+2)+β/|x|~(β+2))x,with 0 α β. Considering periodic solutions with zero angular momentum, we prove that the corresponding problem degenerates to 1-dimensional and possesses infinitely many periodic solutions which must be oscillating line solutions or constant solutions. Considering solutions with non-zero angular momentum, we compute Morse indices of the circular solutions first, and then apply the mountain pass theorem to show the existence of non-circular solutions with non-zero topological degrees. We further prove that besides circular solutions the system possesses in fact countably many periodic solutions with arbitrarily large topological degree, infinitely many quasi-periodic solutions, and infinitely many asymptotic solutions.     

Several classes of exact solutions to the 1 + 1 Born–Infeld equation

We obtain closed-form exact solutions to the 1 + 1 Born–Infeld equation arising in nonlinear electrodynamics. In particular, we obtain general traveling wave solutions of one wave variable, solutions of two wave variables, similarity solutions, multiplicatively separable solutions, and additively separable solutions. Then, putting the Born–Infeld model into correspondence with the minimal surface equation using a Wick rotation, we are able to construct complex helicoid solutions, transformed catenoid solutions, and complex analogues of Scherk’s first and second surfaces. Some of the obtained solutions are new, whereas others are generalizations of solutions in the literature. These exact solutions demonstrate the fact that solutions to the Born–Infeld model can exhibit a variety of behaviors. Exploiting the integrability of the Born–Infeld equation, the solutions are constructed elegantly, without the need for complicated analytical algorithms.     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

一类推广的KdV方程的新行波解

本文研究了一类推广的Kd V方程的行波解求解的问题.利用新的G展开法,并借助Mathematica计算软件,获得了该方程的含有多个任意参数的新的行波解,分别为三角函数解、双曲函数解、有理函数解和指数函数解,扩大了该类方程的解的范围.     

Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey–Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow‐up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended. Copyright © 2013 John Wiley & Sons, Ltd.     

New exact solutions to the -dimensional Konopelchenko–Dubrovsky equation

In this paper, the extended tanh method, the sech–csch ansatz, the Hirota’s bilinear formalism combined with the simplified Hereman form and the Darboux transformation method are applied to determine the traveling wave solutions and other kinds of exact solutions for the (2+1)-dimensional Konopelchenko–Dubrovsky equation and abundant new soliton solutions, kink solutions, periodic wave solutions and complexiton solutions are formally derived. The work confirms the significant features of the employed methods and shows the variety of the obtained solutions.     

Darboux transformation and Rogue waves of the Kundu–nonlinear Schrödinger equation

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n‐fold Darboux transformation. From known solution Q, the determinant representation of n‐th new solutions of Q^[n] are obtained by the n‐fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third‐order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.