一类广义Sine—Gordon方程的解的存在唯一性

本文利用非线性Ｇａｌｅｒｋｉｎ方法,证明了一类广义Ｓｉｎｅ－Ｇｏｒｄｏｎ方程在Ｄｉｒｉｃｈｌｅｔ边值条件下的整体解的存在唯一性．     

广义非线性Sine-Gordon方程的两个隐式差分格式

本文对一类非线性Sine-Gordon方程的初边值问题提出了两个隐式差分格式．两个隐式差分格式的精度均为O(τ~2 h~2)．我们用离散泛函分析的方法证明了格式的收敛性和稳定性,并证明了求解格式的追赶迭代法的收敛性,最后给出了数值结果．结果表明本文的格式是有效的和可靠的．     

有阻尼Sine－Gordon方程的全局吸引子的维数

本文通过引入新范数,得到有阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的Ｄｉｒｉｃｈｌｅｔ问题的全局吸引子的维数的一个估计．结果表明：当“阻尼”与“扩散”同时增大或正弦项系数减小时,吸引子的维数减小．特别地,得到了零维吸引子存在的参数条件．     

阻尼Sine-Gordon方程的H1-Galerkin混合元方法数值解

利用H1-Galerkin混合有限元方法讨论阻尼Sine-Gordon方程,得到一维情况下半离散和全离散格式的最优阶误差估计,并且推广应用到二维和三维情况,而且不用验证LBB相容性条件.     

有阻尼Sine－Gordon方程的全局吸引子的维数

本文通过引入新范数，得到有阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的Ｄｉｒｉｃｈｌｅｔ问题的全局吸引子的维数的一个估计．结果表明：当“阻尼”与“扩散”同时增大或正弦项系数减小时，吸引子的维数减小．特别地，得到了零维吸引子存在的参数条件．     

Sine-Gordon方程的动态边界反馈镇定

考虑一端固定,控制输入在另一端的Sine-Gordon方程的动态边界反馈镇定问题.我们给出闭环系统的适定性,并利用乘子方法得到在动态边界反馈控制下闭环系统的多项式衰减率.     

二维广义非线性Sine-Gordon方程的一个ADI格式

本文对二维广义非线性Sine-Gordon方程提出了一个带参数的ADI格式,其精度为O(τ2 h1),有效的降低了计算量,并证明了格式的稳定性与收敛性,最后通过参数的不同选取给出了数值算例,结果表明本文的格式是有效的和可靠的.     

带非均匀项sine-Gordon方程的Bcklund变换及对易定理

Bcklund变换(BT)的对易定理是一个很重要的性质,但其数学证明在文献中讨论得较少。文[1]给出了一类保谱的MKdv-SG方程的BT及对易定理的证明.本文研究非保谱的带非均匀项的sine-Gordon方程u_(xt)=sinu (axu_x)_x,(a为参数),(1)它的特征值满足关系λ_1=αλ。我们给出了方程(1)的BT,并证明了对易定理.     

Weak solutions to quasilinear wave equations of Klein-Gordon or Sine-Gordon type and relaxation to reaction-diffusion equations

This paper regards the existence of weak solutions for a quasilinear wave equation of Klein-Gordon and Sine-Gordon type with the presence of a linear damping term and the relaxation to the reaction-diffusion equation when the momentum relaxation time tends to zero. In the limit process is fundamental the celebrated Div-curl Lemma of Tartar and Murat. Received February 5, 1996     

非齐次边界条件下Sine-Gordon型二阶非线性系统的全局吸引子及能稳性

本文在非齐次边界条件下,证明了有阻尼Sine-Gordon型二阶非线性系统的全局吸引子的存在性．同时证明了当时,该全局吸引子为—个平衡点,而且该平衡点具有指数吸引性．并将所得结果应用到系统能稳性的研究中．     

Time Discretisation of Parabolic Problems with the Variable 3-Step BDF

In this paper the stability of the 3-step backward differentiation formula (BDF) on variable grids for the numerical integration of time-dependent parabolic problems is analysed. A stability inequality with a stability constant depending in a controllable way on the mesh is obtained. In particular if the ratios r _j of adjacent mesh-sizes of the underlying grid satisfy the bound r _j r¯ < 1.199 then any mixture of the j-step BDF for j {1, 2, 3} is stable provided the number of changes between increasing and decreasing mesh-sizes is uniformly bounded. From the stability inequality error estimates can be obtained.     

Long Time Asymptotic Behavior of Solution of Difference Scheme for a Semilinear Parabolic Equation

In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as $t\rightarrow\infty$. For the discrete solution of nonlinear parabolic problem, we get its long time asymptotic behavior which is similar to that of the continuous solution. For simplicity, we consider one-dimensional problem.     

Convergence to equilibrium for a time semi-discrete damped wave equation

We prove that the solution of the backward Euler scheme applied to a damped wave equation with analytic nonlinearity converges to a stationary solution as time goes to infinity. The proof is based on the {\L}ojasiewciz-Simon inequality. It is much simpler than in the continuous case, thanks to the dissipativity of the scheme. The framework includes the modified Allen-Cahn equation and the sine-Gordon equation.     

高维抛物型方程差分格式的稳定性

本文提出了一个改进抛物型方程差分格式稳定性条件的新方法,给出并证明新方法稳定的充要条件,数值例子显示了本方法的计算优越性.     

半线性抛物型积分—微分方程的谱方法

本文在讨论带有半线性记忆项抛物型方程半离散格式的基础上,构造了空间方向谱离散,时间方向后欧拉方法的全离散格式,并给出了误差估计。对于记忆项的数值积分,采用了梯形公式与矩形公式结合的方法,并估计了数值积分的影响。     

拟线性抛物方程的弱解

该文给出了拟线性退化抛物方程pa_t{u}+pa_x{f(u)}=pa_xx{A(u(x,t))}∈R^2_+×(0,+∞) ,u(x,0)=u_0(x),x∈R 一种弱解的新定义, 利用Div Curl引理证明了解的存在性．     

关于一类拟线性退化抛物方程

对拟线性退化抛物方程θ_(xx)u uθ_yu-θ_tu=f(·,u)进行了研究,得到了在[O,T]×Ω上的初边值问题解的存在唯—性,这里要求T充分小．     

A note on “New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation”

Exact solutions of the Nizhnik-Novikov-Veselov equation by Li [New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation, Appl. Math. Comput. 215 (2009) 3777-3781] are analyzed. We have observed that fourteen solutions by Li from 30 do not satisfy the equation. The other 16 exact solutions by Li can be found from the general solutions of the well-known solution of the equation for the Weierstrass elliptic function.