基于Shannon-Cosine小波精细积分法的壁画降噪修复方法

为修复受破损且噪声点众多的壁画图像,提出了用偏微分方程（partial differential equation,PDE）扩散的方法对图像进行降噪修复。针对PDE法求解精度较低的问题,提出了一种Shannon-Cosine小波精细积分法,运用小波数值方法对偏微分方程进行离散处理,降低其方程组规模,并采用精细积分法求解,有效提高了计算速度。试验结果表明,采用该算法对受损壁画降噪处理后,视觉上,图像边界更清晰,且噪声点得到有效减少,达到了保边降噪的效果,更符合人眼的视觉效果;客观上,与中值滤波、均值滤波和维纳滤波方法相比,采用本算法处理后的图像其PSNR值和SSIM值均最大。因此,运用Shannon-Cosine小波精细积分法求解图像的PDE模型是可行的,取得了较好的图像降噪效果。     

具有非线性阻尼的记忆型抽象发展方程的时间依赖吸引子

运用时间依赖空间中的过程理论和收缩函数方法以及更多细节性估计,研究了具有非线性阻尼和衰退记忆的抽象发展方程的解在时间依赖空间中的渐近性态,证明了时间依赖吸引子在空间E_t^{\theta }中的存在性。     

统一的对流扩散型可压缩流体力学方程与解法

流体力学的动量方程、能量方程、湍动能方程和耗散方程都具有对流扩散方程的形式,但连续方程却不是对流扩散型的。对于可压缩问题,本文通过合理的数学推导,不作任何近似、假定与简化,得到一个全新的连续方程形式．该连续方程以压力为未知变量,并具有对流扩散型形式,使得所有的流体动力学方程组都具有完全统一的方程形式,给出了这种三维对流扩散方程组的有限精确差分计算格式。对流体力学的进一步发展具有一定意义．     

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。     

Spinor Green’s functions via spherical means on products of space forms

We explicitly compute the Green’s function of the spinor Klein–Gordon equation on the Riemannian and Lorentzian manifolds of the form _M0×?×_MN$M_0\times ?\times M_N$, with each factor being a space of constant sectional curvature. Our approach is based on an extension of the method of spherical means to the case of spinor fields and on the use of Riesz distributions.     

Numerical methods for computing ground states and dynamics of nonlinear relativistic Hartree equation for boson stars

Efficient and accurate numerical methods are presented for computing ground states and dynamics of the three-dimensional (3D) nonlinear relativistic Hartree equation both without and with an external potential. This equation was derived recently for describing the mean field dynamics of boson stars. In its numerics, due to the appearance of pseudodifferential operator which is defined in phase space via symbol, spectral method is more suitable for the discretization in space than other numerical methods such as finite difference method, etc. For computing ground states, a backward Euler sine pseudospectral (BESP) method is proposed based on a gradient flow with discrete normalization; and respectively, for computing dynamics, a time-splitting sine pseudospectral (TSSP) method is presented based on a splitting technique to decouple the nonlinearity. Both BESP and TSSP are efficient in computation via discrete sine transform, and are of spectral accuracy in spatial discretization. TSSP is of second-order accuracy in temporal discretization and conserves the normalization in discretized level. In addition, when the external potential and initial data for dynamics are spherically symmetric, the original 3D problem collapses to a quasi-1D problem, for which both BESP and TSSP methods are extended successfully with a proper change of variables. Finally, extensive numerical results are reported to demonstrate the spectral accuracy of the methods and to show very interesting and complicated phenomena in the mean field dynamics of boson stars.     

Rogue waves, dissipation, and downshifting

We investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one-dimensional Dysthe equation. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. Using the inverse spectral theory of the NLS equation, simulations of the damped Dysthe equation for sea states characterized by JONSWAP spectrum consistently show that rogue wave events are well-predicted by proximity to homoclinic data, as measured by the spectral splitting distance δ. The cut off distance δ_cutoff decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur.     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

New Exact Solutions of Multi-component mKdV Equation

In this paper, new Jacobi elliptic function solutions of multi-component mKdV equation are obtained directly in a unified way. When the modulus m→1, those periodic solutions degenerate as the corresponding hyperbolic function solutions. Then, to the three-component mKdV equation, five types of effective solution are presented in detail.