对称连续分布函数的最优不变估计

该文考虑了未知对称连续分布函数的不变估计问题.连续分布函数在单调变换群下是不变的^[1], 但这个变换群不能保证对称分布函数的不变性.于是, 所要研究的判决问题在单调变换群下不再是不变的. 为了保证判决问题不变性, 考虑一个新的变换群—单调奇变换群, 它确保了所研究的判决问题的不变性.注意到对称分布函数零点的特殊性质, 即, 对任一对称分布函数F, 均有F(0)=1/2,通过视零点为一伪观察值, 得到了所有的非随机化不变估计, 并在不变估计中找到了最优不变估计.     

热方程的非古典势对称群与不变解

主要研究了热方程与波方程的非古典势对称群生成元及相应的群不变解．研究表明对于守恒形式的偏微分方程,可通过其伴随系统求得的非古典势对称群生成元来构造其显式解．这些显式解不能由方程本身的Lie对称群生成元或Lie-B?cklund对称群生成元构造得到．     

在Lie群作用下不变的Markov过程

本文的目的是,依照经典的L′evy-Khinchin三参数表示的精神,论述在Lie群作用下不变的Markov过程的表示理论.对不变Markov过程,按其一般性,我们将在三个层面中进行讨论.首先是Lie群里平移不变的Markov过程,然后是一般流形中在可迁群作用下不变的Markov过程.这两类过程都称L′evy过程,具有三参数表示.第三类过程为在不可迁群作用下不变的Markov过程.在一定条件下,这类过程可分解为一横截群轨道的径过程和一沿着群轨道的角过程.后者为时间非齐次的不变Markov过程,具有依赖于时间的三参数表示.     

一类非线性发展方程的对称分析及级数解

基于李对称理论分析了广义Burgers方程的推广方程,获得其有限维李对称.进一步,研究向量场的伴随表示构造优化系统.最终基于对称约化,获得了方程的约化系统及包含级数解在内的群不变解.     

具有拟不变测度的交换拓扑群上的正定函数

<正> §1.引言如所周知,拓扑群上的正定函数是研究拓扑群的酉表示的重要工具.对于交换的、具有平移不变测度(即 Haar 测度)的拓扑群,例如局部紧的拓扑群,群上的连续正定函数可以用连续特征标关于对偶群上某个测度的积分表示.对于不具有平移不变测度的交换拓扑群,并不是每个连续正定函数都可以用上述积分表示.因而对于不具有平移不变的交换拓扑群,对群上的连续正定函数较难研究,关于抽象调和分析中其余的问题也有类似情     

关于内顺从群的一些性质

本文给出了离散群的可数内不变平均的具体的表达式．证明了由可数内不变平均生成的向量空间的维数等于极小内不变平均的基数，并得到了内顺从群的一些新的刻划．     

mKdV方程的对称与群不变解

主要考虑mKdV方程的一些简单对称及其构成的李代数,并利用对称约化的方法将mKdV方程化为常微分方程,从而得到该方程的群不变解,这是对该方程群不变解的进一步扩展.     

关于对偶的拟不变测度

<正> 关于局部紧的交换拓扑群上,基于不变测度的富里埃变换理论是大家都知道的.与此相联系的相应于局部紧群上的不变测度,在对偶群上有对偶的不变测度,这两个测度是由 Parseval 等式所联结的.这些都是群上调和分析的基本结果.我们知道在某些相当重     

高校应用数学学报──第9卷（1994年）B辑（英文版）第4期目次和提要

高校应用数学学报──第９卷（１９９４年）Ｂ辑（英文版）第４期目次和提要微分方程的对称和群不变解田畴（中国科技大学数学系）该文首先给出微分方程的对称和单参数不变群的关系的一个证明；其次给出微分方程的对称和群不变解的关系；最后，作为应用对Ｋｄｖ方程讨论了...     

(3+1)维波方程新的精确解和不变子空间(英文)

本文利用不变集方法研究(3+1)维波方程,结果表明存在一类波方程关于集合E_0={u:u_x=v_xF(u),u_y=v_yF(u),u_z=v_zF(u)}不变.通过取特殊的v,得到一些特殊的波方程在伸缩群、旋转群以及推广的伸缩和旋转群下不变的精确解,并将该方法推广到(N+1)维波方程的情形.     

Prohibitions caused by nonlocality for nonlocal Boussinesq‐KdV type systems

It is found that two different celebrate models, the Korteweg de‐Vrise (KdV) equation and the Boussinesq equation, are linked to a same model equation but with different nonlocalities. The nonlocal KdV equation can be derived in two ways, via the so‐called consistent correlated bang companied by the parity and time reversal from the local KdV equation and via the parity and time reversal symmetry reduction from a coupled local KdV system which is a two‐layer fluid model. The same model can be called as the nonlocal Boussinesq system if the nonlocality is changed as only one of parity and time reversal. The nonlocal Boussinesq equation can be derived via the parity or time reversal symmetry reduction from the local Boussinesq equation. For the nonlocal Boussinesq equation, with help of the bilinear approach and recasting the multisoliton solutions of the usual Boussinesq equation to an equivalent novel form, the multisoliton solutions with even numbers and the head on interactions are obtained. However, the multisoliton solutions with odd numbers and the multisoliton solutions with even numbers but with pursuant interactions are prohibited. For the nonlocal KdV equation, the multisoliton solutions exhibit many more structures because an arbitrary odd function of can be introduced as background waves of the usual KdV equation.     

Upper bounds of the rates of decay for solutions of the Boussinesq equations

In this paper,upper bounds of the L2-decay rate for the Boussinesq equations are considered.Using the L2 decay rate of solutions for the heat equation,and assuming that the solutions of the Boussinesq equations are smooth,we obtain the upper bounds of L2 decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data.The decay results may then be obtained by passing to the limit of approximating sequences of solutions.The main tool is the Fourier splitting method.     

Generalized Cauchy Matrix Approach for Lattice Boussinesq-Type Equations

The authors generalize the Cauchy matrix approach to get exact solutions to the lattice Boussinesq-type equations:lattice Boussinesq equation,lattice modified Boussinesq equation and lattice Schwarzian...     

一个经典的可积Neumann系统和Boussinesq族Lax对的非线性化

本文在位势与特征函数之间的Neumann约束条件下,经典Boussinesq族的Lax对被非线性化成为自然相容的Lax系统;而且,其为Liouville完全可积的Hamiltonian系统,同时获得了Boussinesq方程解的对合表示。     

光滑区域上二维无黏性无热传导Boussinesq方程组与三维轴对称不可压Euler方程组的指数增长全局光滑解

研究二维无黏性无热传导Boussinesq方程组和三维轴对称不可压Euler方程组光滑解的增长情况,找各种区域使其上的方程组有快增长的解。对Boussinesq方程组,通过选取初始温度和速度的一个分量,可以把方程去耦为两部分。从关于涡量的部分求出涡量、速度场和使结论成立的区域,从关于温度的部分,可见温度的高阶导的增长仅依赖于速度场的一个分量。通过适当选取该分量,得到温度高阶导有指数增长的全局光滑解。对轴对称Euler方程组做类似的处理,适当选取速度场的径向分量,可把方程组去耦,最终得到一类光滑区域,在其上方程组有指数增长全局光滑解。该研究把Chae、Constantin、Wu对一个二维锥形区域上无黏性无热传导Boussinesq方程的结果,推广到一类光滑区域上, 并把他们的方法应用到三维轴对称不可压Euler方程组, 得到了类似的结果。     

On the nonlocal Boussinesq equation: Multiple-soliton solutions

In this work we investigate the first-order nonlocal Boussinesq system. We use the simplified form of Hirota’s direct method to determine multiple-soliton solutions for this system.     

Solitary-wave solutions to a dual equation of the Kaup–Boussinesq system

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the travelling-wave solutions to a dual equation of the Kaup–Boussinesq system. The expressions for smooth solitary-wave solutions are obtained.     

Boussinesq方程温和解的全局适定性北大核心CSCD

Boussinesq方程作为描述许多地球物理现象的模型,是Navier-Stokes方程与热力学方程之间耦合的零阶近似.利用隐函数定理,研究带黏性高维Boussinesq系统,并得到了小初值位于尺度不变空间时温和解的全局适定性.     

Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system

In this paper, we first address the space‐time decay properties for higher‐order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument, and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions for small initial data in some scaling invariant function spaces. The smallness conditions are somehow weaker than those presented by Brandolese and Schonbek. We further investigate the asymptotic profiles and decay properties of these strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

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This paper deals with a variant Boussinesq equations which describes the propagation of shallow water waves in a lake or near an ocean beach. We derive out two hetero-B\"{a}cklund transformations between the variant Boussinesq equations and two linear parabolic equations by using the extended homogeneous balance method. We also obtain two hetero-B\"{a}cklund transformations between the variant Boussinesq equations and Burgers equations. Furthermore, we obtain two hetero-B\"{a}cklund transformation between the variant Boussinesq equations and heat equations. By using these B\"{a}cklund transformations and so-called "seed solution", we obtain a large number of explicit exact solutions of the variant Boussinesq equations. Especially, The infinite explicit exact singular wave solutions of variant Boussinesq equations are obtained for the first time. It is worth noting that these singular wave solutions of variant Boussinesq equations will blow up on some lines or curves in the $(x,t)$ plane. These facts reflect the complexity of the structure of the solution of variant Boussinesq equations. It also reflects the complexity of shallow water wave propagation from one aspect.