导出方程

Further, with the help of Riccati equations, an infinite number of conservation laws for the solton hierarchy are deduced. For the sake of simplicity, taking the general TD hierarchy as an illustrative example, we prove that its 2×2 Lenard pair of operators forms a Hamiltonian pair. Thus the isospectral evolution TD hierarchy is the general Hamiltonian system and possesses the Bi-Hamiltonian structures and Multi-Hamiltonian structures. By using the method of derivation of functional under some constraint condition, a complete one-to-one correspondence between the Hamiltonian functions of the hierarchy and its conservation density functions can be built. These results can also be applied to the isospectral evolution soliton hierarchy of this paper. Finally, there's a gauge transformation between the spectral problem of this paper and the AKNS system. Moreover, the potentials in these spectral problems satisfy the general Miura transformation, the corresponding relationship between the two soliton hierarchies is also given.

进一步本文还通过特征函数的组合关系所满足的Riccati方程，得到了该等谱方程族的无穷多个守恒律；为简便起见，本文以广义TD族为例，由它的2×2 Lenard算子对的性质证明了此算子对为Hamilton算子对，这说明广义TD族是广义Hamilton系统且具有Bi-Hamilton结构和Multi-Hamilton结构；进而利用它的依赖于谱参数的一般守恒密度的积分在约束条件下求泛函导数的方法，得到了广义TD族的Hamilton函数与守恒密度之间的对应关系，这些性质对于由本文提出的2×2谱问题所导出的等谱孤子族仍成立；另外此谱问题与AKNS系统存在着规范变换，位势之间有广义Miura变换，而孤子方程之间也满足一定的等价关系。

After descibing the concepts, definitionsand operation approaches of quantum group and SLq(2) Lie algebra, the physicalmeaning of quantum group particles is explained in the present thesis. The gaugeinvariance in SLq(2) gauge field and its thermodynamics model is considered basedon the idea of q-deformation. The zero energy gap equation and q-deformed energygap equation are derived under the assumption of q-quantization and semi-classical q-quantization, then the zero-temperature energy gap A(0,q) and crititical temperature〓 can be calculated from the q-energy gap equation.

在介绍了量子群和SLq(2)李代数的基本概念，定义和运算方法后，文中进一步阐述了量子群粒子的物理意义；在q变形思想基础上，本文研究了SLq(2)规范场及其热力学模型中的规范不变性质，并在q量子化和半经典q量子化的假设下导出了零能隙方程和q变形能隙方程；然后从q能隙方程计算了与零温度能隙△(0，q)和临界温度〓。

In this work, the equations describing circular Gaussian beams propagating inside absorptive Keer medium have been derived from nonlinear paraxial equation. Based on the equations, an evolution equation of beam waist lias been obtained.

本文从非线性近轴波方程出发，导出了在吸收克尔介质中具有二次位相因子的圆对称高斯光束的传输方程，得到了光束束腰的演化方程。

From the Maxwell′s equation,the quasi static electromagnetic field equations in a perfect conductor have been deduced in this paper.The equations are used to analyze two eddy current field engineering problems,and good results are obtained.

本文从麦克斯韦方程出发，推导出良导体内的似稳电磁场方程，将导出的电磁场方程用于分析两个电磁工程实例，获得了两个工程问题的涡流场解析计算结果。

The Fourier transfermation about the azimuthal angle and Hankeltransfermation about the radial dirction have been used to turn the basic dis-placement equatioons and constitutive equations under cylindrical coordinatesystem into a system of second order ordinary differential equations in thewavenumber domain,by use of the initial prarameter method to solve differ-ential equations.

利用关于方位角的Fourier变换及关于径向的Hankel变换，将柱坐标系下位移基本方程和本构方程转化为波数域内二阶常微分方程组，利用求解微分方程的初参数法，建立了介质层的传递矩阵，导出了层状弹性半空间在地表作用任意静荷载情形下的解析解。

And by using the initial conditions as well as the end conditions, the dynamic problem is then transferred to a second kind Volterra integral equation about the function of the axial strain with respect to time which can also be solved successfully by the interpolation method. For piezoelectric and pyroelectric hollow cylinders, by following the solving procedure for elastic hollow cylinder and by using the electric boundary conditions, the dynamic problems are transferred to two Volterra integral equations about two functions of time, one is axial strain and the other is related to electric displacement, which can also be solved efficiently and quickly by employing interpolation method. The elastodynamic solutions of hollow spheres, which are made of elastic, piezoelectric and pyroelectric materials, respectively, for spherically symmetric problems are also obtained.

对于弹性空心圆柱，通过引入一特定函数将非齐次边界条件化为齐次边界条件，然后利用正交展开技术，导出关于时间函数的方程，再结合初始条件和端部边界条件，将原问题转化为关于一个时间函数的第二类Volterra积分方程，运用插值法可给出此积分方程的解；对于压电和热释电空心圆柱，利用求解弹性空心圆柱相似的方法，再结合电学边界条件，原问题转化为关于两个时间函数（轴向应变和与电位移有关的函数）的第二类Volterra积分方程组，同样可用插值法来构造相应的递推公式高效地求解此积分方程组。

The weak formulation permit is satisfied not point by point, finite element method balances only at node neighborhood, in unit balance is not required, in analytical method single check from infinite series does not satisfy the equation, but infinite number of points satisfy the equation, and then the weak formulation of dual system of Hamilton equations was educed.

基于基本方程和边界条件的弱形式，给出相应的广义变分和不同形式的有限元所采用的弱条件，说明弱形式允许不是逐点满足，有限元法只在节点邻域平衡，在单元内不要求平衡，解析法用无穷级数中单取一项也不满足方程，也是无穷多项在积分下满足，进而导出对偶体系弱形式哈密尔顿方程。

The energy balance of flue gas, radiative heat transfer equations and zero dimensional model of heat transfer are derived in the multiple combustion boiler combining a chain grate with a pulverized coal, considering the radiation heat between the lower and upper zone in the furnace.

将链条炉排加煤粉复合燃烧锅炉的炉膛分成上下两部分，导出了上下炉膛的烟气热平衡方程、辐射换热方程及炉膛传热计算的零维模型，方程中考虑了上下炉膛间的辐射换热。

In this paper, by means of finding the solutions of Maxwells equitions which have a tensor dielectric permittivity,the wave equitions were obtained in rotation dielectric under the condition of longitudinal direct current magnetic field.

本文求解张量介质电常数条件下的麦克斯韦方程，导出了纵向场作用下旋光材料的波动方程。由于电场与磁场是互相耦合不能单独存在，因而引入了孪生标量波的概念，并证明了该标量波满足亥姆霍兹方程。

Synthesizing the degrees of freedom of both rotor and helicopter body, and involving time derivatives of flap motion coefficient and induced velocity model derived from vortex theory, the governing equations of helicopter maneuvering flight are derived properly, with combination of kinematic equations and dynamic equations.

集旋翼和机体之自由度并计及挥舞运动系数对时间的导数和涡流理论的诱速模型，把运动学方程和动力学方程组合起来，从而导出了机动飞行的控制方程。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。