守恒律

Firstly, using strategy of introducing associated auxiliary equation to a givenpartial differential equation without admitting Lagrangian, we make the extendedequations possess a variational principle. Based on this, the concepts of extendedconservation law and extended symmetry to given partial differential equationare put forwarded.

首先，我们给出了引入伴随方程扩充原方程的策略，使给定偏微分方程的扩充方程组具有对应泛函，即称为Lagrange系统的方法，以此为基础提出了作为偏微分方程传统守恒律和对称概念的一种推广—偏微分方程扩充对称和扩充守恒律的概念；其次，先给出以得到的Lagrange系统为基础，利用N？

Two classes of numerical methods based on FD-WENO schemes were recommended for solving nonconservative compressible ideal fluid dynamics equations.

以求解双曲守恒律组的FD-WENO格式为基础提出了两类用于求解非守恒可压编理想流体力学方程组的数值方法。

Chapter two proposes the unified form of Hojman"s conservation law and Lutzky"s conservation law. Firstly, the author introduces the general Lie group of transformations that the variations of both the time and the generalized coordinates are considered, derives the determining equation of Lie symmetry for the system, presents a new conservation law, which contains the Hojman"s and the Lutzky"s conservation law as two special cases, and obtains a condition to exclude trivial Hojmans conserved quantities.

第二章，Hojman定理和Lutzky定理的统一形式：首先，引入一般意义下的Lie变换群(即位型变量q_s和时间变量t同时变换)，给出系统的Lie对称性确定方程，提出一个新的守恒律，Hojman定理与Lutzky定理则分别是这个新守恒律在两个特殊情况下的推论，导出一个可排除平凡Hojman守恒量的定理，并分别讨论了Birkhoff系统和非完整系统的Lie对称性和Hojman守恒量，最后，讨论了Hamilton系统的梅对称性与Lie对称性的关系，给出了由梅对称性求Hojman守恒量的方法。

Compared with the initial value problems of scalar conservation laws with smooth flux function, the global weak entropy solutions for the initial-boundary value problems of scalar conservation laws with weak discontinuous flux function include the following new interaction types: a rarefaction wave collides with the boundary and is absorbed compltetely or partially by the boundary; a rarefaction wave collides with the boundary and the boundary will reflect a contact or non-contact shock wave; a contact or non-contact shock wave collides with the boundary and is absorbed by the boundary; a contact or non-contact shock wave collides with the boundary and a new non-contact shock will rebound from the boundary simultaneously or later.

与具有光滑流函数的单个守恒律的初始值问题相比，具有弱间断流函数的单个守恒律初边值问题的整体弱熵解中包括下列新的相互作用类型：稀疏波碰到边界并被边界部分或全部吸收；稀疏波与边界相撞，边界反射出一个接触或非接触激波；接触或非接触激波碰到边界并被边界吸收；接触或非接触激波与边界相撞，边界同时或稍后反射出一个新的非接触激波。

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。

Conservation of mass.conservation of moment muni and conservation of energy are basic regularity obeyed by all movement in nature.For a certain problem in natrue .if we quantify the corresponding conversation law . we can deduce partial differential equation reflecting this problem .and every differential equation reflecting a special physical phenomenon on certain codition.

质量守恒、动量守恒和能量守恒是自然界一切运动所遵循的基本规律，对于自然界的某一特定问题，如果把相应的守恒律数量化，就导出刻划这个问题的微分方程，每一个微分方程都在一定条件下刻划了某一特定的物理现象。

Based on this, a new WENO difference scheme which based on Dispersion-Relation-Preserving relation is developed, and representative test cases with this scheme for computational aeroacoustics problems has been implemented and compared in order to test capability of wave capturing; In addition, WENO schemes generally do not converge at high order in the presence of contact discontinuity of Euler equations, so a conservative front tracking technique coupling WENO schemes and Level Set method to simulate the translating density profile is presented here, and numerical simulation with this technique for representative test case has been implemented and results show the desired accuracy.

本文研究了高阶精度加权基本无振荡格式及其在双曲守恒律方程中的应用，在此基础上作了两个方面的工作：一是针对高频声波问题构造出一种基于保色散关系的WENO有限差分格式，并对计算气动声学问题的代表性算例进行了大量数值实验，比较了该格式捕捉波数的能力；另外，针对高阶WENO格式在处理Euler方程的接触间断时精度有所降低的问题，研究了利用界面追踪技术Level Set方法和高阶激波捕捉WENO格式相结合的一种守恒追踪方法，并且给出有代表性的密度滑移面问题的算例，得到一致高阶精度的数值模拟结果。

Based on Wu's elimination method and "divide-and-conquer" strategy, the undetermined coefficient method for constructing polynomial conservation laws of nonlinear differential-difference equations was improved. Furthermore, a Maple package CLawDDEs was developed to automatically derive conserved densities and associated fluxes of polynomial differential-difference equations.

基于吴消元法和"分治"策略，改进了基于标度不变性构造非线性微分差分方程多项式形式守恒律的待定系数算法，并在计算机代数系统Maple上实现了改进后的算法，其中的软件包CLawDDEs可自动推导出微分差分方程的守恒密度及连带流。

They are consistent with those in breathers, demonstrating that the breather is actually a bounded pair of bright and dark solitons. Finally, we find the first conservation law should be modified by adding a phase.

最后，我们发现非零边值DNLSE的无穷多个守恒律中，第一个守恒律当亮孤子与暗孤子数目不相等时需要补上一个位相。

For a gauge-invariant system with a higher-order Lagrangian,the quantal Noether identities under the local and non-local transformation in configuration space have also been derived.

指出在某些情形下，由量子Noether恒等式可导致系统的量子守恒律。这种求守恒律的程式与量子Noether定理不同。

Breath, muscle contraction of the buttocks; arch body, as far as possible to hold his head, right leg straight towards the ceiling (peg-leg knee in order to avoid muscle tension).

呼气，收缩臀部肌肉；拱起身体，尽量抬起头来，右腿伸直朝向天花板（膝微屈，以避免肌肉紧张）。

The cost of moving grain food products was unchanged from May, but year over year are up 8%.

粮食产品的运输费用与5月份相比没有变化，但却比去年同期高8％。