Rota-Baxter Operators on 3-dimensional Lie Algebras and Solutions of the Classical Yang-Baxter Equation

In this paper,we compute Rota-Baxter operators on the 3-dimensional Lie algebra g whose derived algebra’s dimension is 2.Furthermore,we give the corresponding solutions of the classical Yang-Baxter equation in the 6-dimensional Lie algebras g ■ _(ad~*) g~* and some new structures of left-symmetric algebra induced from g and its Rota-Baxter operators.

POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION

In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.

三维单左对称代数上的Nijenhuis算子和经典Yang-Baxter方程解

为了研究三维复单左对称代数上的Nijenhuis算子,通过求解结构常数构成的二次方程组,对三维复单左对称代数上的Nijenhuis算子进行了刻画.此外,利用所得结果,给出了半直积邻接李代数上的经典Yang-Baxter方程的解.

GLOBAL CLASSICAL SOLUTION TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

In this paper, we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ)provided that the initial energy is small in some sense. In our result, we give a relation between the initial energy and the viscosity coefficient μ, and it shows that the initial energy can be large if the coefficient of the viscosity μ is taken to be large, which implies that large viscosity μ means large solution.

Classical and nonclassical symmetry classifications of nonlinear wave equation with dissipation

A complete classical symmetry classification and a nonclassical symmetry classification of a class of nonlinear wave equations are given with three arbitrary parameter functions. The obtained results show that such nonlinear wave equations admit richer classical and nonclassical symmetries, leading to the conservation laws and the reduction of the wave equations. Some exact solutions of the considered wave equations for particular cases are derived.

SOLVING COLORED YANG-BAXTER EQUATION BY WU’S METHOD

In this article, we discuss nonsymmetric solutions of the colored Yang-Baxter equation dependent on spectral as well as colored parameters and give all seven-vertex solutions by Wu＇s method. It is also proved that the solutions are composed of six groups of basic solutions up to five solution transformations. Moreover, al l solutions can be classified into two categories called Baxter type and free-fermion type.

GLOBAL CLASSICAL SOLUTIONS TO THE 3-D ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH GENERAL INITIAL ENERGY

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ｜｜po｜｜L∞ is appropriate small and 1 〈 γ 〈 6/5. Here the initial density could have vacuum and we do not require that the initial energy is small.

Darboux Transformations and Soliton Solutions for Classical Boussinesq-Burgers Equation

Two basic Darboux transformations of a spectral problem associated with a classical Boussinesq-Burgersequation are presented in this letter.They are used to generate new solutions of the classical Boussinesq-Burgersequation.

Conditional Similarity Reductions of Jimbo—Miwa Equation via the Classical Lie Group Approach

Recently, the Clarkson and Kruskal direct method has been modified to find new similarity reductions (conditional similarity reductions) of nonlinear systems and the results obtained by the modified direct method cannot be obtained by the current classical and/or non-classical Lie group approach. In this paper, we show that the conditional similarity reductions of the Jimbo-Miwa equation can be reobtained by adding an additional constraint equation to the original model to form a conditional equation system first and then solving the model system by means of the classical Lie group approach.

MAXIMAL ATTRACTORS OF CLASSICAL SOLUTIONS FOR REACTION DIFFUSION EQUATIONS WITH DISPERSION

The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norm estimate for the attractor. This estimate is optimal for the attractor under Neumann boundary condition. Next, the same problem is discussed for reaction diffusion system with uniformly contracting rectangle, and it reveals that the maximal attractor of classical solution for such system in the whole space is only necessary to be established in some invariant region. Finally, a few examples of application are given.

A new supersymmetric classical Boussinesq equation

In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlevé test and we also calculate its one- and two-soliton solutions.

Non-Noether Conserved Quantity of Poincaré-Chetaev Equations of a Generalized Classical Mechanics

在谎言对称的 non-Nocther 保存了的现在的纸，在谎言组的一般无穷小的转变下面的概括古典力学的 Poincare-Chetaev 方程的数量被讨论。首先，我们建立方程的 Liesymmetry 的决定方程。第二，谎言对称的 non-Noether 保存了方程的数量被推出。最后，一个例子被给说明结果的申请。

QUANTUM COMODULE APPROACH THE SOLUTION OF THE QUANTUM YANG-BAXTER EQUATION

In this paper, the relationship between solutions of the Quantum Yang-Baxter Equation and quantum comodules, and some properties of the quantum comodule category are characterized here. These results make it possible to give some set-theoretical solutions of the Quantum Yang-Baxter Equation.

The Singular Upper Triangle Type Solutions with Spin 1/2 of Quantum Yang-Baxter Equation （Ⅱ）

In these two papers (Ⅰ) and (Ⅱ),the singular upper triangle type solutions with spin 1/2 of quantum Yang-Baxter equation are given.In the Paper (Ⅱ),we give the general solutions of the Yang-Baxter equation in this case.

EQUATION OF MATHEMATICAL PHYSICS AND OTHER AREAS OF APPLICATION: A CLASSICAL INTEGRABLE SYSTEM AND THE INVOLUTIVE REPRESENTATION OF SOLUTIONS OF THE HIGHER ORDER KAUP-NEWELL EQUATION

Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be a completely integrable system (R2N, Adp AND dq, H = H-1) with the Hamiltonian H-1 = -[A3q, p]-1/2[A2p, p][A2q, q]. while the nonlinearization of the time part leads to its N-involutive system {H(m)}. The involutive solution of the compatible fsystem (H-1), (H(m)) is mapped by into the solution of the higher order Kaup-Newell equation.

Equation of Motion of a Mass Point in Gravitational Field and Classical Tests of Gauge Theory of Gravity

一个系统的方法被开发在保持的重力的计量器学习一个集体点的古典运动。由使用 Mathematica，首先，重力的计量器地的领域方程的一个球形的对称的解决方案被获得，它只是 traditionalSchwarzschild 解决方案。联合计量器协变性和牛顿的原则“运动的 s 秒法律，一个集体点在的运动的方程重力删除被推出。基于领域方程的球形的对称的解决方案和一个集体点的运动的方程在重力删除，我们能讨论严肃的计量器理论的古典测试包括由阳光的光的 thedeGection ，内部行星的轨道和传递太阳的雷达回响的时间延期的 perihelia 的领前。严肃的计量器理论给的这些古典测试的理论预言被一般相关性完全与那些一样给，这被发现。

Applications of Non-Classical Equations and Their Approaches to the Solution of Some of Classes Equations Arise in the Kelvin-Helmholtz Mechanism and Instability

In the presented work, we consider applications of non-classical equations and their approaches to the solution of some classes of equations that arise in the Kelvin-Helmholtz Mechanism (KHM) and instability. In all areas where the Kelvin-Helmholtz instability (KHI) problem is investigated with the corresponding data unchanged, the solution can be taken directly in a specific form (for example, to determine the horizontal structure of a perturbation in a barotropic rotational flow, which is a boundary condition taken, as well as other types of Kelvin-Helmholtz instability problems). In another example, the shear flow along the magnetic field in the Z direction, which is the width of the contact layer between fast and slow flows, has a velocity gradient along the X axis with wind shear. The most difficult problems arise when the above unmentioned equation has singularities simultaneously at points and in this case, our results also remain valid. In the case of linear wave analysis of Kelvin-Helmholtz instability (KHI) at a tangential discontinuity (TD) of ideal magneto-hydro-dynamic (MHD) plasma, it can be attributed to the presented class, and in this case, as far as we know, solutions for eigen modes of instability KH in MHD plasma that satisfy suitable homogeneous boundary conditions. Based on the above mentioned area of application for degenerating ordinary differential equations in this work, the method of functional analysis in order to prove the generalized solution is used. The investigated equation covers a class of a number of difficult-to-solve problems, namely, generalized solutions are found for classes of problems that have analytical and mathematical descriptions. With the aid of lemmas and theorems, the existence and uniqueness of generalized solutions in the weight space are proved, and then general and particular exact solutions are found for the considered problems that are expressed analytically explicitly. Obtained our results may be used for all the difficult-to-solve processes of KHM and instabilities and instabilities, which cover widely studied areas like galaxies, Kelvin-Helmholtz instability in the atmospheres of planets, oceans, clouds and moons, for example, during the formation of the Earth or the Red Spot on Jupiter, as well as in the atmospheres of the Sun and other stars. In this paper, also, a fairly common class of equations and examples are indicated that can be used directly to enter data for the use of the studied suitable tasks.

Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

This paper concerns the sonic-supersonic structures of the transonic crossflow generated by the steady supersonic flow past an infinite cone of arbitrary cross section.Under the conical assumption,the three-dimensional(3-D)steady Euler equations can be projected onto the unit sphere and the state of fluid can be characterized by the polar and azimuthal angles.Given a segment smooth curve as a conical-sonic line in the polar-azimuthal angle plane,we construct a classical conical-supersonic solution near the curve under some reasonable assumptions.To overcome the difficulty caused by the parabolic degeneracy,we apply the characteristic decomposition technique to transform the Euler equations into a new degenerate hyperbolic system in a partial hodograph plane.The singular terms are isolated from the highly nonlinear complicated system and then can be handled successfully.We establish a smooth local solution to the new system in a suitable weighted metric space and then express the solution in terms of the original variables.

李代数sl(2,C)上的经典Yang-Baxter方程的解及其应用

针对复数域上特殊线性李代数sl(2,C)的经典Yang-Baxter方程解的问题,利用sl(2,C)的基元素,通过计算Yang-Baxter算子在其基元素上的作用的方法,得到了sl(2,C)的经典Yang-Baxter方程的一些解,进而给出了sl(2,C)上的某些左对称代数结构.

GLOBAL EXISTENCE AND BLOW-UP PHENOMENA OF CLASSICAL SOLUTIONS FOR THE SYSTEM OF COMPRESSIBLE ADIABATIC FLOW THROUGH POROUS MEDIA

By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blow-up phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘small ’ solution.