Invariant Sets and Exact Solutions to Higher-Dimensional Wave Equations

The invariant sets and exact solutions of the （1 ＋ 2）-dimensional wave equations are discussed. It is shown that there exist a class of solutions to the equations which belong to the invariant set E0 = {u ： ux = vxF（u）,uy = vyF（u） }. This approach is also developed to solve （1 ＋ N）-dimensional wave equations.     

Variable Separation and Derivative-Dependent Functional Separable Solutions to Generalized KdV Equations

Using the generalized conditional symmetry approach, a complete list of canonicalforms for the Kortewegde-Vries type equations with which possessing derivative-dependent functional separable solutions (DDFSSs) is obtained.The exact DDFSSs of the resulting equations are explicitly exhibited.     

Conditional Lie-Bäcklund Symmetry and New Variable Separation Solutions of the Third Order KdV-Type Equations

The functionally generalized variable separation solutions of a general KdV-type equations u_t=u_xxx + A(u, u_x)u_xx + B(u, u_x) are investigated by developing the conditional Lie-Bäcklund symmetry method. A complete classification of the considered equations, which admit multi-dimensional invariant subspaces governed by higher-order conditional Lie-Bäcklund symmetries, is presented. As a result, several concrete examples are provided to construct functionally generalized variable separation solutions of some resulting equations.     

Variable Separation and Exact Separable Solutions for Equations of Type uxt=A(u,ux)uxx+B(u,ux)

The generalized conditional symmetry is developed to study the variable separation for equations of type u_xt=A(u,u_x)u_xx+B(u,u_x). Complete classification of those equations which admit derivative-dependent functional separable solutions is obtained and some of their exact separable solutions are constructed.     

Functional Separable Solutions to Nonlinear Diffusion Equations by Group Foliation Method

We consider the functional separation of variables to the nonlinear diffusion equation with source and convection termut = (A(x)D(u)ux)x B(x)Q(u),Ax ≠ 0.The functional separation of variables to this equation is studied by using the group foliation method.A classification is carried out for the equations which admit the function separable solutions.As a consequence,some solutions to the resulting equations are obtained.     

Exotic Localized Coherent Structures of the （2＋1）—Dimensional Dispersive Long—Wave Equation

This article is concerned with the extended homogeneous balance method for studying the abundant localized solution structures in the (2 1)-dimensional dispersive long-wave equations uty ηxx (u^2)xy/2=0,ηt (uη u uxy)x=0.Starting from the homogeneous balance method,we find that the richness of the localized coberent structures of the model is caused by the entrance of two variable-separated arbitrary functions.for some special selections of the arbitrary functions,it is shown that the localized structures of the model may be dromions,lumps,breathers,instantons and ring solitons.     

On Differential Equations Describing 3-Dimensional Hyperbolic Spaces

In this paper, we introduce the notion of a (2+1)-dimensional differential equation describing three-dimensional hyperbolic spaces (3-h.s.). The (2+1)-dimensional coupled nonlinear Schrödinger equation and its sister equation, the (2+1)-dimensional coupled derivative nonlinear Schrödinger equation, are shown to describe 3-h.s. The (2+1)-dimensional generalized HF model: S_t={(1/2i)[S,S_y]+2iσS}_x, σ_x=-(1/4i)tr(SS_xS_y), in which S∈[GL_C(2)]/[GL_C(1)×GL_C(1)], provides another example of (2+1)-dimensional differential equations describing 3-h.s. As a direct consequence, the geometric construction of an infinite number of conservation laws of such equations is illustrated. Furthermore we display a new infinite number of conservation laws of the (2+1)-dimensional nonlinear Schrödinger equation and the (2+1)-dimensional derivative nonlinear Schrödinger equation by a geometric way.     

Three New (2+1)-dimensional Integrable Systems and Some Related Darboux Transformations

We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A₁, then under the framework of zero curvature equations we generate two (2+1)-dimensional integrable hierarchies, in-cluding the (2+1)-dimensional shallow water wave (SWW) hierarchy and the (2+1)-dimensional Kaup-Newell (KN) hierarchy. Through reduction of the (2+1)-dimensional hierarchies, we get a (2+1)-dimensional SWW equation and a (2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the (2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the (2+1)-dimensional KN equation could be deduced. Finally, with the help of the spatial spectral matrix of SWW hierarchy, we generate a (2+1) heat equation and a (2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang-Mills equations.     

多面体共替代对Sr_2(Al_(1–x) Mg_x)(Al_(1–x) Si_(1+x))O_7:Eu~(2+)晶体结构和发光颜色的影响

采用高温固相法合成Sr_(1.98)(Al_(1–x) Mg_x)(Al_(1–x) Si_(1+x))O_7: 2%Eu~(2+)荧光粉完全固溶体,利用X射线衍射、光致发光光谱和光学显微镜进行晶体结构和发光性能的研究. Sr_2Al_2SiO_7和Sr_2MgSi_2O_7同构化合物中包含[MgO_4]、[SiO_4]和[AlO_4]四面体,较大体积的[MgO_4]和较小体积的[SiO_4],共同替代体积相似的[AlO_4],导致[(Si/Al)O_4]收缩和[(Mg/Al)O_4]膨胀,晶胞参数c减少, a和V增大,使Eu~(2+)周围的环境发生改变,晶体场劈裂程度减小,发射峰位从503 nm蓝移至467 nm,实现发光光谱从绿色(0.2384, 0.3919)到蓝色(0.1342,0.1673)的转变.当x为0时,发射峰的半高宽为120 nm, x从0.25增加到1时,半高宽由89 nm逐渐减小至50 nm,多面体的替代会改变荧光粉的发光性能.     

Invariant Sets and Exact Solutions to Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

In this paper, we introduce a new invariant set ˜E₀={u:u_x=f^ˊ(x)F(u)+ε [g^ˊ(x) -f^ˊ(x)g(x)]F(u)exp(-∫^u(1/F(z))dz), where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact solutions to nonlinear diffusion equation u_t=(D(u)u_x)_x+Q(x,u)u_x+P(x,u), are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set ˜E₀.     

Symmetries and Infinite Dimensional Lie Algebra of the Variable Coefficient KDV Equation

With the help of the conditional strong symmetry, we obtain the K symmetries, r symmetries, infinite dimensional Lie algebra and the corresponding infinite conservation quantities of the variable coefficient KdV equation u_t = h₁(t)(u_xxx +6uu_x) +4h₂(t)u_x - h₀(t)(2u+ xu_x) with h₀, h₁ and h₂ being three arbitrary functions of t.     

NFX(X=-1,0,+1)分子离子基态的结构与势能函数

用密度泛函理论的B3LYP方法,分别以6-311++g(df,3pd),6-311g(3d,3p)和6-311++g(3df,3pd)为基函数对NF分子、NF⁺和NF^-离子基态进行几何优化和频率计算,并进行单点能扫描计算.用最小二乘法拟合得到NF^X(X=-1,0,+1)分子离子基态的Murrell-Sorbie势能函数.利用得到的解析势能函数计算出的NF分子和NF⁺离子基态光谱常数(B_e,α_e,ω_e,ω_eχ_e)与实验值符合很好.首次得到NF^-离子基态的光谱常数(B_e,α_e,ω_e,ω_eχ_e)和力常数(f₂,f₃,f₄),为NF-离子基态的后期研究提供理论参考.     

New Similarity Reductions and Compacton Solutions for Boussinesq-Like Equations with Fully Nonlinear Dispersion

In this paper, similarity reductions of Boussinesq-like equations with nonlinear dispersion (simply called B(m,n) equations) u_tt=(uⁿ)_xx+(u^m)_xxxx, which is a generalized model of Boussinesq equation u_tt=(u²)_xx+u_xxxx and modified Bousinesq equation u_tt=(u³)_xx+u_xxxx, are considered by using the direct reduction method. As a result, several new types of similarity reductions are found. Based on the reduction equations and some simple transformations, we obtain the solitary wave solutions and compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they re-emerge with the same coherent shape) of B(1,n) equations and B(m,m) equations, respectively.     

Abundant New Explicit Exact Soliton—Like Solutions and Painleve Test for the Generalized Burgers Equation in （2＋1）—Dimensional Space

We obtain Backlund transformation and some new kink-like solitary wave solutions for the generalized Burgers equation in (2 1)-dimensional space,ut 1/2(uδy^-1ux)x-uxx=0,by using the extended homogeneous balance method.As is well known,the introduction of the concept of dromions (the exponentially localized solutions in (2 1)-dimensional space)has triggered renewed interest in (2 1)-dimensional soliton systems.The solutions obtained are used to show that the variable ux admits exponentially localized solutions rather than the physical field u(x,y,t) itself.In addition,it is shown that the equation passes Painleve test.     

(3 + 1)-Dimensional Integrable Models Possessing Infinite Dimensional Virasoro-Type Symmetry Algebra

Using a series of concrfte reGzations of the centerless Virasoro-type symmetry algebra [σ(ƒ₂(t), σ(ƒ₁(t))] = σ(ƒ₁ƒ₂ - ƒ₂ƒ₁), various (3+1)-dimensional integrable equations under the condition that they possess Kac-Moody-Virasoro-type infinite dimensional symmetry algebra are obtained.     

General Solution and Localized COherent Soliton Structures of the （2＋1）—Dimensional Generalized Davey—Stewarson Equations

In this paper,the variable separation approach is used to obtain localized coherent structures of the (2 1)-dimensional generalized Davey-Stewarson equations:iqt 1/2(qxx=qyy) (R S)q=0,Rx=-σ/2|q|y^2,Sy=-σ/2|q|2/x.Applying a special Baecklund transformation and introducing arbitrary functions of the seed solutions.and abundance of the localized structures of this model is derived,By selceting the arbitrary functions appropriately,some special types of localized excitations such as dromions,dromion lattice,breathers,and instantons are constructed.     

Gd1-xEuxAl3(BO3)4纳米晶的合成及其发光性能

以高温固相反应法合成了Gd_1-xEu_xAl₃(BO₃)₄:Eu³⁺纳米荧光粉。使用XRD分析确定了样品的物相,并根据谢乐公式计算出其微晶的纳米粒度。采用了“粉末悬浮法”以甘油为分散介质,在RF540荧光光度计上测试了纳米晶荧光粉的激发光谱和发射光谱。GdAl₃(BO₃)₄基质本身发光,GdAl₃(BO₃)₄:Eu纳米荧光粉表现了Eu³⁺的特征发射光谱,其中最强峰为⁵D₀→⁷F₂发射,表明晶体结构中没有对称中心格位。实验表明在GdAl₃(BO₃)₄:Eu纳米晶荧光粉中,存在Gd³⁺对Eu³⁺发光的基质敏化作用。     

氟哌酸1HNMR的测定及过渡金属离子对其影响

氟哌酸^*[即1-乙基-6-氟-4-氧-7-(1-哌嗪)-3-(1,4-二氢化喹啉甲酸]是一种广谱、安全、有效、可供口服的抗感染药。我们测定了它的¹H自旋-晶格弛豫时间T₁及其在Cr³⁺、Mn²⁺、和Fe³⁺等过渡金属离子存在下的T₁值。实验结果表明,在加入过渡金属离子前后,氟哌酸不同位¹H的T₁值发生了一定的变化。据此可揭示过渡金属离子和氟哌酸形成配合物的配位点,从而可确定所形成配合物的结构。     

外电场作用下黄曲霉素B1的分子结构和光谱

利用密度泛函（DFT）理论,在B3LYP/6-311G基组水平上研究外电场（0~0.015 a.u.）作用下黄曲霉素B₁（Aflatoxin B₁,AFB₁）分子的总能量、偶极矩、键长、HOMO-LUMO能隙、红外光谱.在优化构型的基础上,利用含时密度泛函（TDDFT）方法研究紫外可见吸收光谱和激发态的变化.结果表明：在CO键方向上加电场,外电场逐渐增大时,AFB₁分子的总能量逐渐增大,偶极矩逐渐减小.电场对不同的键长影响不同.能隙随外电场逐渐增大而减小,红外光谱吸收峰出现红移或蓝移,紫外可见吸收光谱的两个吸收峰发生不同的移动,但总的谱线集中在近紫外光区.分子各个激发态的激发能在外电场逐渐增大时逐渐减小,表明分子结构越来越不稳定.