Nonlocal symmetry and exact solutions of the(2+1)-dimensional modified Bogoyavlenskii–Schiff equation

In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the（2＋1）-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system.In addition,the（2＋1）-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion（CRE） solvable.As a result,the soliton–cnoidal wave interaction solutions of the equation are explicitly given,which are difficult to find by other traditional methods.Moreover figures are given out to show the properties of the explicit analytic interaction solutions.

Symmetry Groups and New Exact Solutions to （2＋1）-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.

Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrdinger equation with variable coefficients

In this paper, the extended symmetry transformation of （3＋1）-dimensional （3D） generalized nonlinear Schrodinger （NLS） equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.

Symmetry analysis and explicit solutions of the (3+1)-dimensional baroclinic potential vorticity equation

This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-（3＋1）- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding （2＋1）-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.

Lie Symmetry Groups of(2+1)-Dimensional BKP Equation and Its New Solutions

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physicssystems.Applying the modified direct method to the well-known (2+1)-dimensional BKP equation we get its symmetry.Furthermore,the exact solutions of (2+1)-dimensional BKP equation are obtained through symmetry analysis.

Truncated series solutions to the(2+1)-dimensional perturbed Boussinesq equation by using the approximate symmetry method

In this paper, the（2＋1）-dimensional perturbed Boussinesq equation is transformed into a series of two-dimensional（2 D） similarity reduction equations by using the approximate symmetry method. A step-by-step procedure is used to acquire Jacobi elliptic function solutions to these similarity equations, which generate the truncated series solutions to the original perturbed Boussinesq equation. Aside from some singular area, the series solutions are convergent when the perturbation parameter is diminished.

Painlevé property, local and nonlocal symmetries, and symmetry reductions for a (2+1)-dimensional integrable KdV equation

The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.

Symmetry Groups and New Exact Solutions of(2+1)-Dimensional Dispersive Long-Wave Equations

Using the modified CK＇s direct method, we derive a symmetry group theorem of （2＋1）-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of （2＋1）- dimensional dispersive long-wave equations are obtained.

On Symmetry Reduction of the (1 + 3)-Dimensional Inhomogeneous Monge-Ampère Equation to the First-Order ODEs

We present the results obtained concerning the classification of symmetry reduction of the (1 + 3)-dimensional inhomogeneous Monge-Ampère equation to first-order ODEs. Some classes of the invariant solutions are constructed.

消石利尿化瘀法对高尿酸血症大鼠血清UA、NO、ET-1、TXB_2及6-keto-PGF1a的影响

目的:研究以消石利尿化瘀法创制的中药复方对高尿酸血症大鼠血清UA、NO、ET-1、TXB2及6-keto-PGF1a含量的影响及其对血管内皮功能的保护作用。方法:将雄性SD大鼠随机分为空白组、模型组、苯溴马隆组及中药复方高、中、低剂量组,共6组。除空白组外,其余各组均以腺嘌呤和酵母膏灌胃,同时各给药组灌胃给予相应的药物,空白组和模型组给予等体积蒸馏水,连续21天。末次给药后,采用紫外分光光度法检测大鼠血清UA及NO的含量,放射免疫分析法检测ET-1、TXB2及6-keto-PGF1a的含量。结果:模型组大鼠血清UA、ET-1、TXB2含量与空白组比较,显著升高(P<0.05);NO、6-keto-PGF1a含量与空白组比较,显著降低(P<0.05);中药复方中剂量组大鼠血清UA、ET-1、TXB2含量较模型组显著降低(P<0.01);中药复方中剂量组大鼠血清NO、6-keto-PGF1a含量较模型组显著升高(P<0.01)。结论:以消石利尿化瘀法创制的中药复方在降低高尿酸血症大鼠血清尿酸的同时,对实验动物血管内皮的舒缩功能有保护作用。

毒性弥漫性甲状腺肿患者IGF-1、UA及FGF-23与骨密度的关系

目的 探讨毒性弥漫性甲状腺肿(GD)患者胰岛素样生长因子(IGF-1)、尿酸(UA)及成纤维生长因子23(FGF-23)与骨密度的关系。方法 选取2017年5月至2021年5月在河南省平顶山市第一人民医院进行治疗的98例GD患者作为观察组,选取同期本院102例健康体检者设为对照组,比较两组IGF-1、UA、FGF-23水平及骨密度(BMD)值。根据T值将观察组分为骨量正常组、骨量减少组,比较两组IGF-1、UA及FGF-23水平,分析三项指标与BMD值的相关性。结果 观察组IGF-1、UA及FGF-23水平均高于对照组,BMD值低于对照组,差异均有统计学意义(P<0.05)。观察组根据T值结果分为:骨量正常组(63例)、骨量异常组(35例)。骨量异常组IGF-1、UA及FGF-23水平高于骨量正常组,差异有统计学意义(P<0.05)。Pearson直线相关结果显示,IGF-1、UA及FGF-23水平与BMD值呈负相关(P<0.05)。结论 GD患者伴有不同的IGF-1、UA、FGF-23水平紊乱,IGF-1、UA、FGF-23与GD患者BMD值呈负相关,临床需加强三项指标检测,以预防、治疗GD。

血清IGF-1、Aβl-42及UA在缺血性卒中后认知功能障碍患者中的变化

目的探讨血清胰岛素样生长因子(IGF-1)、β淀粉样蛋白(Aβl-42)、尿酸(UA)在缺血性卒中后认知功能障碍患者中的变化,为缺血性卒中后认知功能障碍的防治提供依据。方法将220例缺血性卒中患者根据认知功能评定结果分为缺血性脑卒中后认知障碍组118例与认知功能正常组102例。分别测定2组血清IGF-1、Aβl-42和UA的浓度。结果认知功能障碍组的定向力、注意和计算、记忆、概念、语言方面评分及总分均低于认知功能正常组,血清IGF-1及Aβl-42均低于认知功能正常组,UA浓度高于认知功能正常组,差异均有统计学意义(P<0.05)。结论联合检测血清IGF-1、Aβl-42及UA浓度可用于缺血性卒中后认知功能障碍的早期评估和预测。

HbA1c、25(OH)D、UA与AIS合并CMBs患者认知功能障碍的关系研究

目的研究糖化血红蛋白(HbA1c)、25-羟基维生素D[25(OH)D]、尿酸(UA)与急性缺血性脑卒中(AIS)合并脑微出血(CMBs)患者认知功能障碍的关系。方法选取2017年8月至2020年8月本院接收140例AIS患者临床资料作为研究对象,根据是否合并CMBs进行分组,将合并CMBs者设为研究组(n=76),未合并CMBs者设为对照组(n=64)。比较两组患者HbA1c、25(OH)D及UA水平,采用蒙特利尔认知评价量表(MOCA)比较两组患者认知功能情况,采用多元Logistic回归分析影响AIS合并CMBs患者认知功能障碍的危险因素。结果研究组患者HbA1c、UA水平明显高于对照组,25(OH)D明显低于对照组,差异均有统计学意义(P<0.05)。研究组患者延迟回忆、语言能力、抽象思维、定向力、注意力与计算力、视空间及执行能力评分明显低于对照组,差异有统计学意义(P<0.05)。根据研究组患者认知功能情况进行分组,认知功能良好组34例,认知功能障碍组42例,经多元Logistic回归显示:脑微出血数量(2级)、HbA1c(异常)、25(OH)D(异常)及UA(异常)为影响AIS合并CMBs患者认知功能障碍的独立危险因素(P<0.05)。结论 HbA1c、25(OH)D及UA水平与急性缺血性脑卒中合并脑微出血患者认知功能障碍密切相关,临床上检测其水平对监测患者认知功能障碍具有重要意义。

New Exact Solutions and Conservation Laws to （3＋1）-Dimensional Potential-YTSF Equation

Using the modified find some new exact solutions to Lie point symmetry groups and also get conservation laws, of the CK＇s direct method, we build the relationship between new solutions and old ones and the （3＋1）-dimensional potentiaial-YTSF equation. Baaed on the invariant group theory, Lie symmetries of the （3＋1）-dimensional potential-YTSF equation are obtained. We equation with the given Lie symmetry.

Exact Solutions to (2+1)-Dimensional Kaup-Kupershmidt Equation

In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the （2＋1）-dimensional Kaup-Kupershmidt （KK） equation are presented. We successfully obtain more general exact travelling wave solutions for （2＋ 1）-dimensional KK equation by the symmetry method and the （G1/G）-expansion method. Consequently, we find some new solutions of （2＋1）-dimensional KK equation, including similarity solutions, solitary wave solutions, and periodic solutions.

A generalized Mei conserved quantity and Mei symmetry of Birkhoff system

This paper studies a new conserved quantity which can be called generalized Mei conserved quantity and directly deduced by Mei symmetry of Birkhoff system. The conditions under which the Mei symmetry can directly lead to generalized Mei conserved quantity and the form of generalized Mei conserved quantity are given. An example is given to illustrate the application of the results.

Variable Separation for(1+1)-Dimensional Nonlinear Evolution Equations with Mixed Partial Derivatives

We present basic theory of variable separation for (1 + 1)-dimensional nonlinear evolution equations withmixed partial derivatives.As an application,we classify equations u_(xt)=A(u,u_x)u_(xxx)+B(u,u_x) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.

Residual symmetry, CRE integrability and interaction solutions of two higher-dimensional shallow water wave equations

Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.

ET-1与UA在轻度认知障碍患者中的意义

目的 探析血管内皮素-1(ET-1)与血清尿酸(UA)在轻度认知障碍(MCI)患者中的检测价值。方法 分析2013-01—2016-11信阳市中心医院院及信阳市职业技术学院附属医院诊治的74例MCI患者的临床资料。根据MCI患者的临床症状分成A组(单纯记忆障碍,无其他认知障碍,32例)和B组(记忆障碍合并其他认知功能障碍,42例)2组。另外选取同期接受健康体检的76例正常健康者作为本次研究的对照组。比较MCI组与对照组的基线资料、血清UA与ET-1水平,比较A、B2组患者的血清UA与ET-1水平。多因素Logistic回归分析MCI的危险因子。结果 MCI组与对照组的基线资料无显著性差异(P>0.05)。MCI组患者的血清UA水平明显低于对照组,ET-1水平明显高于对照组(P<0.01);A组患者血清UA明显高于B组(P<0.01),ET-1水平与B组相比差异无统计学意义(P>0.05)。年龄、血压、血糖、UA、ET-1、TC、LDL-C是MCI患者的独立危险因子。结论 高ET-1、低UA水平均是MCI患者发病的独立危险因子,检测ET-1与UA水平在MCI患者中具有重要的临床价值。

Spontaneous U（1） symmetry breaking and Bose-Einstein condensation

Based on Bogoliubov＇s truncated Hamiltonian HB for a weakly interacting Bose system, and adding a U（1） symmetry breaking term √V（λα0＋λα0^＋） to HB, we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose-Einstein condensation（BEC） occurs if and only if the U（1） symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schroedinger eigenvalue equation and using the self-consistent condition.