完整系统三阶Lagrange方程的一种推导与讨论

从牛顿运动方程出发，推导了完整系统关于广义加速度的Lagrange方程.讨论了该方程与传统分析力学中的Lagrange方程的相容性问题.结果显示，三阶Lagrange方程可以通过对Lagrange方程求一阶时间导数得到，表明它们是相容的.因此三阶Lagrange方程提供了一种不同于传统Lagrange方程方法的求解物体运动方程的途径.     

Nonlocal representation of the sl(2, R) algebra for the Chazy equation

A demonstration of how the point symmetries of the Chazy equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy equation under a generalized transformation, and find the point symmetries of the Chazy equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy equation which is given by a right Painlevé series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a left Painlevé series and one given by a right Painlevé series where the leading-order behaviors and the resonances are explicitly those of the Chazy equation.     

完整系统三阶Lagrange方程的一种推导与讨论

从牛顿运动方程出发,推导了完整系统关于广义加速度的Lagrange方程.讨论了该方程与传统分析力学中的Lagrange方程的相容性问题.结果显示,三阶Lagrange方程可以通过对Lagrange方程求一阶时间导数得到,表明它们是相容的.因此三阶Lagrange方程提供了一种不同于传统Lagrange方程方法的求解物体运动方程的途径.     

Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation

We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation and the four-dimensional Martínez Alonso–Shabat equation. Finally, we construct extensions of the integrable hierarchies associated to the hyper-CR equation for Einstein–Weyl structures, the reduced quasi-classical self-dual Yang–Mills equation, the four-dimensional universal hierarchy equation, and the four-dimensional Martínez Alonso–Shabat equation.     

THE NUMERICAL SOLUTION OF FIRST KIND INTEGRAL EQUATION FOR THE HELMHOLTZ EQUATION ON SMOOTH OPEN ARCS

1. IntroductiouThe mathewtical tratod of the scattering Of theharmonic acoustic or electromagnoticwaves by an Mtely lOng sethecylindrical obstacle with a 8mooth opeu coDtour crewSeCtboF C Rs Ieads to unbounded boundare wtue problems for the Helmhltz equabo I3lwith wave nUmer h > 0.In the singtelayer Woach one Seeks the solutbo in the formwhere d8. is the element of arc length, and the fundamental solUbo to the Helmholtz equatfonis giveu byin terms Of the Hds fUnction H6') of order zero…     

一类非线性波动方程的显式精确解

本文用直接方法和假设的一种结合求出了一类较广泛的非线性波动方程ｕｔｔ－ａ１ｕｘｘ＋ａ２ｕｔ＋ａ３ｕ＋ａ４ｕＳ＾２＋ａ５ｕ＾３＝０的一些显式精确行波解,贱个有重要的非线性数学物理方程,如φ＾４方程,Ｋｌｅｉｎ－Ｇｏｒｄｏｎ方程,Ｓｉｎｅ－Ｇｏｒｄｏｎ方程,及Ｓｉｎｈ－Ｇｏｒｄｏｎ方程的近似,Ｌａｎｄａｕ－Ｇｉｎｚｂｕｒｇ－Ｈｉｇｇｓ方程,Ｄｕｆｆｉｎｇ方程,非线性电报方程等都可作为该方程的特殊情形得     

Exact master equations describing reduced dynamics of the Wigner function

Master equations of different types describe the evolution (reduced dynamics) of a subsystem of a larger system generated by the dynamic of the latter system. Since, in some cases, the (exact) master equations are relatively complicated, there exist numerous approximations for such equations, which are also called master equations. In the paper, we develop an exact master equation describing the reduced dynamics of the Wigner function for quantum systems obtained by a quantization of a Hamiltonian system with a quadratic Hamilton function. First, we consider an exact master equation for first integrals of ordinary differential equations in infinite-dimensional locally convex spaces. After this, we apply the results obtained to develop an exact master equation corresponding to a Liouville-type equation (which is the equation for first integrals of the (system of) Hamilton equation(s)); the latter master equation is called the master Liouville equation; it is a linear first-order differential equation with respect to a function of real variables taking values in a space of functions on the phase space. If the Hamilton equation generating the Liouville equation is linear, then the vector fields that define the first-order linear differential operators in the master Liouville equations are also linear, which in turn implies that for a Gaussian reference state the Fourier transform of a solution of the master Liouville equation also satisfies a linear differential equation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 203–219, 2005.     

非稳定非线性薛定谔方程的马尔钦科方程

该文基于对非稳定非线性薛定愕方程作反散射变换得到的Ｚａｋｈａｒｏｖ－Ｓｈａｂａｔ方程,直接对积分核作变换,导出马尔钦科方程．得到的马尔钦科方程在形式上与一般非线性薛定谔方程得到的一样简单明了,且不存在逆变换的自洽困难．     

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.     

Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions

In this paper, we propose complete radiation boundary conditions (CRBCs) for solutions of the convected Helmholtz equation with a uniform mean flow in a waveguide. We first study CRBCs for the Helmholtz equation in a waveguide. Noting that the convected Helmholtz equation is associated with the Helmholtz equation via the Prandtl–Glauert transformation, CRBCs for the convected Helmholtz equation is derived from CRBCs for the Helmholtz equation. We analyse well-posedness and convergence of approximate solutions satisfying CRBCs for the convected Helmholtz equation. In addition, simple numerical experiments will be presented to confirm the theoretical results.     

Symmetry reductions and soliton-like solutions for the variable coefficient MKdV equations

Two types of symmetry reductions are derived for the variable coefficient MKdV equation, which contain well-known Painleve II type equation and Jacobian elliptic equation. In addition, soliton-like solutions of the variable coefficient MKdV equation are also obtained. Finally, a transformation between the variable coefficient MKdV equation and the MKdV equation are also found.     

Explicit Peakon solutions to a family of wave-breaking equations

The singular traveling wave solutions of a general 4-parameter family equation which unifies the Camass-Holm equation, the Degasperis-Procesi equation and the Novikov equation are investigated in this paper. At first, we obtain the explicit peakon solutions for one of its specific case that $a=(p+2)c$, $b=(p+1)c$ and $c=1$, which is referred to a generalized Camassa-Holm-Novikov (CHN) equation, by reducing it to a second-order ordinary differential equation (ODE) and solving its associated first-order integrable ODE. By observing the characteristics of peakon solutions to the CHN equation, we construct the peakon solutions for the general 4-parameter breaking wave equation. It reveals that singularities of the peakon solutions come up only when the solutions attain singular points of the equation, which might be a universal principal for all singular traveling wave solutions for wave breaking equations.     

A generalized Fitzhugh–Nagumo equation

In this paper we investigate mappings of the classical Fitzhugh–Nagumo equation to a generalized Fitzhugh–Nagumo equation. These mappings are invertible and transform the solutions of the classical Fitzhugh–Nagumo equation into solutions of the generalized Fitzhugh–Nagumo equation considered here. These mappings are found by considering the Lie point symmetries admitted by the classical Fitzhugh–Nagumo equation and the generalized Fitzhugh–Nagumo equation considered here. A particular example of a generalized Fitzhugh–Nagumo equation that satisfies the boundary conditions of the classical Fitzhugh–Nagumo equation is considered. Numerical solutions of the generalized Fitzhugh–Nagumo equation that do not satisfy the boundary conditions of the classical Fitzhugh–Nagumo equation are obtained by implementing the Method of Lines.     

Rangwala-Rao方程和几个非线性导数Schrdinger型方程的一类显式精确孤波解

首先求出了Lienard方程的显式精确解,进而求出了Rangwala-Rao方程,Ablowitz方程,Chen-Lee-Lin方程,以及Gerdjikov-Ivanov方程的型如的显式精确孤波解。     

Exact and traveling-wave solutions for convection-diffusion-reaction equation with power-law nonlinearity

Based on the simplest equation method, we propose exact and traveling-wave solutions for a nonlinear convection-diffusion-reaction equation with power law nonlinearity. Such equation can be considered as a generalization of the Fisher equation and other well-known convection-diffusion-reaction equations. Two important cases are considered. The case of density-independent diffusion and the case of density-dependent diffusion. When the parameters of the equation are constant, the Bernoulli equation is used as the simplest equation. This leads to new traveling-wave solutions. Moreover, some wavefront solutions can be derived from the traveling-wave ones. The case of time-dependent velocity in the convection term is studied also. We derive exact solutions of the equations by using the Riccati equation as simplest equation. The exact and traveling-wave solutions presented in this paper can be used to explain many biological and physical phenomena.     

一维半导体流体动力学模型的局部有界解

本文讨论了能量方程是压力一密度关系的一维半导体流体动力学模型方程,通过把欧拉－泊松方程变成拟线性波动方程,利用拟线性波动方程的局部解存在性,得到一维半导体流体动力学模型的局部解,并且解是有界的。     

Reduction method for studying localized solutions of neural field equations on the Poincaré disk

We present a reduction method to study localized solutions of an integrodifferential equation defined on the Poincaré disk. This equation arises in a problem of texture perception modeling in the visual cortex. We first derive a partial differential equation which is equivalent to the initial integrodifferential equation and then deduce that localized solutions which are radially symmetric satisfy a fourth order ordinary differential equation.     

New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations

In this paper we consider a special fifth-order KdV equation with constant coefficients and we obtain traveling wave solutions for it, using the projective Riccati equation method. By mean of a scaling, exact solutions to general Kaup-Kupershmidt (KK) equation are obtained. As a particular case, exact solutions to standard KK equation can be derived. Using the same method, we obtain exact solutions to standard Ito equation. By mean of scaling, new exact solutions to general Ito equation are formally derived.     

Exact solutions of a two-dimensional nonlinear Schrödinger equation

In the present study, we converted the resulting nonlinear equation for the evolution of weakly nonlinear hydrodynamic disturbances on a static cosmological background with self-focusing in a two-dimensional nonlinear Schrödinger (NLS) equation. Applying the function transformation method, the NLS equation was transformed to an ordinary differential equation, which depended only on one function ξ and can be solved. The general solution of the latter equation in ζ leads to a general solution of NLS equation. A new set of exact solutions for the two-dimensional NLS equation is obtained.     

Symmetry reduction,exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.