Generalized master equations under delocalized initial conditions

The initial condition term that must be appended to the generalized master equation (GME) when the density matrix is not initially diagonal in the representation chosen is studied and explicit expressions are obtained for several cases. The term is shown to vanish for initial occupation of a Bloch state of arbitrary wave vector if the system is a crystal and the representation is that of site states, despite the violation of the initial diagonality condition. It is pointed out how one is to use the expressions for the initial term in transport calculations.This article is dedicated to Prof. Max Dresden on the occasion of his sixtieth birthday.     

Approximate Generalized Conditional Symmetries for Perturbed Evolution Equations

The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via their A GCSs is illustrated with examples.     

Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations

In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive effects and strictly following the arguments in the asymptotic derivation of the classical CH equation. The resulting equations generalize the CH equation in two different ways. The first generalization replaces the quadratic nonlinearity of the CH equation with a general power-type nonlinearity while the second one replaces the dispersive terms of the CH equation with fractional-type dispersive terms. In the absence of both higher-order nonlinearities and fractional-type dispersive effects, the generalized equations derived reduce to the classical CH equation that describes unidirectional propagation of shallow water waves. The generalized equations obtained are compared to similar equations available in the literature, and this leads to the observation that the present equations have not appeared in the literature.     

时间无卷积广义主方程中核函数的精确解和高阶微扰展开:在自旋-玻色模型和激发态能量转移中的应用

The time-convolutionless (TCL) quantum master equation provides a powerful tool to simulate reduced dynamics of a quantum system coupled to a bath. The key quantity in the TCL master equation is the so-called kernel or generator, which describes effects of the bath degrees of freedom. Since the exact TCL generators are usually hard to calculate analytically, most applications of the TCL generalized master equation have relied on approximate generators using second and fourth order perturbative expansions. By using the hierarchical equation of motion (HEOM) and extended HEOM methods, we present a new approach to calculating the exact TCL generator and its high order perturbative expansions. The new approach is applied to the spin-boson model with different sets of parameters, to investigate the convergence of the high order expansions of the TCL generator. We also discuss circumstances where the exact TCL generator becomes singular for the spin-boson model, and a model of excitation energy transfer in the Fenna-Matthews-Olson complex.     

New Periodic Wave Solutions to Generalized Klein-Gordon and Benjamin Equations

By making use of extended mapping method and auxiliary equation for finding new periodic wave solu tions of nonlinear evolution equations in mathematical physics, we obtain some new periodic wave solutions for generalized Klein-Cordon equation and Benjamin equation, which cannot be found in previous work. This method also can be used to find new periodic wave solutions of other nonlinear evolution equations.     

广义量子主方程：教程综述和最新进展

广义量子主方程(GQME)为模拟嵌入在量子环境中的开放量子体系的约化动力学提供了一种通用且严格的计算方法. 开放量子体系的动力学在能量、电荷以及量子相干转移过程和光化学反应中至关重要. 量子系统通常被定义为我们感兴趣的自由度,例如捕光分子的电子态或凝聚态体系中的特定振动模式. 系统周围的环境也被称为热浴,必须考虑它对系统的影响. 例如,广义量子主方程理论中用投影算符方法对其进行描述. 本综述总结了广义量子主方程的两种标准形式,即时间卷积形式的Nakajima-Zwanzig GQME和无卷积形式的广义量子主方程. 在更流行的NZ-GQME形式中,记忆核刻画了非马尔可夫和非微扰效应,给出了约化密度矩阵的精确量子动力学. 总结了几种通过含有分子信息但无投影算符的时间关联函数作为输入信息,进而求解含投影算符的记忆核的方法. 特别值得一提的是近期提出的NZ-GQME改进版方法,该方法是基于将哈密顿量划分为更通用的对角和非对角部分. 上述系统相关的热浴时间关联函数可以通过数值精确或近似量子动力学方法计算. 本文将有助于理解广义量子主方程的理论背景,并且展望通过GQME与量子计算技术的结合解决使用当今最先进的经典超级计算机无法解决的与量子动力学和量子信息相关的复杂问题.     

Conservation Laws of The Generalized Riemann Equations

Two special classes of conserved densities involving arbitrary smooth functions are explicitly formulated for the generalized Riemann equation at arbitrary N. The particular case when N = 2 covers most of the known conserved densities of the equation, and the result is also extended to the famous Gurevich-Zybin, Monge-Ampere and two-component Hunter-Saxton equations by considering certain reductions.     

Stochastic analysis of a Hopf bifurcation: Master equation approach

The effect of inhomogeneous fluctuations in a reaction-diffusion system exhibiting a Hopf bifurcation is analyzed using the master equation approach. A Taylor expansion of the logarithm of the stationary probability, known as the stochastic potential, is calculated. This procedure displays marked analogies with the theory of normal forms. The critical potential, reduced to its local expansion around an arbitrary point of the limit cycle, brings out the essential role played by the phase of the oscillating variables. A comparison with the Langevin analysis of Walgraefet al. [J. Chem. Phys. 78(6):3043 (1983)] is performed.     

A Modified Homogeneous Balance Method and Its Applications

A modified homogeneous balance method is proposed by improving some key steps in the homogeneous balance method. Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneous balance method. Generalized Boussinesq equation, KP equation, and mKdV equation are chosen as examples to llustrate our method. This approach is also applicable to a large variety of nonlinear evolution equations.     

Generalized master equations for continuous-time random walks

An equivalence is established between generalized master equations and continuous-time random walks by means of an explicit relationship between(t), which is the pausing time distribution in the theory of continuous-time random walks, and(t), which represents the memory in the kernel of a generalized master equation. The result of Bedeaux, Lakatos-Lindenburg, and Shuler concerning the equivalence of the Markovian master equation and a continuous-time random walk with an exponential distribution for(t) is recovered immediately. Some explicit examples of(t) and(t) are also presented, including one which leads to the equation of telegraphy.This study was partially supported by ARPA and monitored by ONR Contract No. (N00014-17-C-0308).For continuity, the reader is directed to the article entitled Random Walks on Lattices. IV. Continuous Time Walks and Influence of Absorbing Boundaries, by E. W. Montroll and H. Scher, which will appear in Volume 9, Number 2, of this journal, and which should precede the following article. Regrettably, the two articles were inadvertently switched during processing.     

On Ground-Traveling Waves for the Generalized Kadomtsev-Petviashvili Equations

As a continuation of our previous work, we improve some results on convergence of periodic KP traveling waves to solitary ones as the period goes to infinity. In addition, we present some qualitative properties of such waves, as well as nonexistence results, in the case of general nonlinearities. We suggest an approach which does not use any scaling argument. This revised version was published online in July 2006 with corrections to the Cover Date.     

Pseudo-differential Operators and Generalized Lax Equations in Symbolic Computation

In this paper, the pseudo-differential operators and the generalized Lax equations in integrable systems are implemented in symbolic software Mathematica. A great deal of differential polynomials which appear in the procedure are dealt with by differential characteristic chain method. Using the program, several classical examples are given.     

Two-Soliton Solutions and Interactions for the Generalized Complex Coupled Kortweg-de Vries Equations

Kortweg-de Vries (KdV)-typed equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Generalized complex coupled KdV(GCCKdV) equations are investigated in this paper. Through the dependent variable transformations and symbolic computation, GCCKdV equations are transformed into their bilinear forms, based on which the one- and two-soliton solutions are obtained. Through the interactions of two solitons, the regular elastic collision are shown. When the wave numbers are complex, three kinds of solitonic collisions are presented: (i) two solitons merge and separate fromeach other periodically; (ii) two solitons exhibit the attraction and repulsion nearly twice, and finally separate from each other after such type of interaction; (iii) two solitons are fluctuant in the central region of the collision. Propagation features ofsolitons are investigated with the effects of the coefficients in the GCCKdV equations considered. Velocity of soliton increase with the α increasing. Amplitude of v increase with the α increasing and decrease with the β increasing.     

Approximate Derivative-Dependent Functional Variable Separation for the Generalized Diffusion Equations with Perturbation

As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized diffusion equations with perturbation. Complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is obtained. As a result, the corresponding approximate derivative-dependent functional separable solutions to some resulting perturbed equations are derived by way of examples.     

Extended Homogeneous Balance Method and Lax Pairs, Backlund Transformation

Using the extended homogeneous balance method, which is very concise and primary, Lax pairs and Backlund transformation for most nonlinear evolution equations, such as the compound KdV-Burgers equation and nonlinear diffusion equation are obtained.     

Approximate Derivative-Dependent Functional Variable Separation for the Generalized Diffusion Equations with Perturbation

As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized diffusion equations with perturbation. Complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is obtained. As a result, the corresponding approximate derivative-dependent functional separable solutions to some resulting perturbed equations are derived by way of examples.     

Backlund Transformations for the Initial Problems of Nizhnich and Nizhnich-Novikov-Veselov Equations

The homogeneous balance method is a method for solving general partial differential equations (PDEs). Inthis paper we solve a kind of initial problems of the PDEs by using the special Backlund transformations of the initialproblem. The basic Fourier transformation method and some variable-separation skill are used as auxiliaries. Two initialproblems of Nizhnich and the Nizhnich-Novikov-Veselov equations are solved by using this approach.     

Relationship Among Solutions of a Generalized Riccati Equation

In this paper, some solutions of a generalized Riccati equation are investigated, which are given in the recent articles [Chaos, Solitons ＆ Fractals 24 （2005） 257; Phys. Lett. A 336 （2005） 463], and the relationship among the solutions is revealed.