Traveling-Wave Solutions for Korteweg–de Vries–Burgers Equations through Factorizations

Traveling-wave solutions of the standard and compound form of Korteweg–de Vries–Burgers equations are found using factorizations of the corresponding reduced ordinary differential equations. The procedure leads to solutions of Bernoulli equations of non-linearity 3/2 and 2 (Riccati), respectively. Introducing the initial conditions through an imaginary phase in the traveling coordinate, we obtain all the solutions previously reported, some of them being corrected here, and showing, at the same time, the presence of interesting details of these solitary waves that have been overlooked before this investigation.     

Effects of gradual spatial variation in resources on population extinction: Analytic calculations for abrupt transitions

We study bifurcations in a spatially extended nonlinear system representing population dynamics with the help of analytic calculations. The result we obtain helps in the understanding of the onset of abrupt transitions leading to the extinction of biological populations. The result is expressed in terms of Airy functions and sheds light on the behavior of bacteria in a Petri dish as well as of large animals such as rodents moving over a landscape.     

Memory effects in Fisher equation with nonlinear convection term

Memory effect in diffusion–reaction equation plays important role in physical, biological and chemical sciences. An exact solutions of Fisher equation in the presence of nonlinear convection term with finite memory transport is obtained. Solutions of corresponding diffusion–reaction equation without memory effect is also obtained and a comparison is made between obtained solutions. In particular, the solitary wave solutions are found.     

Anomalous biennial oscillations in a Fisher equation with a discretized verhulst term

The dynamics of a biological population governed by a modified Fisher equation is studied by means of Monte Carlo simulations. Reproduction of the population occurs at discrete times, while transport caused by diffusion and conduction takes place on shorter time scales. The discrete reproduction, modeled with a set of coupled logistic maps, exhibits phenomena which are not evident in the usual continuum version of the Fisher equation. Several mechanisms for biennial oscillations of the total population are investigated. One of these shows an ordered coupling between random diffusive motion and the chaotic attractor of the logistic map.     

A family of evolution equations with nonlinear diffusion,Verhulst growth,and global regulation: Exact time-dependent solutions

A family of evolution equations describing a power-law nonlinear diffusion process coupled with a local Verhulst-like growth dynamics, and incorporating a global regulation mechanism, is considered. These equations admit an interpretation in terms of population dynamics, and are related to the so-called conserved Fisher equation. Exact time-dependent solutions exhibiting a maximum nonextensive q$q$-entropy shape are obtained.     

Fisher方程的有界衰减振荡解

为了研究非线性发展方程的有界衰减振荡解,特选取Fisher方程为例. Fisher方程在描述激发介质的非数值模型(如Belousov-Zhabotinsky (BZ)反应)中, 其解的振幅取负值是有意义的.应用平面动力系统理论,研究了Fisher方程有界行波解存在的条件, 利用LS解法和线性化解法给出了其有界衰减振荡解的近似解析表达式,并进行了误差估计.     

Quantum Fisher Information and Uncertainty Relations

It is well known that the Cramér–Rao inequality places a lower bound for quantum Fisher information in terms of the variance of any quantum measurement. We establish an upper bound for quantum Fisher information of a parameterized family of density operators in terms of the variance of the generator. These two bounds together yield a generalization of the Heisenberg uncertainty relations from statistical estimation perspective.     

LS解法和Fisher方程行波系统的定性分析

提出了求解非线性发展方程的新方法——LS解法.LS解法是基于(G’/G)展开法和扩展的双曲正切函数展开法.并引入了Poincar定性理论的思想,然后以Fisher方程为例进行了试验.通过定性分析首先获得了Fisher方程行波系统积分曲线的性质,然后解得了Fisher方程作为耗散系统时单调减少的波前解和作为扩张系统时单调递增的波前解.一些试验结果与Ablowitz所得结果一致.也得到了Fisher方程作为扩张系统时的新结果.LS解法是在定性理论指导下,在已获知解曲线性质的情况下进行精确求解的,求解目标明确.LS解法揭示了线性系统也可以用作辅助方程来求解非线性系统.     

Weighted Fisher informations, their derivation and use

Fisher information has been used to derive many laws of physics, including its differential equations. In many of these Fisher-based derivations the information in the time measurement is required to be negative. Yet by its expression in the standard derivation of the Cramer-Rao (CR) inequality, a negative Fisher seems impossible. To the contrary, we show that in fact the standard CR derivation allows the Fisher to be meaningfully regarded as either negative or positive. The mathematics allow it, and the choice to be made depends upon the physical parameter whose information level is sought. For example, the rules of special relativity require an imaginary time parameter ict, , with c the speed of light and t the time. The squared time parameter is then negative, requiring (as shown) the Fisher information to be negative as well. Further generalizations of the CR derivation are also made, which allow the estimation of general parameters using arbitrarily weighted mean-squared error criteria. The weights are found to define a family of correspondingly weighted Fisher informations. Finally, a condition is found for achieving efficient estimation in the presence of any weight function.     

Transition between extinction and blow-up in a generalized Fisher–KPP model

Stationary solutions of the Fisher–KPP equation with general nonlinear diffusion and arbitrary reactional kinetic orders terms are characterized. Such stationary (separatrix-like) solutions disjoint the blow-up solutions from those showing extinction. In addition a criterion for general parameter values is presented, which allows determining the blow-up or vanishing character of the solutions.     

Information,Quantum Mechanics,and Gravity

This is a basically expository article, with some new observations, tracing connections of the quantum potential to Fisher information, to Kähler geometry of the projective Hilbert space of a quantum system, and to the Weyl-Ricci scalar curvature of a Riemannian flat spacetime with quantum matter.Á Denise     

Investigations of information quantifiers for the Tavis–Cummings model

In this article, a system of two two-level atoms interacting with a single-mode quantized electromagnetic field in a lossless resonant cavity via a multi-photon transition is considered. The quantum Fisher information, negativity, classical Fisher information, and reduced von Neumann entropy for the two atoms are investigated. We found that the number of photon transitions plays an important role in the dynamics of different information quantifiers in the cases of two symmetric and two asymmetric atoms. Our results show that there is a close relationship between the different quantifiers. Also, the quantum and classical Fisher information can be useful for studying the properties of quantum states which are important in quantum optics and information.     

Logarithm Versus Square Root： Comparing Quantum Fisher Information

In classical statistics, the Fisher information is unique in the sense that it is essentially the only monotone Riemannian metric on the space of probability densities. In quantum theory, this uniqueness breaks down, and there are many natural quantum analogues of the Fisher information, among which two particular versions distinguish themselves by their intuitive and informational significance： The first has its origin in the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurement, and is defined via the square root of the density operator. The second arises from Helstrom＇s study of quantum detection in 1967, and is defined via the symmetric logarithmic derivative. The aim of this paper Js to compare these two versions of quantum Fisher information, and to establish two informational inequalities relating them.