New classes of exact solutions of three-dimensional Navier-Stokes equations

New classes of exact solutions of the three-dimensional unsteady Navier-Stokes equations containing arbitrary functions and parameters are described. Various periodic and other solutions, which are expressed through elementary functions are obtained. The general physical interpretation and classification of solutions is given.Comment: 11 page

Non-analytical power law correction to the Einstein-Hilbert action: gravitational wave propagation

We analyze the features of the Minkowskian limit of a particular non-analytical f(R) model, whose Taylor expansion in the weak field limit does not hold, as far as gravitational waves (GWs) are concerned. We solve the corresponding Einstein equations and we find an explicit expression of the modified GWs as the sum of two terms, i.e. the standard one and a modified part. As a result, GWs in this model are not transverse, and their polarization is different from that of General Relativity. The velocity of the GW modified part depends crucially on the parameters characterizing the model, and it mostly results much smaller than the speed of light. Moreover, this investigation allows one to further test the viability of this particular f(R) gravity theory as far as interferometric observations of GWs are concerned.Comment: 18 pages, 3 figure

Exact Solvability of the two-photon Rabi Hamiltonian

Exact spectrum of the two-photon Rabi Hamiltonian is found, proceeding in full analogy with the solution of standard (one-photon) Rabi Hamiltonian, published by Braak in Phys. Rev. Lett. 107, 100401 (2011). The Hamiltonian is rewritten as a set of two differential equations. Symmetries that get hidden after further treatment are found. One can plainly see, how the Hilbert space splits into four disjunct subspaces, categorized by four values of the symmetry parameter $c=\pm1,\pm i$. There were only two values $\pm1$ for the standard Rabi model. Four analytic functions are introduced by a recurrence scheme for the coefficients of their series expansion. All their roots yield the complete spectrum of the Hamiltonian. Eigenstates in Bargmann space are also at disposal

Young's modulus of Graphene: a molecular dynamics study

The Young's modulus of graphene is investigated through the intrinsic thermal vibration in graphene which is `observed' by molecular dynamics, and the results agree quite well with the recent experiment [Science \textbf{321}, 385 (2008)]. This method is further applied to show that the Young's modulus of graphene: 1. increases with increasing size and saturation is reached after a threshold value of the size; 2. increases from 0.95 TPa to 1.1 TPa as temperature increases in the region [100, 500]K; 3. is insensitive to the isotopic disorder in the low disorder region ($< 5$), and decreases gradually after further increasing the disorder percentage.Comment: accepted by PRB, brief report, discussion on Poisson ratio adde

Network growth model with intrinsic vertex fitness

© 2013 American Physical SocietyWe study a class of network growth models with attachment rules governed by intrinsic node fitness. Both the individual node degree distribution and the degree correlation properties of the network are obtained as functions of the network growth rules. We also find analytical solutions to the inverse, design, problems of matching the growth rules to the required (e.g., power-law) node degree distribution and more generally to the required degree correlation function. We find that the design problems do not always have solutions. Among the specific conditions on the existence of solutions to the design problems is the requirement that the node degree distribution has to be broader than a certain threshold and the fact that factorizability of the correlation functions requires singular distributions of the node fitnesses. More generally, the restrictions on the input distributions and correlations that ensure solvability of the design problems are expressed in terms of the analytical properties of their generating functions

Nonlinear Schroedinger Equation in the Presence of Uniform Acceleration

We consider a recently proposed nonlinear Schroedinger equation exhibiting soliton-like solutions of the power-law form $e_q^{i(kx-wt)}$, involving the $q$-exponential function which naturally emerges within nonextensive thermostatistics [$e_q^z \equiv [1+(1-q)z]^{1/(1-q)}$, with $e_1^z=e^z$]. Since these basic solutions behave like free particles, obeying $p=\hbar k$, $E=\hbar \omega$ and $E=p^2/2m$ ($1 \le q<2$), it is relevant to investigate how they change under the effect of uniform acceleration, thus providing the first steps towards the application of the aforementioned nonlinear equation to the study of physical scenarios beyond free particle dynamics. We investigate first the behaviour of the power-law solutions under Galilean transformation and discuss the ensuing Doppler-like effects. We consider then constant acceleration, obtaining new solutions that can be equivalently regarded as describing a free particle viewed from an uniformly accelerated reference frame (with acceleration $a$) or a particle moving under a constant force $-ma$. The latter interpretation naturally leads to the evolution equation $i\hbar \frac{\partial}{\partial t}(\frac{\Phi}{\Phi_0}) = - \frac{1}{2-q}\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} [(\frac{\Phi}{\Phi_0})^{2-q}] + V(x)(\frac{\Phi}{\Phi_0})^{q}$ with $V(x)=max$. Remarkably enough, the potential $V$ couples to $\Phi^q$, instead of coupling to $\Phi$, as happens in the familiar linear case ($q=1$).Comment: 4 pages, no figure

Exact solution for a non-Markovian dissipative quantum dynamics

We provide the exact analytic solution of the stochastic Schr\"odinger equation describing an harmonic oscillator interacting with a non-Markovian and dissipative environment. This result represents an arrival point in the study of non-Markovian dynamics via stochastic differential equations. It is also one of the few exactly solvable models, for infinite dimensional systems. We compute the Green's function; in the case of a free particle, and with an exponentially correlated noise, we discuss the evolution of Gaussian wave functions.Comment: to appear in Phys. Rev. Let

Dissipative collapse models with non-white noises

We study the generalization of the QMUPL model which accounts both for memory and dissipative effects. This is the first model where both features are combined. After having derived the non-local Action describing the system, we solve the equation for a quantum harmonic oscillator via the path integral formalism. We give the explicit expression for the Green's function of the process. We focus on the case of an exponential correlation function and we analyze in detail the behavior Gaussian wave functions. We eventually study the collapse process, comparing the results with those of previous models.Comment: 17 pages, 2 figure

Two electrons on a hypersphere: a quasi-exactly solvable model

We show that the exact wave function for two electrons, interacting through a Coulomb potential but constrained to remain on the surface of a $\mathcal{D}$-sphere ($\mathcal{D} \ge 1$), is a polynomial in the interelectronic distance $u$ for a countably infinite set of values of the radius $R$. A selection of these radii, and the associated energies, are reported for ground and excited states on the singlet and triplet manifolds. We conclude that the $\mathcal{D}=3$ model bears the greatest similarity to normal physical systems.Comment: 4 pages, 0 figur