The q-PushASEP: A New Integrable Model for Traffic in 1+1 Dimension

We introduce a new interacting (stochastic) particle system q-PushASEP which interpolates between the q-TASEP introduced by Borodin and Corwin (see arXiv:1111.4408, and also arXiv:1207.5035; arXiv:1305.2972; arXiv:1212.6716) and the q-PushTASEP introduced recently by Borodin and Petrov (arXiv:1305.5501). In the q-PushASEP, particles can jump to the left or to the right, and there is a certain partially asymmetric pushing mechanism present. This particle system has a nice interpretation as a model of traffic on a one-lane highway in which cars are able to accelerate or slow down. Using the quantum many body system approach, we explicitly compute the expectations of a large family of observables for this system in terms of nested contour integrals. We also discuss relevant Fredholm determinantal formulas for the distribution of the location of each particle, and connections of the model with a certain two-sided version of Macdonald processes and with the semi-discrete stochastic heat equation.Comment: 22 pages; 4 figures. v2: minor improvements of presentation and discussions. To appear in Journal of Statistical Physic

Comparison of finite-difference schemes for analysis of shells of revolution

Several finite difference schemes are applied to the stress and free vibration analysis of homogeneous isotropic and layered orthotropic shells of revolution. The study is based on a form of the Sanders-Budiansky first-approximation linear shell theory modified such that the effects of shear deformation and rotary inertia are included. A Fourier approach is used in which all the shell stress resultants and displacements are expanded in a Fourier series in the circumferential direction, and the governing equations reduce to ordinary differential equations in the meridional direction. While primary attention is given to finite difference schemes used in conjunction with first order differential equation formulation, comparison is made with finite difference schemes used with other formulations. These finite difference discretization models are compared with respect to simplicity of application, convergence characteristics, and computational efficiency. Numerical studies are presented for the effects of variations in shell geometry and lamination parameters on the accuracy and convergence of the solutions obtained by the different finite difference schemes. On the basis of the present study it is shown that the mixed finite difference scheme based on the first order differential equation formulation and two interlacing grids for the different fundamental unknowns combines a number of advantages over other finite difference schemes previously reported in the literature

Nearest neighbor Markov dynamics on Macdonald processes

Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin (arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters (q,t), where 0<= q, t < 1. Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes. The classification unites known examples of such dynamics and also yields many new ones. When t = 0, one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a q-deformation of the PushTASEP (= long-range TASEP). When q = t, the Macdonald processes become the Schur processes of Okounkov and Reshetikhin (arXiv:math/0107056 [math.CO]). In this degeneration, we discover new Robinson--Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.Comment: 90 pages; 13 figure

Dynamics of interlacing peakons (and shockpeakons) in the Geng-Xue equation

We consider multipeakon solutions, and to some extent also multishockpeakon solutions, of a coupled two-component integrable PDE found by Geng and Xue as a generalization of Novikov's cubically nonlinear Camassa-Holm type equation. In order to make sense of such solutions, we find it necessary to assume that there are no overlaps, meaning that a peakon or shockpeakon in one component is not allowed to occupy the same position as a peakon or shockpeakon in the other component. Therefore one can distinguish many inequivalent configurations, depending on the order in which the peakons or shockpeakons in the two components appear relative to each other. Here we are in particular interested in the case of interlacing peakon solutions, where the peakons alternatingly occur in one component and in the other. Based on explicit expressions for these solutions in terms of elementary functions, we describe the general features of the dynamics, and in particular the asymptotic large-time behaviour. As far as the positions are concerned, interlacing Geng-Xue peakons display the usual scattering phenomenon where the peakons asymptotically travel with constant velocities, which are all distinct, except that the two fastest peakons will have the same velocity. However, in contrast to many other peakon equations, the amplitudes of the peakons will not in general tend to constant values; instead they grow or decay exponentially. Thus the logarithms of the amplitudes (as functions of time) will asymptotically behave like straight lines, and comparing these lines for large positive and negative times, one observes phase shifts similar to those seen for the positions of the peakons. In addition to these K+K interlacing pure peakon solutions, we also investigate 1+1 shockpeakon solutions, and collisions leading to shock formation in a 2+2 peakon-antipeakon solution.Comment: 59 pages, 6 figures. pdfLaTeX + AMS packages + hyperref + TikZ. Changes in v2: minor typos corrected, reference list updated and enhanced with hyperlink