auxiliary matrices for the six vertex model and the algebraic bethe ansatz we connect two alternative concepts of solving integrable models baxter s method of auxiliary matrices or q operators and the algebraic bethe ansatz the main steps of the calculation are performed in a general setting and a formula for the bethe eigenvalues of the q operator is derived a proof is given for states which contain up to three bethe roots further evidence is provided by relating the findings to the six vertex fusion hierarchy for the xxz spin chain we analyse the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity

stochastic turing patterns in the brusselator model a stochastic version of the brusselator model is proposed and studied via the system size expansion the mean field equations are derived and shown to yield to organized turing patterns within a specific parameters region when determining the turing condition for instability we pay particular attention to the role of cross diffusive terms often neglected in the heuristic derivation of reaction diffusion schemes stochastic fluctuations are shown to give rise to spatially ordered solutions sharing the same quantitative characteristic of the mean field based turing scenario in term of excited wavelengths interestingly the region of parameter yielding to the stochastic self organization is wider than that determined via the conventional turing approach suggesting that the condition for spatial order to appear can be less stringent than customarily believed

a q operator identity for the correlation functions of the infinite xxz spin chain an explicit construction for q operators of the finite xxz spin chain with twisted boundary conditions is presented the massless and the massive regimes are considered as well as the root of unity case it is explained how these results yield an alternative expression for the trace function employed in the description of the correlation functions of the inhomogeneous xxz model on the infinite lattice by boos jimbo miwa smirnov and takeyama

auxiliary matrices on both sides of the equator the spectra of previously constructed auxiliary matrices for the six vertex model at roots of unity are investigated for spin chains of even and odd length the two cases show remarkable differences in particular it is shown that for even roots of unity and an odd number of sites the eigenvalues contain two linear independent solutions to baxter s tq equation corresponding to the bethe ansatz equations above and below the equator in contrast one finds for even spin chains only one linear independent solution and complete strings the other main result is the proof of a previous conjecture on the degeneracies of the six vertex model at roots of unity the proof rests on the derivation of a functional equation for the auxiliary matrices which is closely related to a functional equation for the eight vertex model conjectured by fabricius and mccoy

patterns in reaction diffusion systems generated by global alternation of dynamics abstract we recently proposed a mechanism for inducing a turing instability by alternation of reaction diffusion dynamics each of which is pattern free herein we shed light on the question of which particular switching schemes produce that instability and which do not we also consider a particular reaction diffusion model to illustrate the mechanism

dynamic parrondo s paradox abstract this paper is focused on studying parrondo s paradox in non linear dynamics specifically how the periodic combination of the individual maps f and g can give rise to chaos or order we construct dynamical systems exhibiting the paradox for several notions of chaos derived from topological dynamics the effect of altering the order of the combination i e considering f g or g f is analyzed as well as the robustness of the parrondo effect under small perturbations conditions for avoiding parrondian dynamics are also obtained placing special emphasis on the notion of chaos given by turbulence

regular patterns in dichotomically driven activator inhibitor dynamics we investigate turing pattern formation in the presence of additive dichotomous fluctuations in the context of an extended system with diffusive coupling and fitzhugh nagumo kinetics the fluctuations vary in space and or time depending on the realization of the dichotomous switching the system is at a given time for spatial disorder at a given position in one of two possible excitable dynamical regimes each of the two excitable dynamics for itself does not support pattern formation with proper dichotomous fluctuations however the homogeneous steady state is destabilized via a turing instability we investigate the influence of different switching rates different correlation length of the spatial disorder on pattern formation we find three distinct mechanisms for slow switching existing boundaries become unstable for high rates the system exhibits effective bistability which allows for a turing instability for medium rates the fluctuations create spatial structures via a new mechanism where the influence of the fluctuations is twofold first they produce local inhomogeneities which then grow again caused by fluctuations until the whole space is covered utilizing a nonlinear map approach we show bistability of a period one and a period two orbit being associated with the steady homogeneous and the turing pattern state respectively finally for purely static dichotomous disorder we find destabilization of homogeneous steady states for finite nonzero correlation length of the disorder resulting again in turing patterns

auxiliary matrices for the six vertex model at qn 1 ii bethe roots complete strings and the drinfeld polynomial the spectra of recently constructed auxiliary matrices for the six vertex model respectively the spin s 1 2 heisenberg chain at roots of unity are investigated two conjectures are formulated both of which are proved for n 3 and are verified numerically for several examples with n 3 the first conjecture identifies an abelian subset of auxiliary matrices whose eigenvalues are polynomials in the spectral variable the zeros of these polynomials are shown to fall into two sets one consists of the solutions to the bethe ansatz equations which determine the eigenvalues of the six vertex transfer matrix the other set of zeros contains the complete strings which encode the information on the degeneracies of the model due to the loop symmetry present at roots of 1 the second conjecture then states a polynomial identity which relates the complete string centres to the bethe roots allowing one to determine the dimension of the degenerate eigenspaces its proof for n 3 involves the derivation of a new functional equation for the auxiliary matrices and the six vertex transfer matrix moreover it is demonstrated in several explicit examples that the complete strings coincide with the classical analogue of the drinfeld polynomial the latter is used to classify the finite dimensional irreducible representations of the loop algebra this suggests that the constructed auxiliary matrices not only enable one to solve the six vertex model but also completely characterize the decomposition of its eigenspaces into irreducible representations of the underlying loop symmetry

numerical methods for the simulation of trapped nonlinear schrodinger systems we propose analyze and compare the efficiency and accuracy of different numerical schemes for the solution of the nonlinear schrodinger equation with a trapping potential in particular we study schemes of finite difference pseudospectral and spectral types for the space discretization together with explicit symplectic multistep split step and standard variable step integrators to solve the time evolution all of these schemes are evaluated comparatively and some recommendations based on their accuracy and computational efficiency are made

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

discrete time ratchets the fokker planck equation and parrondo s paradox parrondo s games manifest the apparent paradox where losing strategies can be combined to win and have generated significant multidisciplinary interest in the literature here we review two recent approaches based on the fokker planck equation that rigorously establish the connection between parrondo s games and a physical model known as the flashing brownian ratchet this gives rise to a new set of parrondo s games of which the original games are a special case for the first time we perform a complete analysis of the new games via a discrete time markov chain analysis producing winning rate equations and an exploration of the parameter space where the paradoxical behaviour occurs

quantum kz equation with q 1 and correlation functions of the xxz model in the gapless regime an integral solution to the quantum knizhnik zamolodchikov qkz equation with q 1 is presented upon specialization it leads to a conjectural formula for correlation functions of the xxz model in the gapless regime the validity of this conjecture is verified in special cases including the nearest neighbour correlator with an arbitrary coupling constant and general correlators in the xxx and xy limits

third neighbour and other four point correlation functions of spin 1 2 xxz chain the correlation functions of the spin 1 2 xxz chain in the ground state were expressed in the form of multiple integrals for 1 1 and 1 in particular adjacent four point correlation functions were given as certain four dimensional integrals we show that these integrals can be reduced to polynomials with respect to specific one dimensional integrals the results give the polynomial representation of the third neighbour correlation functions

symbolic calculation in development of algorithms split step methods for the gross pitaevskii equation we employ symbolic calculation to perform a systematic study of the accuracy of split step fourier transform methods for the time dependent gross pitaevskii equation gpe provided the most recent approximation for the wave function is always used in the nonlinear atom atom interaction energy every split step algorithm we have tried has the same order time stepping error for the nonlinear gpe and for the linear schroedinger equation

high order split step exponential methods for solving coupled nonlinear schrodinger equations numerical integration schemes for coupled time dependent nonlinear schrodinger equations are examined using exponential splitting step methods exponentiation of the nonlinear potential term is reduced to the exponential of a kinetic energy term which can be calculated by fast fourier transforms high order iteration schemes involving a minimum number of product operators are shown to yield highly accurate amplitude and phase these new splitting methods are shown to be highly efficient both with respect to accuracy and integration time

integral representations for correlation functions of the xxz chain at finite temperature we derive a novel multiple integral representation for a generating function of the z z correlation functions of the spin xxz chain at finite temperature and finite longitudinal magnetic field our work combines algebraic bethe ansatz techniques for the calculation of matrix elements with the quantum transfer matrix approach to thermodynamics

new developments in the eight vertex model we demonstrate that the q matrix introduced in baxter s 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for q v which both contains the bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues

switching induced turing instability we propose a mechanism for inducing a turing instability in systems whose only stable state is pattern free and homogeneous global alternation between two dynamics each of which has the same homogeneous stable state may induce a turing instability that leads to pattern formation we determine what kind of alternation can drive the system to a turing instability and show that the appearance of the induced spatiotemporal structure depends on the ratio of two characteristic times one determined by the external forcing and the other by the instability that drives the system at short times the mechanism is illustrated by means of theoretical calculations and numerical simulations on two well known biological models that are relevant in morphogenesis

higher order splitting algorithms for solving the nonlinear schrodinger equation and their instabilities since the kinetic and potential energy terms of the real time nonlinear schroedinger equation can each be solved exactly the entire equation can be solved to any order via splitting algorithms we verified the fourth order convergence of some well known algorithms by solving the gross pitaevskii equation numerically all such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved a detail error analysis reveals that the noise or elementary excitations of the nonlinear schroedinger equation obeys the bogoliubov spectrum and the instability is due to the exponential growth of high wave number noises caused by the splitting process for a continuum wave function this instability is unavoidable no matter how small the time step for a discrete wave function the instability can be avoided only for delta tk sub max sup 2 or approx 2 pi where k sub max pi delta x

any order propagation of the nonlinear schrodinger equation we derive an exact propagation scheme for nonlinear schrodinger equations this scheme is entirely analogous to the propagation of linear schrodinger equations we accomplish this by defining a special operator whose algebraic properties ensure the correct propagation as applications we provide a simple proof of a recent conjecture regarding higher order integrators for the gross pitaevskii equation extend it to multicomponent equations and to a new class of integrators