bethe ansatz solution of zero range process with nonuniform stationary state the eigenfunctions and eigenvalues of the master equation for zero range process with totally asymmetric dynamics on a ring are found exactly using the bethe ansatz weighted with the stationary weights of particle configurations the bethe ansatz applicability requires the rates of hopping of particles out of a site to be the q numbers n q this is a generalization of the rates of hopping of noninteracting particles equal to the occupation number n of a site of departure the noninteracting case can be restored in the limit q ensuremath rightarrow 1 the limiting cases of the model for q 0 ensuremath infty correspond to the totally asymmetric exclusion process and the drop push model respectively we analyze the partition function of the model and apply the bethe ansatz to evaluate the generating function of the total distance traveled by particles at large time in the scaling limit in case of nonzero interaction q ensuremath ne 1 the generating function has the universal scaling form specific for the kardar parizi zhang universality class

reciprocity laws and galois representations recent breakthroughs suppose f x is a monic irreducible polynomial with integer coefficients if p is a prime number then reducing the coefficients of f x modulo p gives a new polynomial fp x which may be reducible we say that f x is split modulo p if fp x is the product of distinct linear factors example 1 1 1 the polynomial f x x2 1 is split modulo 5 since f5 x x 2 x 3 mod 5 but it is not split modulo 7 since f7 x is irreducible nor is it split modulo 2 since f2 x x 1 2 mod 2 has a repeated factor the first few p for which x2 1 is split modulo p are 5 13 17 29 37 41 53

iterations of the non classical symmetries method and conditional lie backlund symmetries iterations of the non classical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the original equation invariant solutions of these heir equations supply new solutions of the original equation we show that particular cases of such invariant solutions correspond to zhdanov s conditional lie backlund symmetries

exact solution of a one parameter family of asymmetric exclusion processes we define a family of asymmetric processes for particles on a one dimensional lattice depending on a continuous parameter ensuremath lambda ensuremath in 0 1 interpolating between the completely asymmetric processes for ensuremath lambda 1 and the n 1 drop push models for ensuremath lambda 0 for arbitrary ensuremath lambda the model describes an exclusion process in which a particle pushes its right neighboring particles to the right with rates depending on the number of these particles using the bethe ansatz we obtain the exact solution of the master equation

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

a q deformed completely integrable bose gas model the authors construct the hamiltonian of a new quantum integrable q boson lattice model in 1 1 dimensions which has q bosons as dynamical variables and solve it for its energy eigenstates and energy eigenvalues under periodic boundary conditions of finite period this model can be regarded as the q deformation of the integrable lattice bose gas the quantum lattice nonlinear schrodinger quantum lattice nls model in appropriate continuum limits both of these lattice models become the quantum repulsive nls model the bose gas in the lattice case the distinction between the q bosons and ordinary bosons leads to significant new features particularly in the energy spectrum for which the q boson lattice model has an optical branch and bound state spectra as well as the one acoustic branch of the quantum lattice nls model it is argued that this sort of distinction is generic for discrete systems with a finite number of degrees of freedom as opposed to continuum field theories like the bose gas where quantization by quantum group quantization or by canonical quantization leads to the same results various properties of the q bosons are given and used in particular a q boson primakov holstein transformation is given which connects the q boson lattice model with the xxz models of arbitrary spin

stochastic higher spin six vertex model and q taseps abstract we present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the macdonald processes hierarchy first we show how macdonald q difference operators with t 0 an algebraic tool crucial for studying the corresponding macdonald processes can be utilized to get q moments of the height function h in the higher spin six vertex model first computed in 21 using bethe ansatz this result in particular implies that for the vertex model with the step bernoulli boundary condition the value of h at an arbitrary point n 1 t z 2 z 1 has the same distribution as the last component n of a random partition under a specific t 0 macdonald measure on the other hand it is known that x n n n can be identified with the location of the nth particle in a certain discrete time q tasep started from the step initial configuration the second construction we present is a coupling of this q tasep and the higher spin six vertex model with the step bernoulli boundary condition along time like paths providing an independent probabilistic explanation of the equality of h n 1 t and x n n in distribution as an illustration of our main results we obtain gue tracy widom asymptotics of a certain discrete time q tasep with the step initial configuration and special jump parameters by means of schur measures which are t q macdonald measures this analysis combines our results with the identification of averages of observables between the stochastic higher spin six vertex model and schur measures obtained recently in 8

a generalized fitzhugh nagumo equation abstract in this paper we investigate mappings of the classical fitzhugh nagumo equation to a generalized fitzhugh nagumo equation these mappings are invertible and transform the solutions of the classical fitzhugh nagumo equation into solutions of the generalized fitzhugh nagumo equation considered here these mappings are found by considering the lie point symmetries admitted by the classical fitzhugh nagumo equation and the generalized fitzhugh nagumo equation considered here a particular example of a generalized fitzhugh nagumo equation that satisfies the boundary conditions of the classical fitzhugh nagumo equation is considered numerical solutions of the generalized fitzhugh nagumo equation that do not satisfy the boundary conditions of the classical fitzhugh nagumo equation are obtained by implementing the method of lines

on the solutions of nonlinear fractional klein gordon equation with modified riemann liouville derivative abstract in this paper fractional variational iteration method fvim is applied for finding approximate analytical solutions of nonlinear fractional klein gordon equation time fractional derivative is described in the riemann liouville sense a new application of fractional variational iteration method fvim is extended to derive analytical solutions in the form of a series for this equation the behavior of the solutions and the effects of different values of fractional order are indicated graphically it is shown that the solutions obtained by the fvim are reliable convenient and effective for strongly nonlinear partial equations with modified riemann liouville derivative

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

exact results for one dimensional totally asymmetric diffusion models several types of totally asymmetric diffusion models with and without exclusion are considered for some models conditional probabilities of finding n particles on lattice sites at time t with initial occupation at time t 0 are expressed in a determinant form on the other hand the q boson totally asymmetric diffusion model is introduced which interpolates the free boson model and the model with exclusion like interaction the effects of the interaction are compared for the case of two particle diffusion

spatial correlations of the 1d kpz surface on a flat substrate we study the spatial correlations of the one dimensional kpz surface for the flat initial condition it is shown that the multi point joint distribution for the height is given by a fredholm determinant with its kernel in the scaling limit explicitly obtained this may also describe the dynamics of the largest eigenvalue in the goe dyson s brownian motion model our analysis is based on a reformulation of the determinantal green s function for the totally asep in terms of a vicious walk problem

large deviation function of the partially asymmetric exclusion process the large deviation function obtained recently by derrida and lebowitz phys rev lett 80 209 1998 for the totally asymmetric exclusion process is generalized to the partially asymmetric case in the scaling limit the asymmetry parameter rescales the scaling variable in a simple way the finite size corrections to the universal scaling function and the universal cumulant ratio are also obtained to the leading order

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

painleve test integrability and exact solutions for density dependent reaction diffusion equations with polynomial reaction functions abstract a painleve test is performed for a general density dependent reaction diffusion equation where the reaction function takes the form of an n th order polynomial in order to determine the member models of this class which are integrable first we determine the equilibrium behavior for the model then truncated laurent expansions relevant to equations having movable branch points at leading order are used to construct special solutions for the three integrable classes of reaction diffusion equations which were found an auto backlund transformation between two solutions is constructed for an equation having a pole at leading order which can be used to find further solutions some of the solutions are new and through certain simplifications we may recover old solutions as well

current distribution and random matrix ensembles for an integrable asymmetric fragmentation process we calculate the time evolution of a discrete time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size dependent rates in the continuous time limit the process turns into the totally asymmetric simple exclusion process only pieces of size 1 break off a given cluster we express the exact solution of the master equation for the process in terms of a determinant which can be derived using the bethe ansatz from this determinant we compute the distribution of the current across an arbitrary bond which after appropriate scaling is given by the distribution of the largest eigenvalue of the gaussian unitary ensemble of random matrices this result confirms universality of the scaling form of the current distribution in the kpz universality class and suggests that there is a link between integrable particle systems and random matrix ensembles

bethe ansatz solution of discrete time stochastic processes with fully parallel update we present the bethe ansatz solution for the discrete time zero range and asymmetric exclusion processes with fully parallel dynamics the model depends on two parameters p the probability of single particle hopping and q the deformation parameter which in the general case q 1 is responsible for long range interaction between particles the particular case q 0 corresponds to the nagel schreckenberg traffic model with v max 1 as a result we obtain the largest eigenvalue of the equation for the generating function of the distance travelled by particles for the case q 0 the result is obtained for arbitrary size of the lattice and number of particles in the general case we study the model in the scaling limit and obtain the universal form specific for the kardar parisi zhang universality class we describe the phase transition occurring in the limit p 1 when q 0

a new generalisation of macdonald polynomials we introduce a new family of symmetric multivariate polynomials whose coefficients are meromorphic functions of two parameters q t and polynomial in a further two parameters u v we evaluate these polynomials explicitly as a matrix product at u v 0 they reduce to macdonald polynomials while at q 0 u v s they recover a family of inhomogeneous symmetric functions originally introduced by borodin

new exact solutions to the fitzhugh nagumo equation abstract by the first integral method a series of new exact solutions of the fitzhugh nagumo equation have been obtained it is shown that this method is one of the most effective approaches to obtain the exact solutions of the nonlinear evolution equations especially for nonintegrable models

approximate potential symmetries for partial differential equations the method of approximate potential symmetries for partial differential equations with a small parameter is introduced by writing a given perturbed partial differential equation r in a conserved form an associated system s with potential variables as additional variables is obtained approximate lie point symmetries admitted by s induce approximate potential symmetries of r as applications of the theory approximate potential symmetries for a perturbed wave equation with variable wave speed and a nonlinear diffusion equation with perturbed convection terms are obtained the corresponding approximate group invariant solutions are also derived

the asymmetric exclusion process comparison of update procedures the asymmetric exclusion process asep has attracted a lot of interest not only because of its many applications e g in the context of the kinetics of biopolymerization and traffic flow theory but also because it is a paradigmatic model for nonequilibrium systems here we study the asep for different types of updates namely random sequential sequential sublattice parallel and parallel in order to compare the effects of the different update procedures on the properties of the stationary state we use large scale monte carlo simulations and analytical methods especially the so called matrix product ansatz mpa we present in detail the exact solution for the model with sublattice parallel and sequential updates using the mpa for the case of parallel update which is important for applications like traffic flow theory we determine the phase diagram the current and density profiles based on monte carlo simulations we furthermore suggest an mpa for that case and derive the corresponding matrix algebra

on the integrability of zero range chipping models with factorized steady states the conditions of the integrability of general zero range chipping models with factorized steady states which were proposed in evans et al 2004 j phys a math gen 37 l275 are examined we find a three parametric family of hopping probabilities for the models solvable by the bethe ansatz which includes most of known integrable stochastic particle models as limiting cases the solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations which is proved as a byproduct we use the bethe ansatz to solve an eigenproblem for the transition matrix of the markov process on its basis we conjecture an integral formula for the green function of the evolution operator for the model on an infinite lattice and derive the bethe equations for the spectrum of the model on a ring

random walk in beta distributed random environment we introduce an exactly solvable model of random walk in random environment that we call the beta rwre this is a random walk in mathbb z which performs nearest neighbour jumps with transition probabilities drawn according to the beta distribution we also describe a related directed polymer model which is a limit of the q hahn interacting particle system using a fredholm determinant representation for the quenched probability distribution function of the walker s position we are able to prove second order cube root scale corrections to the large deviation principle satisfied by the walker s position with convergence to the tracy widom distribution we also show that this limit theorem can be interpreted in terms of the maximum of strongly correlated random variables the positions of independent walkers in the same environment the zero temperature counterpart of the beta rwre can be studied in a parallel way we also prove a tracy widom limit theorem for this model

some conditional probabilities in the tasep with second class particles in this paper we consider the totally asymmetric simple exclusion process tasep with second class particles that consists of k first class particles and n k second class particles we assume that all first class particles are initially located to the left of the leftmost second class particle under this assumption we find the probability that the first class particles are at x x 1 x k 1 and these positions are still to the left of the leftmost second class particle at time t if we additionally assume that the initial positions of the particles are 1 n that is step initial condition then the formula of the probability does not depend on k and is very similar to a formula for the tasep without second class particles with step initial condition

matrix product formula for u _ q a_ 2 1 zero range process the zero range processes introduced recently by mangazeev maruyama and the authors are integrable discrete and continuous time markov processes associated with the stochastic r matrix derived from the well known quantum r matrix by constructing a representation of the relevant zamolodchikov faddeev algebra we present for n 2 a matrix product formula for the steady state probabilities in terms of q boson operators

factorized weight functions vs factorized scattering starting from an extensive class of factorized weight functions w p on the n dimensional torus we construct an orthonormal base of symmetric n variable polynomials for l 2 s w p dp via lexicographic ordering of the monomial symmetric functions free boson states and the gram schmidt procedure we show that the dominant asymptotics of these polynomials is factorized as a corollary we obtain a large class of quantum integrable soliton systems on the symmetric subspace of l 2 n the class of weight functions contains in particular the weight function yielding macdonald polynomials for that special case the quantum soliton system can be viewed as the dual relativistic calogero sutherland system

pairwise balance and invariant measures for generalized exclusion processes we characterize the steady state of a driven diffusive lattice gas in which each site holds several particles and the dynamics is activated and asymmetric using a quantum hamiltonian formalism we show that for arbitrary transition rates the model has product invariant measure in the steady state a pairwise balance condition is shown to hold configurations and leading respectively into and out of a given configuration are matched in pairs so that the flux of transitions from to is equal to the flux from to pairwise balance is more general than the condition of detailed balance and holds in the non equilibrium steady state of a number of stochastic models

anisotropic 2 1 d growth and gaussian limits of q whittaker processes we consider a discrete model for anisotropic 2 1 dimensional growth of an interface height function owing to a connection with q whittaker functions this system enjoys many explicit integral formulas by considering certain gaussian stochastic differential equation limits of the model we are able to prove a space time limit of covariances to those of the 2 1 dimensional additive stochastic heat equation or edwards wilkinson equation along characteristic directions in particular the bulk height function converges to the gaussian free field which evolves according to this stochastic pde

approximate conditional symmetries and approximate solutions of the perturbed fitzhugh nagumo equation we consider a perturbation of the fitzhugh nagumo equation the perturbation is proportional to the electric potential across the cell membrane the purpose of this investigation is to determine the effects of a change in electric potential across the cell membrane exact solutions of the perturbed equation are easily obtained from the well known solutions of the unperturbed fitzhugh nagumo equation the method of approximate conditional symmetries is used to obtain first order approximate solutions of the perturbed fitzhugh nagumo equation the approximate solutions are compared with the exact solutions of the perturbed equation the exact solutions of the perturbed equation do not indicate a change in the wave front connecting one constant state to another there is only a proportional increase or decrease in the constant nonzero state the approximate solutions do show a change in the shape of the wave front connecting two constant states as well as a proportional increase or decrease in the constant nonzero state

total current fluctuations in the asymmetric simple exclusion process a limit theorem for the total current in the asymmetric simple exclusion process asep with step initial condition is proven this extends the result of johansson on tasep to asep

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

the q deformed binomial distribution and its asymptotic behaviour the q analogue of the binomial distribution is defined by virtue of the q binomial theorem which takes the euler distribution as its limiting form and is new to the literature

exact solution of the master equation for the asymmetric exclusion process using the bethe ansatz we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one dimensional lattice we derive explicit expressions for the conditional probabilitiesp x1 xn t y1 yn 0 of findingn particles on lattices sitesx1 xn at timet with initial occupationy1 yn at timet 0

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

integral formulas for the asymmetric simple exclusion process in this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process asep on the integer lattice mathbb z with nearest neighbor hopping rates p to the right and q 1 p to the left for the most part we consider an n particle system but for certain of these formulas we can take the n to infty limit first we obtain for the n particle system a formula for the probability of a configuration at time t given the initial configuration for this we use bethe ansatz ideas to solve the master equation extending a result of schutz for the case n 2 the main results of the paper derived from this are integral formulas for the probability for given initial configuration that the mth left most particle is at x at time t in one of these formulas we can take the n to infty limit and it gives the probability for an infinite system where the initial configuration is bounded on one side for the special case of the totally asymmetric simple exclusion process tasep our formulas reduce to the known ones

a jacobi gauss lobatto collocation method for solving generalized fitzhugh nagumo equation with time dependent coefficients in this paper we propose a new jacobi gauss lobatto collocation method for solving the generalized fitzhugh nagumo equation the jacobi gauss lobatto points are used as collocation nodes for spatial derivatives the proposed method has the advantage of obtaining the solution in terms of the jacobi parameters a and b in addition the problem is reduced to a system of ordinary differential equations in time this system can be solved by any standard numerical techniques numerical solutions obtained by this method when compared with the exact solution reveal that the obtained solution produces high accurate results numerical results show that the proposed method is of high accuracy and is efficient for solving the generalized fitzhugh nagumo equation also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations