finite size scaling in the driven lattice gas we present a monte carlo study of the high temperature phase of the two dimensional driven lattice gas at infinite driving field we define a finite volume correlation length verify that this definition has a good infinite volume limit independent of the lattice geometry and study its finite size scaling behavior the results for the correlation length are in good agreement with the predictions based on the field theory proposed by janssen schmittmann leung and cardy the theoretical predictions for the susceptibility and the magnetization are also well verified we show that the transverse binder parameter vanishes at the critical point in all dimensions d geqslant 2 and discuss how such result should be expected in the theory of janssen et al in spite of the existence of a dangerously irrelevant operator our results confirm the gaussian nature of the transverse excitations

long range spatial correlations for anisotropic zero range processes the spatial correlations are investigated for a homogeneous system of indistinguishable particles undergoing stochastic anisotropic hopping dynamics on the d dimensional lattice d or 2 the interaction is zero range i e the rate at which particles leave a given site only depends on the occupation number at that site a series expansion around the independent particle system is given for the equal time correlations and is shown to converge for small times t the formal t to infinity limiting expansion is analysed termwise from which a quadrupole type decay approximately r d is derived for the stationary two points function eta 0 eta r eta 0 2 this phenomenon of self organized criticality is a direct consequence of the anisotropy causing the system to violate the condition of detailed balance combined with the conservation law forcing a diffusive decay approximately t d2 of the temporal correlations

phase separation and coarsening in one dimensional driven diffusive systems local dynamics leading to long range hamiltonians a driven system of three species of particles diffusing on a ring is studied in detail the dynamics is local and conserves the three densities a simple argument suggesting that the model should phase separate and break the translational symmetry is given we show that for the special case where the three densities are equal the model obeys detailed balance and the steady state distribution is governed by a hamiltonian with asymmetric long range interactions this provides an explicit demonstration of a simple mechanism for breaking of ergodicity in one dimension the steady state of finite size systems is studied using a generalized matrix product ansatz the coarsening process leading to phase separation is studied numerically and in a mean field model the system exhibits slow dynamics due to trapping in metastable states whose number is exponentially large in the system size the typical domain size is shown to grow logarithmically in time generalizations to a larger number of species are discussed

understanding the edge effect in tasep with mean field theoretic approaches we study a totally asymmetric simple exclusion process tasep with one defect site hopping rate q 1 near the system boundary regarding our system as a pair of uniform tasep s coupled through the defect we study various methods to match a finite tasep and an infinite one across a common boundary several approximation schemes are investigated utilizing the finite segment mean field fsmf method we set up a framework for computing the steady state current j as a function of the entry rate and q for the case where the defect is located at the entry site we obtain an analytical expression for j q which is in good agreement with monte carlo simulation results when the defect is located deeper in the bulk we refined the scheme of macdonald et al 1968 biopolymers 6 1 and find reasonably good fits to the density profiles before the defect site we discuss the strengths and limitations of each method as well as possible avenues for further studies

an existence theorem for quasiconcave functions with applications on demontre un theoreme d existence pour certaines familles de fonctions quasi concaves sur un ensemble compact convexe et on donne des applications de ce theoreme aux espaces localement convexes normes de hilbert et de dimension finie

new integrable systems related to the relativistic toda lattice the bi hamiltonian structure of the relativistic toda lattice is exploited to introduce some new integrable lattice systems their integrable discretizations are obtained by means of the general procedure proposed recently by the author backlund transformations between the new systems and the relativistic toda lattice in both the continuous and discrete time formulations are established

driven diffusive system with non local perturbations we investigate the impact of non local perturbations on driven diffusive systems two different problems are considered here in one case we introduce a non local particle conservation along the direction of the drive and in another case we incorporate a long range temporal correlation in the noise present in the equation of motion the effect of these perturbations on the anisotropy exponent or on the scaling of the two point correlation function is studied using renormalization group analysis

universality classes of driven lattice gases motivated by some recent criticisms to our alternative langevin equation for driven lattice gases dlg under an infinitely large driving field we revisit the derivation of such an equation and test its validity as a result an additional term coming from a careful consideration of entropic contributions is added to the equation this term heals all the recently reported generic infrared singularities the emerging equation is then identical to that describing randomly driven diffusive systems this fact confirms our claim that the infinite driving limit is singular and that the main relevant ingredient determining the critical behavior of the dlg in this limit is the anisotropy and not the presence of a current different aspects of our picture are discussed and it is concluded that it constitutes a very plausible scenario to rationalize the critical behavior of the dlg and variants of it

twenty five years after kls a celebration of non equilibrium statistical mechanics when lenz proposed a simple model for phase transitions in magnetism he couldn t have imagined that the ising model was to become a jewel in field of equilibrium statistical mechanics its role spans the spectrum from a good pedagogical example to a universality class in critical phenomena a quarter century ago katz lebowitz and spohn found a similar treasure by introducing a seemingly trivial modification to the ising lattice gas they took it into the vast realms of non equilibrium statistical mechanics an abundant variety of unexpected behavior emerged and caught many of us by surprise we present a brief review of some of the new insights garnered and some of the outstanding puzzles as well as speculate on the model s role in the future of non equilibrium statistical physics

n soliton solutions and elastic interaction of the coupled lattice soliton equations for nonlinear waves with the aid of symbolic computation a coupled set of the lattice soliton equations is investigated via darboux transformation dt method the n fold dt and conservation laws are constructed based on its lax representation the n soliton solutions in terms of the vandermonde like determinants are derived structures of the one two three and four soliton solutions are shown graphically elastic interactions among the four solitons are discussed solitonic shapes and amplitudes have not changed after the interaction results in this paper might be helpful for understanding the propagation of nonlinear waves

competition between multiple totally asymmetric simple exclusion processes for a finite pool of resources using monte carlo simulations and a domain wall theory we discuss the effect of coupling several totally asymmetric simple exclusion processes taseps to a finite reservoir of particles this simple model mimics directed biological transport processes in the presence of finite resources such as protein synthesis limited by a finite pool of ribosomes if all taseps have equal length we find behavior which is analogous to a single tasep coupled to a finite pool for the more generic case of chains with different lengths several unanticipated regimes emerge a generalized domain wall theory captures our findings in good agreement with simulation results

transverse fluctuations in the driven lattice gas we define a transverse correlation length suitable to discuss the finite size scaling behaviour of an out of equilibrium lattice gas whose correlation functions decay algebraically with the distance by numerical simulations we verify that this definition has a good infinite volume limit independent of the lattice geometry we study the transverse fluctuations as they can select the correct field theoretical description by means of a careful finite size scaling analysis without tunable parameters we show that they are gaussian in agreement with the predictions of the field theory proposed by janssen schmittmann leung and cardy

a discrete schrodinger spectral problem and associated evolution equations a recently proposed discrete version of the schrodinger spectral problem is considered the whole hierarchy of differential difference nonlinear evolution equations associated with this spectral problem is derived it is shown that a discrete version of the kdv sine gordon and liouville equations is included and that the so called inverse class in the hierarchy is local the whole class of related darboux and backlund transformations is also exhibited

a possible classification of nonequilibrium steady states we propose a general classification of nonequilibrium steady states in terms of their stationary probability distribution and the associated probability currents the stationary probabilities can be represented graph theoretically as directed labelled trees closing a single loop in such a graph leads to a representation of probability currents this classification allows us to identify all choices of transition rates based on a master equation which generate the same nonequilibrium steady state we explore the implications of this freedom e g for entropy production

integrable hierarchies of nonlinear difference difference equations and symmetries we construct the hierarchy of nonlinear difference difference equations associated with the discrete schrodinger spectral problem as examples of equations contained in this hierarchy we obtain the discrete time toda and volterra lattice equations in the case of the time discrete toda lattice we construct its lie point and generalized symmetries finally we present its backlund transformations and relate it to the already constructed symmetries

on the knaster kuratowski mazurkiewicz shapley theorem 1 it is well known that the celebrated k k m theorem knaster et al 4 on closed coverings of a simplex is equivalent to brouwer s fixed point theorem recently shapley 8 used the pathfollwing technique of lemke and howson 5 and scarf 6 7 to establish a generalization of the k k m theorem the generalized result is here called the k k m s theorem which is interesting in its own right the purpose of the present note is to show that the k k m s theorem is a simple consequence of a theorem of ky fan on a generalization of kakutani s fixed point theorem the present author would like to acknowledge that the essential idea of this note is borrowed from kannai 3

a trace identity and its applications to the theory of discrete integrable systems a general theory for studying discrete integrable systems is developed based on a trace identity that the author previously proposed a scheme for generating hierarchies of discrete integrable systems is presented under broad assumptions the resulting hierarchies are shown to consist of liouville integrable hamiltonian systems

lattice gas automata with self organization abstract self organization or dynamic phase transitions occur in computer simulations of a lattice gas model with strictly local collision dynamics conserving number of particles momentum and lattice symmetries the microscopic collision processes are biased i e they have asymmetric transition rates in this model a spatially uniform initial state is unstable at the onset of instability long wavelength sound modes drive the system into a state with very long range order with dynamics similar to spinodal decomposition the structure factor s k t and the selected wavelength for maximal growth max t are measured as a function of time the state with long range order is highly organized into moving stable spatial patterns of triangles or parallel strips of macroscopic size with two or one broken symmetry respectively the entropy suddenly drops at self organization both the onset of metastability and the detailed structure of the final patterns are explained by mean field theory

on the geometry of darboux transformations for the kp hierarchy and its connection with the discrete kp hierarchy we tackle the problem of interpreting the darboux transformation for the kp hierarchy and its relations with the modified kp hierarchy from a geometric point of view this is achieved by introducing the concept of a darboux covering we construct a darboux covering of the kp equations and obtain a new hierarchy of equations which we call the darboux kp hierarchy dkp we employ the dkp equations to discuss the relationships among the kp equations the modified kp equations and the discrete kp equations our approach also handles the various reductions of the kp hierarchy we show that the kp hierarchy is a projection of the dkp the mkp hierarchy is a dkp restriction to a suitable invariant submanifold and that the discrete kp equations are obtained as iterations of the dkp ones

field theory of critical behavior in a driven diffusive system we use a field theoretic renormalization group method to study the critical properties of a diffusive system with a single conserved density subject to a constant uniform external field a fixed point stable belowd c 5 is found to govern the critical behavior scaling forms of density correlation functions are derived and critical exponents are obtained to all orders in 5 d spatial correlations are found to be very anisotropic with elongated correlations along the external field long wavelength transverse fluctuations are suppressed completely to yield mean field transverse exponents

solution of the boundary value problem for the integrable discrete srs system on the semi line the complete solution of the initial boundary value problem for the integrable discrete version of the stimulated raman scattering system on the semi line is constructed by means of the inverse spectral transform the spectral data obey a riccati time evolution equation which allows for soliton generation out of a medium initially at rest it is also proved that the construction of the solution at any finite distance actually results in solving an algebraic system

shift in critical temperature for random spatial permutations with cycle weights we examine a phase transition in a model of random spatial permutations which originates in a study of the interacting bose gas permutations are weighted according to point positions the low temperature onset of the appearance of arbitrarily long cycles is connected to the phase transition of bose einstein condensates in our simplified model point positions are held fixed on the fully occupied cubic lattice and interactions are expressed as ewens type weights on cycle lengths of permutations the critical temperature of the transition to long cycles depends on an interaction strength parameter for weak interactions the shift in critical temperature is expected to be linear in with constant of linearity c using markov chain monte carlo methods and finite size scaling we find c 0 618 0 086 this finding matches a similar analytical result of ueltschi and betz we also examine the mean longest cycle length as a fraction of the number of sites in long cycles recovering an earlier result of shepp and lloyd for non spatial permutations

mixed function method for obtaining exact solution of nonlinear differential difference equations and coupled nddes in this paper we construct a new mixed function method for the first time by using this new method we study the two nonlinear differential difference equations named the generalized hybrid lattice and two component volterra lattice equations some new exact solutions of mixed function type such as discrete solitary wave solutions discrete kink and anti kink wave solutions and discrete breather solutions with kink and anti kink character are obtained and their dynamic properties are also discussed by using software mathematica we show their profiles

crossover between field theories with short range and long range exchange or correlations the relation between scalar field theories with short range and long range exchange or correlations is studied it is shown to all orders in perturbation theory that the critical exponents of fields and composite operators are continuous functions of the parameters alpha xi characterising the decay rate of the long range exchange correlations with power like falloff 1 rd 2 2 alpha 1 rd 2 xi the scaling law d 2 xi v 2 where v is the correlation length exponent as well as the harris criteria 1970 are shown to be exact to all orders in perturbation theory a discrepancy between two widely used approaches to the crossover problem is pointed out

distribution of structure factors and phase transitions in a driven lattice gas using monte carlo methods we study the full distribution of structure factors in the standard driven lattice gas at half filling the time evolution of the structure factors associated with the two lowest non trivial wavevectors is used to construct histograms for saturation drive at temperatures well above both are exponentially distributed but with different widths these results are well described by a continuum field theory with a dynamics which violates the usual fluctuation dissipation theorem as t is lowered one of these distributions remains essentially unaltered by contrast the other develops a maximum away from the origin while staying single peaked throughout this behaviour is an unmistakable signal for a continuous transition

biased diffusion of two species abstract a stochastic lattice gas with equal numbers of oppositely charged particles diffusing under the influence of a uniform external electric field and the excluded volume condition exhibits an order disorder phase transition controlled by particle density and field strength ordered configurations are spatially inhomogeneous allowing for both zero and nontrivial winding numbers around toroidal lattices with periodic boundary conditions we explore the transition and the structure of the ordered state using both simulational and analytic tools

a maximum entropy mean field method for driven diffusive systems we introduce a method of generating systematic mean field mf approximations for the nonequilibrium steady state of ferromagnetic ising driven diffusive systems dds based on the maximum entropy principle due to jaynes in the phase coexistence region mf approximations to the master equation do not provide a closed system of equations in the mf variables this can be traced to the conservation of the order parameter by the stochastic dynamics our maximum entropy mean field memf approximation method is applicable to high temperatures as well to the low temperature phase coexistence region it is based on a derivation of a generalized variational free energy from the maximum entropy principle with the mf evolution equations playing the role of constraints in the phase coexistence region this free energy is nonconvex and is interpreted by means of a maxwell construction we use a pair level variant of the memf approximation to calculate quantities of interest for the ferro magnetic ising dds on a square lattice results of calculations with several different choices of transition rates satisfying local detailed balance are discussed and compared with those obtained by other methods

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

an integrable lattice hierarchy associated integrable coupling darboux transformation and conservation laws abstract a new integrable lattice hierarchy is constructed from a discrete matrix spectral problem some related properties of the new hierarchy are discussed the hamiltonian structures and liouville integrability of the new hierarchy are established by using the discrete trace identity a kind of integrable coupling for the new hierarchy is constructed through enlarging spectral problems a darboux transformation dt with two variable parameters and the infinitely many conservation laws for a typical lattice equation in the new hierarchy are constructed based on its lax representation the explicit solutions are obtained via the dt the structures for those solutions are graphically investigated all these properties might be helpful to understanding some physical phenomena

exact diffusion constant for one dimensional asymmetric exclusion models the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions is considered on a ring of size n the steady state of this system is known all configurations have equal weight which allows for easy computation of the average velocity of a particle in the steady state here an exact expression for the diffusion constant of a particle is obtained for arbitrary number of particles and system size by using a matrix formulation two limits of infinite system size n are discussed firstly when the number of particles remains finite as n to infinity the diffusion constant remains dependent on the exact number of particles due to correlations between successive collisions secondly when the density p of particles is non zero i e when the number of particles is equal to n rho as n to infinity the diffusion constant scales as n 12 the exponent 1 2 is related to the dynamic exponent z 3 2 of the kpz equation in 1 1 dimensions

pattern recognition using asymmetric attractor neural networks the asymmetric attractor neural networks designed by the monte carlo mc adaptation rule are shown to be promising candidates for pattern recognition in such a neural network with relatively low symmetry when the members of a set of template patterns are stored as fixed point attractors their attraction basins are shown to be isolated islands embedded in a chaotic sea the sizes of these islands can be controlled by a single parameter we show that these properties can be used for effective pattern recognition and rejection in our method the pattern to be identified is attracted to a template pattern or a chaotic attractor if the difference between the pattern to be identified and the template pattern is smaller than a predescribed threshold the pattern is attracted to the template pattern automatically and thus is identified as belonging to this template pattern otherwise it wanders in a chaotic attractor for ever and thus is rejected as an unknown pattern the maximum sizes of these islands allowed by this kind of neural networks are determined by a modified mc adaptation rule which are shown to be able to dramatically enlarge the sizes of the islands we illustrate the use of our method for pattern recognition and rejection with an example of recognizing a set of chinese characters

phase separation dynamics in driven diffusive systems we study phase separation dynamics in a driven diffusive system our simulations are based on the cahn hilliard equation with an additional flux term due to an external field we study the dynamical scaling parallel and perpendicular to the field a crossover is observed from isotropic domains at early times to extremely anisotropic domains at later times we find that the inverse interfacial density an isotropic measure of the domain size increases ast with 1 3 from early times independent of the field strength even though we do not observe dynamical scaling during these times our results indicate that a growth exponent 1 3 may be more universal than previously expected we analyze the dynamics in terms of surface driven instabilities and one dimensional solitary waves

a generalized ablowitz ladik hierarchy multi hamiltonian structure and darboux transformation starting from a discrete spectral problem we derive a hierarchy of nonlinear discrete equations which includes the ablowitz ladik hierarchy and a new hierarchy as special cases especially we investigate in detail the integrability and resolvability of the new hierarchy it is shown that the new hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure a darboux transformation is established for a typical discrete system in the new hierarchy with the help of the gauge transformation of its lax pair as applications of the darboux transformation new exact solutions of the discrete system are explicitly given

inverse spectral transform for the ragnisco tu equation with heaviside initial condition abstract the ragnisco tu equation with the heaviside initial condition is studied in this paper through the inverse spectral transform method the procedure based on the regularization is applied to construct the relation between potentials and dressing matrices the explicit solutions are obtained and their dynamic behaviors are discussed

generic singularities in a driven diffusive system abstract an ising model in the lattice gas representation is driven far from equilibrium by an external electric field into a steady state apart from displaying a new class of singularities associated with the second order phase transition multi particle correlations are singular for all temperatures above criticality further for all t below t c there are anomalies in interfacial properties which may explain the suppression of roughness being consequences of the violation of the fluctuation dissipation theorem these generic features are studied analytically and by simulations

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

a class of coupled kdv systems and their bi hamiltonian formulation a hamiltonian pair is proposed and thus a type of hereditary operators results all the corresponding systems of evolution equations possess local bi hamiltonian formulation and a special choice of the systems leads to the kdv hierarchy illustrative examples are given

comment on transverse fluctuations in the driven lattice gas extensive simulation results of the transverse fluctuations in two driven lattice gases the classical one with current and a modified version without current are in agreement with the field theory proposed by garrido et al gsm based on the facts that results from both models are indistinguishable and they obey excellent scaling only by using gsm exponents i concluded that the conclusions of the recent letter by caracciolo et al are flawed

an exact solution of a one dimensional asymmetric exclusion model with open boundaries a simple asymmetric exclusion model with open boundaries is solved exactly in one dimension the exact solution is obtained by deriving a recursion relation for the steady state if the steady state is known for all system sizes less thann then our equation 8 gives the steady state for sizen using this recursion we obtain closed expressions 48 for the average occupations of all sites the results are compared to the predictions of a mean field theory in particular for infinitely large systems the effect of the boundary decays as the distance to the power 1 2 instead of the inverse of the distance as predicted by the mean field theory

antiferromagnetic potts models on the square lattice a high precision monte carlo study we study the antiferromagnetic q state potts model on the square lattice for q 3 and q 4 using the wang swendsen koteck wsk monte carlo algorithm and a powerful finite size scaling extrapolation method for q 3 we obtain good control up to correlation length 5000 the data are consistent with ae2 p 1 a1e as with p 1 the staggered susceptibility behaves as stagg 5 3 for q 4 the model is disordered 2 even at zero temperature in appendices we prove a correlation inequality for potts antiferromagnets on a bipartite lattice and we prove ergodicity of the wsk algorithm at zero temperature for potts antiferromagnets on a bipartite lattice

a modified toda spectral problem and its hierarchy of bi hamiltonian lattice equations starting from a modified toda spectral problem a hierarchy of generalized toda lattice equations with two arbitrary constants is constructed through discrete zero curvature equations it is shown that the hierarchy possesses a bi hamiltonian structure and a hereditary recursion operator which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals two cases of the involved constants present two specific integrable sub hierarchies one of which is exactly the toda lattice hierarchy

the finite band solution of the jaulent miodek equation a method to construct the finite band solution to the soliton equation is presented we take the jaulent miodek equation as an example with the use of the nonlinearization of lax pair the jaulent miodek equation is decomposed into two finite dimensional integrable systems whose properties are studied from the view of an r matrix then through solving these finite dimensional integrable systems the finite band solution of the jaulent miodek equation is obtained

phase transition in the abc model recent studies have shown that one dimensional driven systems can exhibit phase separation even if the dynamics is governed by local rules the mathrm abc model which comprises three particle species that diffuse asymmetrically around a ring shows anomalous coarsening into a phase separated steady state in the limiting case in which the dynamics is symmetric and the parameter q describing the asymmetry tends to one no phase separation occurs and the steady state of the system is disordered in the present work we consider the weak asymmetry regime q mathrm exp ensuremath ensuremath beta n where n is the system size and study how the disordered state is approached in the case of equal densities we find that the system exhibits a second order phase transition at some nonzero ensuremath beta c the value of ensuremath beta c 2 ensuremath pi sqrt 3 and the optimal profiles can be obtained by writing the exact large deviation functional for nonequal densities we write down mean field equations and analyze some of their predictions

discrete third order spectral problems and a new toda type equation in this paper we construct the class of equations associated with a discrete spectral problem of third order among the equations of this class we find an integrable differential equation on the lattice with exponential nonlinearity by considering its associated darboux transformation we construct an explicit solution

is there a true model d critical dynamics we show that non locality in the conservation of both the order parameter and a non critical density model d dynamics leads to new fixed points for critical dynamics depending upon the parameters characterizing the non locality in the two fields we find four regions i model a like where both conservations are irrelevant ii model b like with the conservation in the order parameter field relevant and the conservation in the coupling field irrelevant iii model c like where the conservation in the order parameter field is irrelevant but the conservation in the coupling field is relevant and iv model d like where both conservations are relevant while the first three behaviours are already known in dynamical critical phenomena the last one is a novel phenomenon due entirely to the non locality in the two fields

interfacial growth in driven ginzburg landau models short and long time dynamics interfacial growth in driven systems is studied from the initial stage to the longtime regime numerical integrations of a ginzburg landan type equation with a new flux term introduced by an external field are presented the interfacial instabilities are induced by the external field from the numerical results we obtain the dispersion relation for the initial growth during the intermediate temporal regime fingers of a characteristic triangular shape could grow depending on the boundary conditions the final state corresponds to strips multifinger states or a one finger state the results for the initial growth are interpreted by means of surface driven and mullins sekerka instabilities the shape of the one finger state is explained in terms of the characteristic length introduced by the external field

a family of completely integrable multi hamiltonian systems explicitly related to some celebrated equations by introducing a spectral problem with an arbitrary parameter we derive a kaup newell type hierarchy of nonlinear evolution equations which is explicitly related to many important equations such as the kundu equation the kaup newell kn equation the chen lee liu cll equation the gerdjikov ivanov gi equation the burgers equation the modified korteweg devries mkdv equation and the sharma tasso olver equation it is shown that the hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure under the bargann constraint between the potentials and the eigenfunctions the spectral problem is nonlinearized as a finite dimensional completely integrable hamiltonian system the involutive representation of the solutions for the kaup newell type hierarchy is also presented in addition an n fold darboux transformation of the kundu equation is constructed with the help of its lax pairs and a reduction technique according to the darboux transformation the solutions of the ku

hamiltonian structure of discrete soliton systems we describe an approach for investigating the hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem by using the so called implicit representations of the isospectral flows we demonstrate the existence of the recursion operator l which is a strong and hereditary symmetry of the flows it is then proven that every equation in the isospectral hierarchy possesses the hamiltonian structure if l has a skew symmetric factorization and the first equation ut k 0 in the hierarchy satisfies some simple condition we obtain related properties such as the implectic symplectic factorization of l the liouville complete integrability and the multi hamiltonian structures of the isospectral hierarchy four examples are given

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

the camassa holm hierarchy n dimensional integrable systems and algebro geometric solution on a symplectic submanifold this paper shows that the camassa holm ch spectral problem yields two different integrable hierarchies of nonlinear evolution equations nlees one is of negative order ch hierachy while the other one is of positive order ch hierarchy the two ch hierarchies possess the zero curvature representations through solving a key matrix equation we see that the well known ch equation is included in the negative order ch hierarchy while the dym type equation is included in the positive order ch hierarchy furthermore under two constraint conditions between the potentials and the eigenfunctions the ch spectral problem is cast in 1 a new neumann like n dimensional system when it is restricted into a symplectic submanifold of 2n which is proven to be integrable by using the dirac poisson bracket and the r matrix process and 2 a new bargmann like n dimensional system when it is considered in the whole 2n which is proven to be integrable by using the standard poisson bracket and the r matrix process

surface driven instability and enhanced relaxation in the dynamics of a nonequilibrium interface we determine the stability of a nonequilibrium interface between two coexisting solid phases in the presence of a weak external field starting at the coarsegrained cahn hilliard level we use the method of matched asymptotics to derive the macroscopic interfacial dynamics we then show that the external field leads to an instability due to flux along the interface in contrast with the more common mullins sekerka type instability which involves fluxes normal to the interface we also find that the external field produces an important modification of the gibbs thomson relation with these results we perform the linear stability analysis for an approximately flat interface if the field is tangent to the interface the modification of the gibbs thomson relation is important and the interface is stabilized if the field is normal to the interface the surface flux is important and the effect can be stabilizing or destabilizing but the orientational dependence is opposite what would be obtained if the mullins sekerka instability dominates numerical simulations are performed to study the effect of the surface current and are in agreement with our analytical results