diversity induced coherence resonance in spatially extended chaotic systems the effect of parameter diversity on coupled chua systems is investigated in the absence of diversity the systems jump back and forth between two variable domains of a chaotic attractor and the residence times within a single domain are uncertain by introducing parameter diversity a combined numerical and analytical approach indicates that the systems can jump regularly from one domain to another at an intermediate range of diversity a signature of coherence resonance furthermore the influences of coupling strength and the number of units are also considered our results provide a possibility for the control of chaos in spatially extended chaotic systems by the manipulation of parameter diversity

coherent regimes of mutually coupled chua s circuits we study the dynamical regimes that emerge from the strong coupling between two chua s circuits with parameters mismatch for the region around the perfect synchronous state we show how to combine parameter diversity and coupling in order to robustly and precisely target a desired regime this target process allows us to obtain regimes that may lie outside parameter ranges accessible for any isolated circuit the results are obtained by following a recently developed theoretical technique the order parameter expansion and are verified both by numerical simulations and on electronic circuits the theoretical results indicate that the same predictable change in the collective dynamics can be obtained for large populations of strongly coupled circuits with parameter mismatches

symmetry reductions exact solutions and conservation laws of the generalized zakharov equations in this paper the generalized zakharov equations which describe interactions between high and low frequency waves in plasma physics are studied from the perspective of lie symmetry analysis and conservation laws based on some subalgebras of symmetries several reductions and numerous new exact solutions are obtained all of these solutions represent modified traveling waves the obtained solutions include expressions involving airy functions bessel functions whittaker functions and generalized hypergeometric functions previously unknown conservation laws are constructed for the generalized zakharov equations using the direct method profiles are presented for some of these new solutions

conservation laws for some systems of nonlinear pdes via the symmetry adjoint symmetry pair method abstract this paper applies the symmetry adjoint symmetry pair method to find conservation laws for nonlinear partial differential equations as examples conservation laws are obtained for a combined potential system with nonlinear telegraph and nonlinear diffusion convection equations as special cases as well as for an extended nonlinear wave equation system in addition for both examples conservation laws are obtained for and nonlocally related equivalent systems the paper concludes with a posed open problem

vertex operator representation of some quantum tori lie algebras we are defining the trigonometric lie subalgebras in bar x infty bar a infty bar b infty bar c infty bar d infty which are the natural generalization of the well known sin lie algebra the embedding formulas into bar x infty are introduced these algebras can be considered as some lie algebras of quantum tori an irreducible representation ofa b series of trigonometric lie algebras is constructed special cases of the trigonometric lie factor algebras which can be considered as a quantum preserving lie algebra structure deformation of the kac moody algebras are considered

new explicit solutions of einstein equations in the framework of gap theory the presented paper develops a new method for solving einstein equations explicitly not only the solutions are new but also the mathematical framework of their construction which is given by a nonstandard function theory built over nonstandard algebras in the following denoted as gaps and will be called here as gap theory it is shown that the gap theory succeeds in regions where group theory fails the group structure is too small to solve einstein equations explicitly

on the initial value problem for the modified benjamin ono equation the initial value problem for the modified benjamin ono equation is solved by exploiting the fokas ablowitz inverse scattering transform for a certain complex extension of the benjamin ono equation the asymptotics of the jost functions for small positive values of the spectral parameter are sharpened and lead to a previously overlooked nongeneric case

derivation of conservation laws from nonlocal symmetries of differential equations an identity is derived which yields a correspondence between symmetries and conservation laws for self adjoint differential equations this identity does not rely on use of a lagrangian as needed to obtain conservation laws by noether s theorem moreover unlike noether s theorem which can only generate conservation laws from local symmetries the derived identity generates conservation laws from nonlocal as well as local symmetries it is explicitly shown how noether s theorem is extended by the identity conservation laws arising from nonlocal symmetries are obtained for a class of scalar wave equations with variable wave speeds the constants of motion resulting from these nonlocal conservation laws are shown to be linearly independent of all constants of motion resulting from local conservation laws

dynamics of a large system of coupled nonlinear oscillators abstract we consider the interaction of a large number of limit cycle oscillators with linear all to all coupling and a distribution of natural frequencies the system exhibits extremely rich dynamics as the coupling strength and the width of the frequency distribution are varied we find a variety of steady behaviors that can be described by a stationary distribution in phase space frequency locking amplitude death incoherence and partial locking an unexpected result is that the system can also exhibit unsteady behavior in which the phase space distribution evolves periodically quasiperiodically or even chaotically the simple form of the model allows us to derive several analytical results the stability boundaries of amplitude death and incoherence are found explicitly rigorous results on the existence and stability of frequency locking are also obtained

effects of microscopic disorder on the collective dynamics of globally coupled maps this paper studies the effect of independent additive noise on the synchronous dynamics of large populations of globally coupled maps our analysis is complementary to the approach taken by teramae and kuramoto j teramae y kuramoto strong desynchronizing effects of weak noise in globally coupled systems phys rev e 63 2001 036210 who pointed out the anomalous scaling properties preceding the loss of coherence we focus on the macroscopic dynamics that remains deterministic at any noise level and differs from the microscopic one using properly defined order parameters an analytical approach is proposed for describing the collective dynamics in terms of an approximate low dimensional system the systematic derivation of the macroscopic equations provides a link between the microscopic features of the population single element dynamics and noise distribution and the properties of the emergent behaviour the macroscopic bifurcations induced by noise are compared to those originating from parameter mismatches within the population

a spectral transform for the intermediate nonlinear schrodinger equation a new spectral transform system is introduced to solve the initial value problem for the intermediate nonlinear schrodinger inls equation describing envelope waves in a deep stratified fluid the spectral system is a combination of the zakharov shabat linear system and a local riemann hilbert problem in a strip of the complex plane the inverse scattering transform technique is developed and the backlund darboux transformation soliton solutions and an infinite number of conservation laws are constructed it is shown that all these properties of the inls equation are closely related to those of the intermediate long wave equation

exact solutions for wave equations of two layered media with smooth transition the wave equation c2 x uxx utt 0 is solved for wave speeds c x corresponding to two layered media with smooth transition from layer to layer the wave speed c x has four free parameters to fit a given medium solutions are constructed from invariant solutions of a related system of first order partial differential equations that admit a four parameter symmetry group these solutions are superposed to solve general initial value problems for data with compact support the computation of the superposition coefficients uses elementary fourier analysis solutions are illustrated for various initial conditions

dynamical systems on quantum tori lie algebras we use quantum tori lie algebras qtla which are a one parameter family of sub algebras ofgl to describe local and non local versions of the toda systems it turns out that the central charge of qtla is responsible for the non locality there are two regimes in the local systems conformal for irrational values of the parameter and non conformal and integrable for its rational values we also consider infinite dimensional analogs of rigid tops some of these systems give rise to quantized magneto hydrodynamic equations of an ideal fluid on a torus we also consider infinite dimensional versions of the integrable euler and clebsch cases

on locally and nonlocally related potential systems for any partial differential equation pde system a local conservation law yields potential equations in terms of some potential variable which normally is a nonlocal variable the current paper examines situations when such a potential variable is a local variable i e is a function of the independent and dependent variables of a given pde system and their derivatives in the case of two independent variables a simple necessary and sufficient condition is presented for the locality of such a potential variable and this is illustrated by several examples as a particular example two dimensional reductions of equilibrium equations for fluid and plasma dynamics are considered it is shown that such reductions with respect to helical axial and translational symmetries have conservation laws which yield local potential variables this leads to showing that the well known johnson frieman kruskal oberman jfko and bragg hawthorne grad shafranov equations are locally related to the corresponding he

simplifying the form of lie groups admitted by a given differential equation rather general results are obtained for determining the nature of infinitesimal generators which can be admitted by a given differential equation simple criteria are given to determine whether or not 1 the infinitesimals of independent variables can depend only on independent variables and 2 the infinitesimal of the dependent variable can depend at most linearly on the dependent variable many examples are given a trivial consequence of this paper is that an admitted lie group for any linear partial differential equation of at least second order must have infinitesimals of independent variables depending only on independent variables and the infinitesimal of the dependent variable depends at most linearly on the dependent variable this latter result previously has only been proved for a second order linear pde in the last century sophus lie developed the theory of continuous groups lie groups of transformations he showed that such transformations can be characterized in terms of infinitesimal generators moreover lie gave an algorithm to find the infinitesimal generators admitted by a given differential equation de for any de the admittance of a lie group leads to the construction of a family of new solutions from any known solution the admittance of a lie group for any ordinary differential equation ode leads constructively to a reduction of its order plus quadratures if a partial differential equation pde is invariant under a lie group one can construct special families of solutions called invariant solutions or similarity solutions if a de can be derived from a variational principle then admittance of a lie group is a necessary condition in order to find conservation laws by means of noether s theorem for recent references on lie s algorithm to find infinitesimal generators admitted by a given de and various uses of lie groups for des see l 4 71 52

order parameter expansion and finite size scaling study of coherent dynamics induced by quenched noise in the active rotator model we use a recently developed order parameter expansion method to study the transition to synchronous firing occurring in a system of coupled active rotators under the exclusive presence of quenched noise the method predicts correctly the existence of a transition from a rest state to a regime of synchronous firing and another transition out of it as the intensity of the quenched noise increases and leads to analytical expressions for the critical noise intensities in the large coupling regime it also predicts the order of the transitions for different probability distribution functions of the quenched variables using numerical simulations and finite size scal ing theory to estimate the critical exponents of the transitions we found values which are consistent with those reported in other scalar systems in the exclusive presence of additive static disorder full detail a particular simple analysis was developed in 4 where the authors used an expansion of the dynamical equa tions of the model up to first order in the deviation of the quenched variables from their mean value and identified a self consistency equation which had to be solved numeri cally the order parameter expansion used in the present pa per expands consistently this analysis up to terms of second order thus reaching a higher accuracy the resulting closed system of three differential equations reproduces the order ing abilities of quenched noise in this system and further more predicts a sharp transition back into the disordered state where no macroscopic order is observed the presented method gives an intuition about the different regimes the system is capable of and how the phase borders depend on the system parameters with its use the change of the nullclines due to diversity in the fitzhugh nagumo model 24 and phase transitions induced by diversity 10 have been investigated to complete the study we have done ex tensive numerical calculations of the full dynamical system in the vicinity of the transitions the order of the transitions has been determined and furthermore we present a detailed finite size scaling analysis of the critical exponents the paper is organized as follows first in sec ii we will define the active rotator model and summarize its main prop erties macroscopic observables that describe the collective behavior are introduced then in sec iii the approximation method is applied and conclusions about steady states are drawn in sec iv we present numerical results that support the previous findings and use the theory of finite size scaling to determine some of the critical exponents characterizing the transitions

generalized intermediate long wave hierarchy in zero curvature representation with noncommutative spectral parameter the simplest generalization of the intermediate long wave hierarchy ilw is considered to show how to extend the zakharov shabat dressing method to nonlocal i e integro partial differential equations the purpose is to give a procedure of constructing the zero curvature representation of this class of equations this result obtains by combining the drinfeld sokolov formalism together with the introduction of an operator valued spectral parameter namely a spectral parameter that does not commute with the space variable x this extension provides a connection between the ilwk hierarchy and the saveliev vershik continuum graded lie algebras in the case of ilw2 the fairlie zachos sinh algebra was found

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

integrable order parameter dynamics of globally coupled oscillators we study the nonlinear dynamics of globally coupled nonidentical oscillators in the framework of two order parameter mean field and amplitude frequency correlator reduction the main result of the paper is the exact solution of a corresponding nonlinear system on a two dimensional invariant manifold we present a complete classification of phase portraits and bifurcations obtain explicit expressions for invariant manifolds a limit cycle among them and derive analytical solutions for arbitrary initial data and different regimes

continuum analogues of contragredient lie algebras lie algebras with a cartan operator and nonlinear dynamical systems we present an axiomatic formulation of a new class of infinitedimensional lie algebras the generalizations ofz graded lie algebras with generally speaking an infinite dimensional cartan subalgebra and a contiguous set of roots we call such algebras continuum lie algebras the simple lie algebras of constant growth are encapsulated in our formulation we pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the cartan operator the generalization of the cartan matrix is a linear operator special examples of these algebras are the kac moody algebras algebras of poisson brackets algebras of vector fields on a manifold current algebras and algebras with differential or integro differential cartan operator the nonlinear dynamical systems associated with the continuum contragredient lie algebras are also considered

theory of collective firing induced by noise or diversity in excitable media large variety of physical chemical and biological systems show excitable behavior characterized by a nonlinear response under external perturbations only perturbations exceeding a threshold induce a full system response firing it has been reported that in coupled excitable identical systems noise may induce the simultaneous firing of a macroscopic fraction of units however a comprehensive understanding of the role of noise and that of natural diversity present in realistic systems is still lacking here we develop a theory for the emergence of collective firings in nonidentical excitable systems subject to noise three different dynamical regimes arise subthreshold motion where all elements remain confined near the fixed point coherent pulsations where a macroscopic fraction fire simultaneously and incoherent pulsations where units fire in a disordered fashion we also show that the mechanism for collective firing is generic it arises from degradation of entrainment originated either by noise or by diversity

noether theorem for μ symmetries we give a version of noether theorem adapted to the framework of symmetries this extends to such case recent work by muriel romero and olver in the framework of symmetries and connects symmetries of a lagrangian to a suitably modified conservation law in some cases this conservation law actually reduces to a standard one we also note a relation between symmetries and conditional invariants we also consider the case where the variational principle is itself formulated as requiring vanishing variation under prolonged variation fields leading to modified euler lagrange equations in this setting symmetries of the lagrangian correspond to standard conservation laws as in the standard noether theorem we finally propose some applications and examples

on invariance properties of the wave equation a complete group classification is given of both the wave equation c2 x uxx utt 0 i and its equivalent system vt ux c2 x vx ut ii when the wave speed c x const equations i and ii admit either a two or four parameter group for the exceptional case c x ax b 2 equation i admits an infinite group equations i and ii do not always admit the same group for a given c x the group for i can have more parameters or fewer parameters than that for ii moreover the groups can be different with the same number of parameters separately for i and ii all possible c x that admit a four parameter group are found explicitly the corresponding invariant similarity solutions are considered some of these wave speeds have realistic physical properties c x varies monotonically from one positive constant to another positive constant as x goes from to

hidden invariances in problems of two dimensional and three dimensional wall jets for newtonian and non newtonian fluids this work is devoted to the investigation of self similar solutions for steady wall jets the problem is considered in the context of two and three dimensional prandtl boundary layer equations and three dimensional parabolized navier stokes equations for newtonian and non newtonian fluids in contrast to dimensional analysis which does not allow the determination of self similar solutions in this case a generating functions approach elaborated by vinogradov soviet math dokl 19 1978 pp 144 148 enables one to derive conservation laws for the above mentioned problems and as a consequence to find new self similarities of the navier stokes equations

phase transitions induced by microscopic disorder a study based on the order parameter expansion based on the order parameter expansion we present an approximate method which allows us to reduce large systems of coupled differential equations with diverse parameters to three equations one for the global mean field variable and two which describe the fluctuations around this mean value the method is based on a systematic perturbation expansion and can be applied around the vicinity of the homogeneous state with this tool we analyze phase transitions induced by microscopic disorder in three prototypical models of phase transitions which have been studied previously in the presence of thermal noise we study how macroscopic order is induced or destroyed by time independent local disorder and analyze the limits of the approximation by comparing the results with the numerical solutions of the self consistency equation which arises from the property of self averaging finally we carry on a finite size analysis of the numerical results and calculate the corresponding critical exponents

darboux transformations infinitesimal symmetries and conservation laws for the nonlocal two dimensional toda lattice the technique of darboux transformation is applied to the nonlocal partner of the two dimensional periodic an 1 toda lattice this system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear systems with a quantized spectral parameter we give the generalization of the darboux transformation technique on linear equations of this type we present the connections between the solutions of overdetermined linear systems and their expansions in series in the neighbourhood of singular points the solutions of the nonlocal toda lattice and infinite hierarchies of the infinitesimal symmetries and conservation laws are obtained

optimized analysis of the critical behavior in polymer mixtures from monte carlo simulations a complete outline is given for how to determine the critical properties of polymer mixtures with extrapolation methods similar to the ferrenberg swendsen techniques recently devised for spin systems by measuring not only averages but the whole distribution of the quantities of interest it is possible to extrapolate the data obtained in only a few simulations neart c over the entire critical region thereby saving at least 90 of the computer time normally needed to locate susceptibility peaks or cumulant intersections and still getting more precise results a complete picture of the critical properties of polymer mixtures in the thermodynamic limit is then obtained with finite size scaling functions since the amount of information extracted from a simulation in this way is drastically increased as compared to conventional methods the investigation of mixtures with long chains or built in asymmetries is now possible as an example the critical points exponents and amplitudes of dense symmetric polymer mixtures with chain lengths ranging fromn 16 up ton 256 are determined within the framework of the 3d bond fluctuation model using grand canonical simulation techniques as an example for an asymmetry the generalization of the method to asymmetric monomer potentials is briefly discussed

generalized internal long waves equations construction hamiltonian structure and conservation laws a general class of the ilw type equations is constructed we introduce a hamiltonian structure and construct an infinite number of conservation laws

on the construction of conservation laws a mixed approach a new approach combining the ibragimov method and the one by anco and bluman with the aim of algorithmically computing local conservation laws of partial differential equations is discussed some examples of the application of the procedure are given the method of course is able to recover all the conservation laws found by using the direct method at the same time we can characterize which symmetry if any is responsible for the existence of a given conservation law some new local conservation laws for the short pulse equation and for the fornberg whitham equation are also determined

nonlocal integrable partners to generalized mkdv and two dimensional toda lattice equation in the formalism of a dressing method with quantized spectral parameter two new hierarchies milw2 and a two dimensional nonlocal toda lattice are constructed the characteristic property of the first one is the connection with the ilw2 hierarchy by means ofgl 2 miura transformation on the other hand milw2 equations turn out to be symmetry equations for a two dimensional nonlocal toda lattice a new version of the dressing technique with quantized spectral parameter is proposed