the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

distributed finite element kalman filter for field estimation the paper deals with decentralized state estimation for spatially distributed systems described by linear partial differential equations from discrete in space and time noisy measurements provided by sensors deployed over the spatial domain of interest a fully scalable approach is pursued by decomposing the domain into possibly overlapping subdomains assigned to different processing nodes interconnected to form a network each node runs a local finite dimensional discrete time kalman filter which exploits the finite element approach for spatial discretization a backward euler method for time discretization and the parallel schwarz method to iteratively enforce continuity of the field predictions over the boundaries of adjacent subdomains numerical stability of the adopted approximation scheme and stability of the proposed distributed finite element kalman filter are mathematically proved the effectiveness of the proposed approach is then demonstrated via simulation experiments concerning the estimation of a bi dimensional temperature field

new exact solutions of the broer kaup equations in 2 1 dimensional spaces abstract with the aid of maple several new kinds of exact solutions for the broer kaup equations in 2 1 dimensional spaces are obtained by using a new ansatz this approach can also be applied to other nonlinear evolution equations

differential difference equations associated with the fractional lax operators we study integrable hierarchies associated with spectral problems of the form p q where p and q are difference operators the corresponding nonlinear differential difference equations can be viewed as inhomogeneous generalizations of the bogoyavlensky type lattices while the latter turn into the korteweg de vries equation under the continuous limit the lattices under consideration provide discrete analogs of the sawada kotera and kaup kupershmidt equations the r matrix formulation and several of the simplest explicit solutions are presented

the supersymmetric camassa holm equation and geodesic flow on the superconformal group we study a family of fermionic extensions of the camassa holm equation within this family we identify three interesting classes a equations which are inherently hamiltonian describing geodesic flow with respect to an h1 metric on the group of superconformal transformations in two dimensions b equations which are hamiltonian with respect to a different hamiltonian structure and c supersymmetric equations classes a and b have no intersection but the intersection of classes a and c gives a system with interesting integrability properties we demonstrate the painleve property for some simple but nontrivial reductions of this system and also discuss peakon type solutions

non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

some results on the caudrey dodd gibbon kotera sawada equation in this paper a hierarchy of bilinear caudrey dodd gibbon kotera sawada equations with a unified structure is proposed a nonlinear superposition formula for the cdgks equation is proved under certain conditions a backlund transformation for a higher order cdgks equation is presented

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

distributed linear estimation over sensor networks we consider a network of sensors in which each node may collect noisy linear measurements of some unknown parameter in this context we study a distributed consensus diffusion scheme that relies only on bidirectional communication among neighbour nodes nodes that can communicate and exchange data and allows every node to compute an estimate of the unknown parameter that asymptotically converges to the true parameter at each time iteration a measurement update and a spatial diffusion phase are performed across the network and a local least squares estimate is computed at each node the proposed scheme allows one to consider networks with dynamically changing communication topology and it is robust to unreliable communication links and failures in measuring nodes we show that under suitable hypotheses all the local estimates converge to the true parameter value

seiberg witten equations in r4 lie symmetries particular solutions integrability it is shown that the 11 parameter automorphism group of lie algebra of four dimensional euclidean group is a maximal lie symmetries group of the seiberg witten equations in r4 particular explicit solutions which are invariant under so 3 subgroups of the maximal lie symmetries group are constructed it is established that seiberg witten equations do not possess the painleve property nevertheless so 3 invariant solutions obtained are turned to admit a characteristic singularity structure

new multiple soliton like solutions to the generalized 2 1 dimensional kp equation a new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations pdes in a unified way the main idea of this method is to take full advantage of the riccati equation which has more new solutions more new multiple soliton like solutions are obtained for the the generalized 2 1 dimensional kp equation

the painleve property and backlund transformations for the sequence of boussinesq equations we investigate the sequence of boussinesq equations by the method of singular manifolds for the boussinesq equation which is known to possess the painleve property a backlund transformation is defined this backlund transformation which is formulated in terms of the schwarzian derivative obtains the system of modified boussinesq equations and the resulting miura type transformation the modified boussinesq equations are found to be invariant under a discrete group of symmetries acting on the dependent variables by linearizing the miura transformation and modified equations the lax pair is readily obtained furthermore by a result of fokas and anderson the recursion operators defining the sequence of higher order boussinesq equations may be constructed from the miura transformation this allows the recursive definition of backlund transformations for this sequence of equations the recursion operator is found to preserve the discrete symmetries of the modified boussinesq equations this leads to the conclusion that the sequences of boussinesq and modified boussinesq equations identically possess the painleve property are meromorphic we also find that by a simple reduction the sequences of caudrey dodd gibbon and kuperschmidt equations are contained within the boussinesq sequence rational solutions are iteratively constructed for the boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems

elliptic solitons and grobner bases we consider the solution of spectral problems with elliptic coefficients in the framework of the hermite ansatz we show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the grobner bases method and others new integrable potentials and corresponding solutions of the sawada kotera kaup kupershmidt boussinesq equations and others are found

painleve property and geometry abstract the painleve property for discrete hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times the augmented manifold is constructed by expanding the hamilton jacobi equation a complete classification of the types of poles allowed in complex time is given for hamiltonians which separate into the direct product of hyperelliptic curves for such systems bounds on the degrees of the polynomial separating variable change and the other integrals in involution can be found from the pole series and the hamilton jacobi equation it is shown how branching can arise naturally in a painleve system

linearity inside nonlinearity exact solutions to the complex ginzburg landau equation abstract systems of two linear partial differential equations pdes with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear pdes for the one dimensional complex ginzburg landau equation the usual representations of the complex field re a or a are multivalued which makes difficult the search for exact solutions in this manner fortunately the painleve analysis naturally introduces an elementary single valued like representation of a by two fields z grad respectively complex abd real uniquely defined by an explicit expression for and a ze i another important feature is the invariance by parity on a which increases the class of expected solutions this allows to retrieve quite easily the four famous solutions of nozaki and bekki represented by the constant coefficients of two linear partial differential equations and a finite set of constants

soliton solutions of two bidirectional sixth order partial differential equations belonging to the kp hierarchy in this letter we analyse two bidirectional sixth order partial differential equations which are reductions in 1 1 dimensions of equations belonging to the kp hierarchy they have fourth order and fifth order lax pairs respectively we derive their backlund transformations and from the nonlinear superposition formula we can build their soliton solutions like a grammian the interesting dynamics of these solitons is that they may describe not only the overtaking collision but also the head on collision of solitary waves of different type and shape

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

symmetrically coupled higher order nonlinear schrodinger equations singularity analysis and integrability the integrability of a system of two symmetrically coupled higher order nonlinear schrodinger equations with parameter coefficients is tested by means of singularity analysis it is proven that the system passes the painleve test for integrability only in ten distinct cases of which two are new for one of the new cases a lax pair and a multi field generalization are obtained for the other one the equations of the system are uncoupled by a nonlinear transformation

generalized hunter saxton equation and the geometry of the group of circle diffeomorphisms we study an equation lying mid way between the periodic hunter saxton and camassa holm equations and which describes evolution of rotators in liquid crystals with external magnetic field and self interaction we prove that it is an euler equation on the diffeomorphism group of the circle corresponding to a natural right invariant sobolev metric we show that the equation is bihamiltonian and admits both cusped and smooth traveling wave solutions which are natural candidates for solitons we also prove that it is locally well posed and establish results on the lifespan of its solutions throughout the paper we argue that despite similarities to the kdv ch and hs equations the new equation manifests several distinctive features that set it apart from the other three

towards classification of dimensional integrable equations integrability conditions i in this paper we attempt to extend the symmetry approach well developed in the case of 1 1 dimensional equations to the 2 1 dimensional case presence of nonlocal terms in symmetries and conservation laws is the main feature of integrable 2 1 dimensional equations we have introduced a concept of quasi local functions to characterize nonlocalities we have found a few first integrability conditions for a class of scalar equations in terms of quasi local functions and have demonstrated that they are suitable for testing integrability

the singular manifold method revisited for many completely integrable partial differential equations pdes the singular manifold method of weiss allows the recovery of the lax pair and darboux transformation dt and so also the backlund transformation from a truncated painleve expansion recently the so called two singular manifold method has been proposed in order to handle pdes such as the modified korteweg de vries mkdv equation here we present a more natural extension of the weiss singular manifold method which makes use of only one singular manifold but is capable of dealing with such pdes in this approach we allow the possibility that the dt might in fact correspond to an infinite painleve expansion for a certain choice of the arbitrary coefficients this then leads us to a new and more consistent definition of singular manifold equation sme this can give smes different from those usually presented the summation of infinite painleve expansions is effected by seeking a truncation in a new riccati variable z the use o

grammian n soliton solutions of a coupled kdv system from the lax pair and the binary darboux transformation of a coupled korteweg de vries system we show that its nonlinear superposition formula is identical to that obtained for the kaup kupershmidt equation therefore the n soliton solution can be associated with a determinant of the gram type

observability of discretized partial differential equations the situation in which one cannot solve a numerical initial value problem for lack of complete initial data is arising ever more frequently in applied sciences and engineering the procedure of estimating the evolving solution by inserting incomplete data into a numerical model as they become available at different instants of time is called data assimilation or filtering we show that if data generated by a linear system of partial differential equations pde are inserted properly then complete observability of the discrete numerical model is necessary and sufficient for asymptotic stability of the data assimilation process this complete observability means that if the data had been generated by the discrete model rather than by the pde system then the data would define the state of the model uniquely after some finite time simple observability criteria for discretizations of linear constant coefficient pde on periodic domains are formulated in terms of properties of the symbol of the numerical schem

lax hopf based incorporation of internal boundary conditions into hamilton jacobi equation part i theory this article proposes a new approach for computing a semi explicit form of the solution to a class of hamilton jacobi hj partial differential equations pdes using control techniques based on viability theory we characterize the epigraph of the value function solving the hj pde as a capture basin of a target through an auxiliary dynamical system called characteristic system the properties of capture basins enable us to define components as building blocks of the solution to the hj pde in the barron jensen frankowska sense these components can encode initial conditions boundary conditions and internal boundary conditions which are the topic of this article a generalized lax hopf formula is derived and enables us to formulate the necessary and sufficient conditions for a mixed initial and boundary conditions problem with multiple internal boundary conditions to be well posed we illustrate the capabilities of the method with a data assimilation problem for reconstruction of highway traffic flow using lagrangian measurements generated from next generation simulation ngsim traffic data

boundary feedback control in networks of open channels this article deals with the regulation of water flow in open channels modelled by saint venant equations by means of a riemann invariants approach we deduce stabilizing control laws for a single horizontal reach without friction the stability condition is extended to a general class of hyperbolic systems which can describe canal networks with more general topologies a control law design based on this condition is illustrated with a simple case study two reaches in cascade the proof of the main stability theorem is based on a previous result from li ta tsien concerning the existence and decay of classical solutions of hyperbolic systems

backlund transformation of partial differential equations from the painleve gambier classification i kaup kupershmidt equation among the 50 nonlinear second order differential equations of painleve and gambier those which are linearizable provide a natural scheme for deriving the lax pair and the darboux transformation of a nonlinear partial differential equation when the order of the scattering problem is three this new method allows one to obtain from singularity analysis only the backlund transformation of the kaup kupershmidt equation a result which was missing until now

generalized kaup kupershmidt solitons and other solitary wave solutions of the higher order kdv equations new types of exact explicit solitary wave solutions of the higher order kdv equations are identified by applying a direct method designed specifically for constructing solitary wave solutions of evolution equations the first type is the generalized kaup kupershmidt gkk solitary waves which unify the structures of the sech2 kdv like soliton and the kaup kupershmidt soliton and also provide solutions of some other equations one of those equations is found to possess the multi soliton solutions which makes it a good candidate for being integrable in terms of the gkk solitons another type of solutions of the higher order kdv equations identified by applying the method represents the steady state localized structures the variety of equations possessing such solutions includes the integrable in terms of the sech2 solitons sawada kotera equation which thus appears to provide the localized solutions of two types

systems of ordinary differential equations with nonlinear superposition principles abstract this work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations x x t x r n t r in terms of a finite number of particular solutions the point of departure is lie s criterion according to which such a rule exists if and only if the vector fields x t generate a finite dimensional lie algebra we provide three different constructive methods for deriving superposition rules and apply them to systems of coupled riccati equations of the projective and conformal types based respectively on the lie algebra sl n 1 r and o p 1 n p 1

the two singular manifold method i modified korteweg de vries and sine gordon equations we address the problem of finding by painleve analysis only the backlund transformation of partial differential equations pdes having two families of movable singularities with opposite principal parts such as the modified korteweg de vries mkdv sine gordon or nonlinear schrodinger equations this first paper gives an almost algorithmic method which extends the singular manifold method of weiss 1986 that is unable to handle these equations first with only one singular manifold at a time we obtain the darboux transformation second we assume that the ratio of two functions defining the singular manifolds satisfies the most general projective riccati system with undetermined coefficients the darboux transformation then generates a very small number of determining equations admitting a unique solution equivalent to the lax pair of the zakharov shabat ablowitz kaup newell segur scheme by the canonical linearization of the riccati system the method is here applied to the mkdv and sine gordon equations

on the n 2 supersymmetric camassa holm and hunter saxton equations we consider n 2 supersymmetric extensions of the camassa holm and hunter saxton equations we show that they admit geometric interpretations as euler equations on the superconformal algebra of contact vector fields on the 1 2 dimensional supercircle we use the bi hamiltonian formulation to derive lax pairs moreover we present some simple examples of explicit solutions as a by product of our analysis we obtain a description of the bounded traveling wave solutions for the two component hunter saxton equation

new applications of the singular manifold method to the 2 1 dimensional burgers equations new applications of the singular manifold method are presented to a 2 1 dimensional burgers equation and a higher order burgers equation by choosing different seed solutions auto backlund transformations cole hopf transformations and general functional separation solutions containing three arbitrary functions are obtained for the equations in question localized coherent structures are for the first time expressed by jacobi elliptic functions not infinite periodic limit and their interaction properties are numerically studied

on the links between some generalized pinney and nonlinear gambier equations two generalized forms of the pinney equation recently derived by rogers schief and winternitz from lie group considerations are here connected with two linearizable second order ode s belonging to the gambier classification

on soliton solutions of the kaup kupershmidt equation ii anomalous n soliton solutions abstract in the second of two articles designated i physica d 137 2000 25 and ii exact soliton solutions of the kaup kupershmidt equation are derived using a simplified approach based in part on hirota s bilinear transform these solutions have not previously been obtained explicitly by analytic methods it is found that at each order the n soliton solution is characterised by an additional parameter which accounts for the anomalous nature of these solutions the first four solitons are constructed explicitly and in a novel departure from hirota s method an iterative procedure is presented for obtaining the general n soliton solution some possible alternative approaches to solving the kaup kupershmidt equation are considered

nonlocal symmetries and the kaup kupershmidt equation computational and geometric aspects of nonlocal infinitesimal symmetries of nonlinear partial differential equations are considered in particular the relation of nonlocal symmetries with classical generalized and internal symmetries is briefly discussed a nonlocal symmetry for the kaup kupershmidt equation is introduced and studied in some detail some explicit particular solutions are found with its help and a darboux like transformation is also obtained

resonant triads for two bidirectional equations in 1 1 dimensions the two soliton solution of the kdv equation is known not to degenerate into a solution describing resonant triads of solitons in 1 1 dimensions however we show that two integrable coupled kdv systems of the drinfeld sokolov class possess solutions which may degenerate into resonant triads these solutions are associated with a resonance relation which generalizes the usual one previously considered by hirota and ito

a new class of euler equation on the dual of the n 1 extended neveu schwarz algebra let g be the n 1 extended neveu schwarz algebra and greg its regular dual in this paper we will study a super euler system with seven parameters s1 s2 c1 c5 associated with greg we will show that the super euler system is 1 local bi superbihamiltonian if s1 14c1 and s2 12c2 2 supersymmetric if s1 c1 and s2 c2 3 local bi superbihamiltonian and supersymmetric if s1 c1 0 and s2 c2 0 by choosing different parameters we could obtain several supersymmetric or bi superhamiltonian generalizations of some well known integrable systems including the ito equation the 2 component camassa holm equation the 2 component hunter saxton equation and especially the whitham broer kaup dispersive water wave system let g be the n 1 extended neveu schwarz algebra and greg its regular dual in this paper we will study a super euler system with seven parameters s1 s2 c1 c5 associated with greg we will show that the super euler system is 1 local bi superbihamiltonian if s1 14c1 and s2 12c2 2 supersymmetric if s1 c1 and s2 c2 3 local bi superbihamiltonian and supersymmetric if s1 c1 0 and s2 c2 0 by choosing different parameters we could obtain several supersymmetric or bi superhamiltonian generalizations of some well known integrable systems including the ito equation the 2 component camassa holm equation the 2 component hunter saxton equation and especially the whitham broer kaup dispersive water wave system

backlund transformation of partial differential equations from the painleve gambier classification ii tzitzeica equation from the existing methods of singularity analysis only we derive the two equations which define the backlund transformation of the tzitzeica equation this is achieved by defining a truncation in the spirit of the approach of weiss et al so as to preserve the lorentz invariance of the tzitzeica equation if one assumes a third order scattering problem this truncation admits a unique solution thus leading to a matrix lax pair and a darboux transformation in order to obtain the backlund transformation bt which is the main new result of this paper one represents the lax pair by an equivalent two component riccati pseudopotential this yields two different bts the first one is a bt for the hirota satsuma equation while the second one is a bt for the tzitzeica equation one of the two equations defining the bt is the fifth ordinary differential equation of gambier

generalized solitary and periodic wave solutions to a 2 1 dimensional zakharov kuznetsov equation abstract in this paper the exp function method is employed to the zakharov kuznetsov equation as a 2 1 dimensional model for nonlinear rossby waves the observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient the obtained results may be useful for better understanding the properties of two dimensional coherent structures such as atmospheric blocking events

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

a generalized hirota satsuma coupled kdv system darboux transformations and reductions a darboux transformation is constructed for the generalized hirota satsuma coupled kdv system and the result is compared with the recent work of geng and his collaborators x g geng et al phys rev e 79 056602 2009 and x g geng and g l he j math phys 51 033514 2010 it is shown that our darboux transformation may be applied to three interesting reductions of the general system in addition the iteration of this darboux transformation is worked out and some solutions to the associated systems are obtained

darboux transformation and solutions for an equation in 2 1 dimensions painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2 1 dimensions this procedure has allowed us to obtain the lax pair darboux transformation and functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively

deformations of n 2 superconformal algebra and supersymmetric two component camassa holm equation this paper is concerned with a link between central extensions of n 2 superconformal algebra and a supersymmetric two component generalization of the camassa holm equation deformations of superconformal algebra give rise to two compatible bracket structures one of the bracket structures is derived from the central extension and admits a momentum operator which agrees with the sobolev norm of a co adjoint orbit element the momentum operator induces via lenard relations a chain of conserved hamiltonians of the resulting supersymmetric camassa holm hierarchy

algorithmic method for deriving lax pairs from the invariant painleve analysis of nonlinear partial differential equations given a partial differential equation its painleve analysis will first be performed with a built in invariance under the homographic group acting on the singular manifold function then assuming an order for the underlying lax pair a multicomponent pseudopotential of projective riccati type the components of which are homographically invariant is introduced if the equation admits a classical darboux transformation a very small set of determining equations whose solution yields the lax pair will be generated in the basis of the pseudopotential this new method will be applied to find the yet unpublished lax pair of the scalar hirota satsuma equation

on higher order riccati equations as backlund transformations in this short paper we illustrate by few examples the special role played by higher order riccati equations in the construction of backlund transformations for integrable systems

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

the painleve property a modified boussinesq equation and a modified kadomtsev petviashvili equation abstract it is shown that the modified boussinesq equation and the modified kadomtsev petviashvili equation both of which are known to be completely integrable possess the painlev e property for differential equations as defined by weiss tabor and carnevale

application of exp function method to the whitham broer kaup shallow water model using symbolic computation in this letter the exp function method is applied to the whitham broer kaup shallow water model with the help of symbolic computation several kinds of new solitary wave solutions are formally derived

nonlinear superposition formula for the kaup kupershmidt partial differential equation abstract the fifth order kaup kupershmidt kk equation is one of the solitonic equations related to the integrable cases of the henon heiles system as opposed to the sawada kotera equation which is its dual equation the construction of the n soliton solutions of kk is not an easy task in using a perturbation scheme or the hirota bilinear formalism from the backlund transformation obtained by singularity analysis considerations we here establish the permutability theorem for the kk equation this allows us to explain the anomalous structure of the n soliton solution empirically obtained by parker physica d 137 2000 25 33 34 48

a bidirectional kaup kupershmidt equation and directionally dependent solitons exact multisoliton solutions are obtained for the bidirectional kaup kupershmidt equation that was reported in a recent paper j math phys 42 2567 2001 this fills the gap that was left in the prior study these solutions are found to possess the same remarkable and unusual property as the solitary wave solution namely their wave profiles are directionally dependent in particular solitons are presented which describe multiple head on collisions between single and double humped solitary waves explicit expressions are given for the first few multisoliton solutions and a procedure is described for constucting the general n soliton by iteration on the solitons of lower order

supersymmetric euler equations associated to the n 3 neveu schwarz algebra we give a systematic study about supersymmetric euler equations on the smooth dual nsreg n of the neveu schwarz algebra ns n for n 3 let a be the inertia operator and c1 c2 r we will show that the n 2 supersymmetric euler equation with a c1 c2d2 is local bi super hamiltonian with the freezing point c1 c2 nsreg 2 which is similar to that of the virasoro algebra ns 0 and the n 3 supersymmetric euler equation with a c2d3 1 is local bi super hamiltonian with the freezing point 0 c2 nsreg 3 which is similar to that of the n 1 neveu schwarz algebra ns 1

the painleve analysis for n 2 super korteweg de vries equations the painleve analysis of a generic multiparameter n 2 extension of the korteweg de vries kdv equation is presented unusual aspects of the analysis pertaining to the presence of two fermionic fields are emphasized for the general class of models considered we find that the only ones which manifestly pass the test are precisely the four known integrable supersymmetric kdv equations including the skdv1 case

auto backlund transformations and superposition formulas for solutions of drinfeld sokolov systems abstract the paper is devoted to constructing auto backlund transformations abt and superposition formulas for the solutions of the drinfeld sokolov ds systems the transformations are derived from pairs of differential substitutions relating different systems of the ds type the nonlinear superposition formulas for solutions of the ds systems are obtained from the assumption of commutativity of the bianchi diagram we indicate a seed solution for each system which can be used to generate multi soliton solutions as an application of the superposition formulas we construct two soliton solutions for each of the ds systems

euler equations on homogeneous spaces and virasoro orbits abstract we show that the following three systems related to various hydrodynamical approximations the korteweg de vries equation the camassa holm equation and the hunter saxton equation have the same symmetry group and similar bihamiltonian structures it turns out that their configuration space is the virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right invariant metrics on this group or on appropriate homogeneous spaces in particular we describe how arnold s approach to the euler equations as geodesic flows of one sided invariant metrics extends from lie groups to homogeneous spaces we also show that the above three cases describe all generic bihamiltonian systems which are related to the virasoro group and can be integrated by the translation argument principle they correspond precisely to the three different types of generic virasoro orbits finally we discuss interrelation between the above metrics and kahler structures on virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits

on soliton solutions of the kaup kupershmidt equation i direct bilinearisation and solitary wave abstract in this first of two papers designated i and ii physica d 137 2000 34 a direct procedure is given for bilinearising the kaup kupershmidt and sawada kotera equations it is found that whereas the sawada kotera equation can be written in bilinear form in two distinct ways which are shown to be equivalent the kaup kupershmidt equation has a single bilinear representation this distinguishing feature which has previously received no explanation has consequences for obtaining the soliton solutions of these two closely related fifth order integrable equations the anomalous solitary wave of the kaup kupershmidt equation is reconstructed and a simplified approach to obtaining its n soliton solutions is proposed this provides the basis for constructing these solutions explicitly in ii

distributing the kalman filter for large scale systems this paper presents a distributed kalman filter to estimate the state of a sparsely connected large scale n dimensional dynamical system monitored by a network of n sensors local kalman filters are implemented on nl dimensional subsystems nl lt n obtained by spatially decomposing the large scale system the distributed kalman filter is optimal under an lth order gauss markov approximation to the centralized filter we quantify the information loss due to this lth order approximation by the divergence which decreases as l increases the order of the approximation l leads to a bound on the dimension of the subsystems hence providing a criterion for subsystem selection the approximated centralized riccati and lyapunov equations are computed iteratively with only local communication and low order computation by a distributed iterate collapse inversion dici algorithm we fuse the observations that are common among the local kalman filters using bipartite fusion graphs and consensus averaging algorithms the proposed algorithm achieves full distribution of the kalman filter nowhere in the network storage communication or computation of n dimensional vectors and matrices is required only nl lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors in other words knowledge of the state is itself distributed

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

supersymmetric extension of the korteweg de vries equation it is shown that among a one parameter family of supersymmetric extensions of the korteweg de vries equation there is a special system that has an infinite number of conservation laws which can be formulated in the second hamiltonian structure and which has a nontrivial lax representation its modified version is also discussed

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

design of consensus and adaptive consensus filters for distributed parameter systems this work establishes an abstract framework that considers the distributed filtering of spatially varying processes using a sensor network it is assumed that the sensor network consists of groups of sensors each of which provides a number of state measurements from sensing devices that are not necessarily identical and which only transmit their information to their own sensor group a modification to the local spatially distributed filters provides the non adaptive case of spatially distributed consensus filters which penalize the disagreement amongst themselves in a dynamic manner a subsequent modification to this scheme incorporates the adaptation of the consensus gains in the disagreement terms of all local filters both the well posedness of these two consensus spatially distributed filters and the convergence of the associated observation errors to zero in appropriate norms are presented their performance is demonstrated on three different examples of a diffusion partial differential equation with point measurements

n soliton solutions to a dimensional integrable equation we give explicitly n soliton solutions of a 2c 1 dimensional equation xtc xxxz 4cxxzcxxz 2c 1 x zzz 4 d 0 this equation is obtained by unifying two directional generalizations of the potential kdv equation the closed ring with the potential kp equation and the calogero bogoyavlenskij schiff equation this equation is also a reduction of the kp hierarchy we also find the miura transformation which yields the same ring of the corresponding modified equations the study of higher dimensional integrable systems is one of the central themes in integrable systems a typical example of a higher dimensional integrable system is obtained by modifying the lax operators of a basic equation the potential kdv p kdv equation in this paper the lax pair of the p kdv equation have the form l x t d 2 xcx x t 1 t x t d l x t 3 2

an exp function method for new n soliton solutions with arbitrary functions of a 2 1 dimensional vcbk system in this paper the exp function method is generalized to construct n soliton solutions of the 2 1 dimensional variable coefficient broer kaup vcbk system as a result new and more general 1 soliton 2 soliton and 3 soliton solutions with arbitrary functions are obtained from which the uniform formulae of n soliton solutions are derived based on one of the obtained 2 soliton solution the fission and fusion phenomena between the peakons and solitary waves are investigated it is shown that the exp function method may provide us with a straightforward effective and alternative mathematical tool for generating n soliton solutions of nonlinear evolution equations in mathematical physics

on nonlinear population waves we discuss a model system of partial differential equations for description of the spatio temporal dynamics of interacting populations we are interested in the waves caused by migration of the populations we assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations for the particular case of one population and one spatial dimension the general model is reduced to analytically tractable pde with polynomial nonlinearity up to 4th order we investigate this particular case and obtain two kinds of solutions i approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and ii exact solutions which describe nonlinear kink and solitary waves in an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node finally we derive conditions for the asymptotic stability of the obtained solutions

backlund transformation and special solutions for the drinfeld sokolov satsuma hirota system of coupled equations using the weiss method of truncated singular expansions we construct an explicit btransformation of the drinfeld sokolov satsuma hirota system into itself then we find all the special solutions generated by this transformation from the trivial zero solution of this system

kullback leibler average consensus on probability densities and distributed state estimation with guaranteed stability this paper addresses distributed state estimation over a sensor network wherein each node equipped with processing communication and sensing capabilities repeatedly fuses local information with information from the neighbors estimation is cast in a bayesian framework and an information theoretic approach to data fusion is adopted by formulating a consensus problem on the kullback leibler average of the local probability density functions pdfs to be fused exploiting such a consensus on local posterior pdfs a novel distributed state estimator is derived it is shown that for a linear system the proposed estimator guarantees stability i e mean square boundedness of the state estimation error in all network nodes under the minimal requirements of network connectivity and system observability and for any number of consensus steps finally simulation experiments demonstrate the validity of the proposed approach

nonclassical symmetries as special solutions of heir equations in phys d 78 1994 124 we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the given equation in the present paper we show that special solutions of the right order heir equation correspond to classical and nonclassical symmetries of the original equations an infinite number of nonlinear equations which possess nonclassical symmetries are derived

non isospectral problem in 2 1 dimensions in this paper the singular manifold method allows us to obtain a non isospectral lax pair darboux transformations and miura transformations for an equation in 2 1 dimensions and its modified version in this way we can iteratively build different kinds of solutions with solitonic behaviour

extended soliton solutions for the kaup kupershmidt equation we extend the n soliton solutions of the kaup kupershmidt equation on a nonzero background decreasing as x 1 a 2 these new solutions describe the interaction of n solitary waves with a static bell shaped wave we give the conditions so that the backlund transformation relating those solutions and the n solitons of the sawada kotera equation will be satisfied

brief paper moving horizon partition based state estimation of large scale systems this paper presents three novel moving horizon estimation mhe methods for discrete time partitioned linear systems i e systems decomposed into coupled subsystems with non overlapping states the mhe approach is used due to its capability of exploiting physical constraints on states and noise in the estimation process in the proposed algorithms each subsystem solves reduced order mhe problems to estimate its own state and different estimators have different computational complexity accuracy and transmission requirements among subsystems in all cases proper tuning of the design parameters i e the penalties on the states at the beginning of the estimation horizon guarantees convergence of the estimation error to zero numerical simulations demonstrate the viability of the approach

the two singular manifold method ii classical boussinesq system for pt i see musette et al ibid vol 24 p 3895 1994 the two singular manifold method a generalization of the singular manifold method of weiss 1983 is applied to the classical boussinesq system also known as the broer kaup system from the point of view of its singularity analysis the important feature of this system is the existence of two principal families with opposite principal parts the usual singular manifold method takes into account only one of these families at a time our generalization takes into account both families and in this way we are able to derive the lax pair and darboux transformation and hence the auto backlund transformation for the classical boussinesq system from its painleve analysis

on n 3 super korteweg de vries equation a one parameter family is constructed of n 3 supersymmetric and so 3 symmetric extensions of the korteweg de vries kdv equation as a hamiltonian flow on a n 3 superconformal algebra and it is argued that it is nonintegrable for any choice of the parameter then a modified n 3 super kdv equation is proposed which possesses the higher order conserved quantities and so is a candidate for an integrable system upon reduction to n 2 it yields the recently discussed would be integrable version of the n 2 super kdv equation in the bosonic core it contains a coupled system of the kdv type equation and a three component generalization of the mkdv equation a hamiltonian formulation of the new n 3 super kdv equation as a flow on some contraction of the direct sum of two n 3 superconformal algebras is given