on the treves theorem for the ablowitz kaup newell segur equation according to a theorem of treves the conserved functionals of the ablowitz kaup newell and segur akns equation vanish on all pairs of formal laurent series q r of a specified form both of them with a pole of the first order we propose a new and very simple proof for this statement based on the theory of backlund transformations using the same method we prove that the akns conserved functionals vanish on other pairs of laurent series the spirit is the same as in our previous paper on the treves theorem for the korteweg de vries hierarchy with some nontrivial technical differences

internal deformation of lie algebroids and symplectic realizations given a lie algebroid and a bundle over its base which is endowed with a localizable poisson structure and a flat connection we construct an extended bundle whose dual is endowed with an almost poisson structure that is a quadratic poisson structure when a certain compatibility property is satisfied this new formalism on lie algebroids describes systems with internal degrees of freedom

the r matrix theory and the reduction of poisson manifolds the relation between the r matrix approach to the construction of bi hamiltonian structures and the reduction of poisson manifolds is discussed the crucial role of the reduction process is emphasized by showing on the basis of specific examples that the knowledge of a r matrix may not be sufficient the reduction of the poisson structures related with r from the entire algebra to the lax submanifold may become an obstruction in constructing the bi hamiltonian structures of a specific hierarchy of integrable equations

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

a geometrical formulation of fermionic integrable systems a fermionic hamiltonian system is formulated on a supermanifold it is shown that if the system possesses a bi hamiltonian structure one can naturally define a lax equation associated with a 1 1 tensor on this supermanifold and this allows one to construct a set of conserved quantities furthermore if the corresponding nijenhuis tensor vanishes it is shown that all these conserved quantities would be in involution which is a sufficient condition for integrability of the system the fermionic extension of the kdv equation with a bi hamiltonian structure within this geometrical approach is studied

an extension of knopp s core theorem abstract in s l devi j london math soc 2 12 1976 397 401 an inequality sharper than that of knopp s core theorem has been established the purpose of this note is to generalize knopp s core theorem as well as the results obtained by devi

backlund transformations for the isospectral and nonisospectral akns hierarchies by converting the usual lax pairs for the isospectral and nonisospectral akns hierarchies into lax pairs in riccati forms a unified explicit form of backlund transformations for these hierarchies of nonlinear evolution equations can be obtained in the isospectral case it is an auto backlund transformation however in the nonisospectral case it is not an auto backlund transformation

σ core and i core of bounded sequences in this paper we characterize matrices that map every bounded sequence into one whose core is a subset of the i core of the original sequence

a fully supersymmetric akns theory we construct a fully supersymmetric bihamiltonian theory in four superfields admitting zero curvature and lax formulation this theory is an extension of the classical akns which can be recovered as a reduction other supersymmetric theories are obtained as reductions of the susy akns namely a nonlinear schrodinger a modified kdv and the manin radul kdv the susy nonlinear schrodinger hierarchy is related to the one of roelofs and kersten we determine its bihamiltonian and lax formulation finally we show that the susy kdv s mentioned before are related through a susy miura map

recursion operator and bi hamiltonian structure for integrable multidimensional lattices the recursion operator and the bi hamiltonian structure for an integrable two dimensional version of the toda chain are algorithmically derived from the corresponding linear problem the intimate relation between two dimensional theories and non abelian one dimensional theories is emphasized the analogies and differences between discrete and continuous integrable two dimensional systems are pointed out and discussed

non perturbative linearization of dynamical systems we consider the classical problem of linearizing a vector field x around a fixed point we adopt a non perturbative point of view based on the symmetry properties of linear vector fields

bi differential calculi bi hamiltonian systems and conformal killing tensors the theory of dimakis and m ller hoissen dimakis a and m ller hoissen f 2000 j phys a math gen 33 957 74 concerning bi differential calculi and completely integrable systems is related to bi hamiltonian systems of the poisson nijenhuis type in the special case where the ambient manifold is a cotangent bundle one is able to recover and elucidate the theory of ibort et al ibort a magri f and marmo g 2000 j geom phys 33 210 23 which is in turn a reworking in the bi hamitonian context of benenti s theory of hamilton jacobi separable systems in particular it is shown that benenti s conformal killing tensor which is central to his theory has an even more special form than has hitherto been realized and that when it is converted into a field of endomorphisms by raising an index with the ambient metric it necessarily has vanishing nijenhuis torsion

r matrices and higher poisson brackets for integrable systems the tri hamiltonian nature of lax equations is revealed starting with an r matrix on an associative algebra g equipped with a trace form there are g compatible poisson brackets with linear quadratic and cubic dependence on the coordinates the invariant functions casimir functions on g are in involution relative to these brackets they yield a hierarchy of integrable tri hamiltonian lax equations the results can be applied to solvable pde s such as the korteweg de vries equation as well as to finite integrable systems such as the toda lattice in these cases the poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well known recursion operators

statistical core theorems abstract based on the recently introduced concept of a statistical cluster point a definition is given for the statistical core of a complex number sequence which is analogous to the knopp core the theorems proved here are statistical analogues of theorems about the knopp core concerning its containment or invariance under a matrix transformation

optimal control of harvesting for single population the aim of this paper is to study the optimal harvesting policies for single population model that is to say we assume that the objective of the harvester is to find a harvesting plan making the harvesting yield maximum or maintaining the constant yield and making the terminal population level maximum when the harvesting has happened in a rigid time interval

schouten tensor and bi hamiltonian systems of hydrodynamic type the necessary conditions are derived for existence of a bi hamiltonian structure for a given hydrodynamic type system one of the conditions is the vanishing of its haantjes tensor a theorem is proved on the canonical forms of the generic bi hamiltonian systems the schouten 2 1 tensor sijk is connected with any hamiltonian system of hydrodynamic type the complete symmetry of the 3 0 tensor sijk is demonstrated the necessary conditions for existence of a single nondegenerate hamiltonian structure are obtained in terms of the special differential k forms p1 pk u1 uk

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

nijenhuis g manifolds and lenard bicomplexes a new approach to kp systems we suggest a method to extend the theory of recursion operators to integrable hamiltonian systems in two space dimensions like kp systems the approach aims to stress the conceptual unity of the theories in one and two space dimensions a sound explanation of the appearance of bilocal operators is also given

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

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

persistence and extinction in general non autonomous logistic model with delays and stochastic perturbation this is a continuation of the paper liu and wang 2012 14 firstly we consider a general non autonomous logistic model with delays and stochastic perturbation then sufficient conditions for extinction are established as well as nonpersistence in the mean weak persistence and stochastic permanence the threshold between weak persistence and extinction is obtained finally numerical simulations are introduced to support the theoretical analysis results

algorithmic construction of o 3 chiral field equation hierarchy and the landau lifshitz equation hierarchy via polynomial bundle we investigate new polynomial hierarchies of lax pairs which contain the polynomial pairs for the system of o 3 chiral field equations and landau lifshitz equation introduced recently and give an algorithmic construction of the corresponding hierarchies of soliton equations we compare the landau lifshitz equation hierarchy obtained via a polynomial bundle with the hierarchy obtained via an elliptic bundle

on a theorem by treves according to a theorem by treves duke math j dumjao 108 251 2001 the conserved functionals of the kdv equation vanish on each formal laurent series 1 x2 u0 k 2 ukxk we propose a new very simple geometrical proof for this statement

optimal harvesting of a stochastic mutualism model with levy jumps in this letter a stochastic mutualism model with levy jumps and harvesting is considered under some simple assumptions sufficient and necessary criteria for the existence of optimal harvesting policy are established the optimal harvesting effort and the maximum of sustainable yield are also obtained the effects of random noises on the optimal harvesting of the model are discussed and some numerical simulations are introduced to illustrate the main results

on the theory of recursion operator the general structure and properties of recursion operators for hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered weak and strong recursion operators are introduced the conditions which determine weak and strong recursion operators are found

optimal harvesting of a stochastic delay logistic model with levy jumps the optimal harvesting problem of a stochastic time delay logistic model with levy jumps is considered in this article we first show that the model has a unique global positive solution and discuss the uniform boundedness of its pth moment with harvesting then we prove that the system is globally attractive and asymptotically stable in distribution under our assumptions furthermore we obtain the existence of the optimal harvesting effort by the ergodic method and then we give the explicit expression of the optimal harvesting policy and maximum yield

a bioeconomic modeling of two prey and one predator fishery model with optimal harvesting policy through hybridization approach abstract this paper introduces an imprecise prey predator model by using interval and fuzzy number in modeling of prey and predator interaction in a prey predator system values of the biological parameter are based on the collection of data by the experts from nature therefore we consider them as interval numbers the proposed prey predator model is consisting of two species which are not only affected by the harvesting of the species but also by the present of predator species all possible existence of the biological and bionomic equilibrium points of the model are discussed under impreciseness we derive the conditions for local and global stability the optimal harvesting policy is studied by considering instantaneous annual rate of discount under fuzziness by logically hybridizing the interval and fuzzy number methodologies we find the optimal equilibrium points and harvesting efforts in the form of interval finally numerical examples are illustrated to support the proposed approach

modelling and analysis of a harvested prey predator system incorporating a prey refuge the present paper deals with a prey predator model incorporating a prey refuge and independent harvesting in either species our study shows that using the harvesting efforts as controls it is possible to break the cyclic behaviour of the system and drive it to a required state the possibility of existence of bionomic equilibria has been considered the problem of optimal harvest policy is then solved by using pontryagin s maximal principle

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

an alternate characterization of integrability this paper will show that the existence of at least three independent symplectic forms related in a simple way on the phase space of a dynamical system is a sufficient condition for the integrability of the system

bi hamiltonian formulation of the o 3 chiral fields equations hierarchy via a polynomial bundle we investigate the hamiltonian structures of some hierarchy of evolution equations related to a polynomial bundle over the algebra sl 4 the bundle contains the polynomial lax pair for the o 3 chiral fields system and for that reason the corresponding hierarchy of nonlinear evolution equations is called the cf hierarchy it is known that the hamiltonian properties of the cf hierarchy may be explained as a consequence of the existence of compatible poisson structures arising from a different lie algebra structure defined over so 4 we show that the generating operator for the cf hierarchy can be regarded as a nijenhuis tensor on the manifold of potentials and then naturally this manifold is equipped with the poisson nijenhuis structure

analysis of non autonomous stochastic gompertz model with delay abstract this paper presents the analysis of behavior of stochastic gompertz model with delay we prove existence and uniqueness of the global positive solution of the considered model besides the conditions for species to be persistent are established as well as the conditions under which population becomes extinct finally numerical illustration with real life example is carried out to confirm our theoretical results

integrable dynamical systems with hierarchy i formulation it is shown that any model with the zero generalized nijenhuis tensor under some additional assumptions will automatically satisfy the hierarchy equation with an infinite number of conserved quantities in involution especially if there is a dual symplectic structure with zero nijenhuis tensor then there exist infinite numbers of poisson brackets and of lagrangians giving the same equation of motion toda lattice is one such example

optimal harvesting policy for stochastic logistic population model abstract in this paper we consider some optimal harvesting policies for single population models in which the harvest effort and the intrinsic growth rate are disturbed by environment noises we choose the maximum sustainable yield and the maximum retained profits as two management objectives and obtain the optimal harvesting policies respectively for the two objectives we give the optimal harvest effort that maximizes the sustainable yield or retained profits the maximum of expectation of sustainable yield or retained profits and the corresponding variance their explicit expressions are determined by the coefficients of equation and the disturbance intensity

lax nijenhuis operators for integrable systems the relationship between lax and bi hamiltonian formulations of dynamical systems on finite or infinite dimensional phase spaces is investigated the lax nijenhuis equation is introduced and it is shown that every operator that satisfies that equation satisfies the lenard recursion relations while the converse holds for an operator with a simple spectrum explicit higher order hamiltonian structures for the toda system a second hamiltonian structure of the euler equation for a rigid body in n dimensional space and the quadratic adler gelfand dickey structure for the kdv hierarchy are derived using the lax nijenhuis equation

block diagonalizability problem for hydrodynamic type systems the necessary and sufficient conditions are derived for the reducibility of a hydrodynamic type system into a block diagonal form with k mutually interacting blocks applications to the perturbations of the benney system and to the hamiltonian systems of partial differential equations are presented the algebraic identities connecting the nijenhuis tensors nb a u v and na u v and haantjes tensors hb a u v and ha u v are discovered

on a new class of completely integrable nonlinear wave equations ii multi hamiltonian structure the multi hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed infinitely many conservation laws had earlier been obtained for these equations starting from a primary hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi hamiltonian structure are obtained and it is shown that the second hamiltonian operator can be constructed solely through a knowledge of the first hamiltonian function the recursion operator which first appears at the level of bi hamiltonian structure gives rise to an infinite sequence of conserved hamiltonians it is found that in general there exist two different infinite sequences of conserved quantities for these equations the recursion relation defining higher hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the k 1 st hamiltonian operator which depends on the kth hamiltonian function the infinite sequence of conserved hamiltonians are common to all the higher hamiltonian structures the equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri hamiltonian structure with two distinct infinite sets of conserved quantities the isothermal case of 1 is an exceptional one that requires separate treatment this corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to poisson s equation in nonlinear acoustics

optimal harvesting policy of logistic population model in a randomly fluctuating environment abstract in this paper we investigate the optimal harvest policy of a stochastic logistic population model the value of this study lies in two aspects mathematically we establish a stochastic threshold theorem to govern whether the population persists or not in the case of population persistence we prove the existence uniqueness and global asymptotic stability of the invariant density of the fokker planck equation associated with the sde model and we further show the relation between these two models in addition we give the unique optimal harvesting effort and the corresponding maximum of expectation of sustainable yield which gives us the profile of the optimal harvesting policy of the sde model ecologically we find that big harvesting effort or big intensity of noise will lead the population to extinct risk almost surely in addition we find that under a fixed randomness strategy and proper harvesting the maximum sustainable yield increases systematically as the harvesting effort increases but overexploitation will reduce the level of maximize sustainable yield and eventually make the whole population extinct with probability one hence in order to obtain the optimal harvesting policy we must decrease the harvesting effort and the intensity of noise furthermore we find that our parameter perturbation method in this paper is more beneficial to the exploitation of renewable resources than the classical one given by beddington and may 1977 the results show that different perturbation method can exhibit different stochastic population dynamics

factorization properties of birational mappings abstract we analyse birational mappings generated by transformations on q q matrices which correspond respectively to two kinds of transformations the matrix inversion and a permutation of the entries of the q q matrix remarkable factorization properties emerge for quite general involutive permutations it is shown that factorization properties do exist even for birational transformations associated with noninvolutive permutations of entries of q q matrices and even for more general transformation which are rational transformations but no longer birational the existence of factorization relations independent of q the size of the matrices is underlined the relations between the polynomial growth of the complexity of the iterations the existence of recursions in a single variable and the integrability of the mappings are sketched for the permutations yielding these properties all these results show that permutations of the entries of the matrix yielding factorization properties are not so rare in contrast the occurrence of recursions in a single variable or of the polynomial growth of the complexity are of course less frequent but not completely exceptional

convergence to normal forms of integrable pdes in an infinite dimensional hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields we prove that one can construct by the method of poincar e normal form a local analytic coordinate transformation near the origin transforming the family into a normal form the result applies to the kdv and nls equations and to the toda lattice with periodic boundary conditions one gets existence of birkhoff coordinates in a neighbourhood of the origin the proof is obtained by directly estimating in an iterative way the terms of the poincar e normal form and of the transformation to it through a rapid convergence algorithm

integrability condition and finite periodic toda lattice a further generalization to construct integrable dynamical systems with hierarchy has been developed based upon a generalization of the zero nijenhuis tensor condition in a symplectic manifold with double symplectic structures as an example the periodic toda lattice hamiltonian has been shown to satisfy these conditions the case of the three dimensional coulomb potential problem has also been analyzed in a similar fashion

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

bi quasi hamiltonian systems a general notion of bi quasi hamiltonian systems is introduced and is related to previous work on various special cases of such systems

poisson quasi nijenhuis manifolds we introduce the notion of poisson quasi nijenhuis manifolds generalizing poisson nijenhuis manifolds of magri morosi we also investigate the integration problem of poisson quasi nijenhuis manifolds in particular we prove that under some topological assumption poisson quasi nijenhuis manifolds are in one one correspondence with symplectic quasi nijenhuis groupoids as an application we study generalized complex structures in terms of poisson quasi nijenhuis manifolds we prove that a generalized complex manifold corresponds to a special class of poisson quasi nijenhuis structures as a consequence we show that a generalized complex structure integrates to a symplectic quasi nijenhuis groupoid recovering a theorem of crainic

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

separable systems with quadratic in momenta first integrals the separability theory of hamiltonian systems on riemannian manifolds is reviewed and developed particular attention is paid to the systems generated by the so called special conformal killing tensors i e benenti systems then infinitely many new classes of separable systems are constructed by appropriate deformations of benenti class systems

on the continuous limit of integrable lattices i the kac moerbeke system and kdv theory kdv theory is constructed systematically through the continuous limit of the kac moerbeke system the infinitely many commuting vector fields the conserved functionals the lax pairs and the bihamiltonian structure are recovered as the limits of suitably defined linear combinations of homologous objects for the kac moerbeke system the combinatorial aspects of this recombination method are treated in detail

recursion operators and bi hamiltonian formulations of the landau lifshitz equation hierarchies using a polynomial pencil of lax pairs we give a new bi hamiltonian formulation for the landau lifshitz equation hierarchy and find a new recursion operator related to it

global asymptotic stability of a stochastic delayed predator prey model with beddington deangelis functional response this paper is concerned with a stochastic predator prey system with beddington deangelis functional response and time delay sufficient conditions for global asymptotic stability of the positive equilibrium point are established the result demonstrates that the delay is harmless for global stability of the stochastic system in some cases some figures are given to validate the analytical results

decoupling problem for systems of quasi linear pde s the necessary and sufficient conditions for the decoupling of a quasi linear system of partial differential equations into k non interacting subsystems are derived several necessary conditions for the decoupling are found and applied to the benney system

r matrix theory formal casimirs and the periodic toda lattice the nonunitary r matrix theory and the associated bi and trihamiltonian schemes are considered the language of poisson pencils and of their formal casimirs is applied in this framework to characterize the bihamiltonian chains of integrals of motion pointing out the role of the schur polynomials in these constructions this formalism is subsequently applied to the periodic toda lattice some different algebraic settings and lax formulations proposed in the literature for this system are analyzed in detail and their full equivalence is exploited in particular the equivalence between the loop algebra approach and the method of differential difference operators is illustrated moreover two alternative lax formulations are considered and appropriate reduction algorithms are found in both cases allowing us to derive the multihamiltonian formalism from r matrix theory the systems of integrals for the periodic toda lattice known after flaschka and henon and their functional relations are recovered through systematic application of the previously outlined schemes