mastersymmetries angle variables and recursion operator of the relativistic toda lattice conserved quantities bi hamiltonian formulation and recursive structure of the relativistic toda lattice rt are obtained in an algorithmic way without making use of the lax representation furthermore for the multisoliton solutions the gradients of the angle variables are described in terms of mastersymmetries a new hierarchy of completely integrable systems is discovered which turns out to correspond to the negative of the hierarchy of rt thus it is shown that the full algebra of time dependent symmetry group generators for each member of the rt hierarchy is isomorphic to the algebra of first order differential operators with laurent polynomials as coefficients the surprising phenomenon is revealed that the members of the rt hierarchy are connected to their negative counterparts by explicit backlund transformation

linear bundles of lie algebras and their applications we consider the algebraic properties of families of lie brackets also called linear bundles of lie algebras and especially ones that can be defined over the classical algebras some applications to the theory of the integrable systems are given as such bundles naturally define compatible poisson lie kirillov tensors

fractals in physics applications and theoretical developments abstract we report about the developments in two subjects in which we have been recently active the first one refers to a reanalysis of the distribution of galaxies and clusters with the methods of modern statistical mechanics contrary to the usual picture of a relatively smooth distribution at large scale we find that fractal and multifractal properties extend up to the observational limits this result should have deep implications also in relation to the recently observed anisotropy of the microwave background radiation the second subject concerns the attempts to develop a theory of fractal growth based on the new concept of the fixed scale transformation we discuss the instructive example of the percolating cluster in which the fst program can be fully implemented we then compare this example with the more complex cases of dla and dbm for which the identification of the scale invariant growth rules is much more complex in this respect the fst points to the key concept that is the scale invariance of screening and noise reduction the state of the art and the main open problems will be critically discussed

poisson structures of calogero moser and ruijsenaars schneider models we examine the hamiltonian structures of some calogero moser and ruijsenaars schneider n body integrable models we propose explicit formulations of the bi hamiltonian structures for the discrete models and field theoretical realizations of these structures we discuss the relevance of these realizations as collective field theory for the discrete models

magri morosi gel fand dorfman s bi hamiltonian constructions in the action angle variables a constructive method of transforming a completely integrable in liouville s sense hamiltonian system into magri morosi gel fand dorfman s mmgd bi hamiltonian form is presented the approach is carried out by making use of the action angle coordinates the classical kepler problem is shown to be a mmgd bi hamiltonian system explicit plethoras of higher order conserved quantities for the kepler problem are derived by employing oevel s method based on the existence of the mmgd bi hamiltonian representation

the hamiltonian structure of the nonabelian toda hierarchy we show that a subset of the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger operator is endowed with a hamiltonian structure and exhibits an infinite number of integrals of motion in involution we also establish the relation between these integrals of motion and the transmission coefficient of the underlying linear problem and show that such a relation implies that for the whole class previously introduced there exists an infinite number of conserved quantities

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

invasion percolation as a fractal growth problem we apply the fixed scale transformation fst method developed for laplacian fractal growth to compute the fractal dimension generated by invasion percolation ip with and without trapping ip is a growth process with quenched random variables we show how to define the corresponding stochastic growth process that corresponds to the quenched average of the growth process with respect to the random variables this stochastic process turns out to have a long memory and to depend on the entire history of the system one can then compute the fst matrix elements for open and closed boundary conditions and from these the fractal dimensions of the clusters up to fourth order terms we obtain d 1 8872 for ip the exact value is d 9148 1 8958 and d 1 8196 for ip with trapping simulations give d1 82 these results provide an important test for the validity of the fst method as a new tool to study fractal structures generated by both equilibrium and irreversible growth processes

upper and lower solution methods for fully nonlinear boundary value problems abstract sufficient conditions are given for the existence of a solution of a fourth order nonlinear boundary value problem with nonlinear boundary conditions the conditions assume the existence of a strong upper solution lower solution pair a concept that is defined in the paper the differential equation has nonlinear dependence on all lower order derivatives of the unknown in particular appropriate nagumo conditions are obtained and employed

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

existence and uniqueness of solutions of second order three point boundary value problems with upper and lower solutions in the reversed order abstract this paper studies the existence and uniqueness of solutions of second order three point boundary value problems with lower and upper solutions in the reversed order obtains the sufficient conditions for the existence and uniqueness of solutions by use of the monotone iterative method and gives the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution

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

from dispersionless to soliton systems via weyl moyal like deformations the formalism of quantization deformation is reviewed and the weyl moyal like deformation is applied to systematic construction of the field and lattice integrable soliton systems from poisson algebras of dispersionless systems

banach poisson lie groups and bruhat poisson structure of the restricted grassmannian the first part of this paper is devoted to the theory of poisson lie groups in the banach setting our starting point is the straightforward adaptation of the notion of manin triples to the banach context the difference with the finite dimensional case lies in the fact that a duality pairing between two non reflexive banach spaces is necessary weak as opposed to a strong pairing where one banach space can be identified with the dual space of the other the notion of generalized banach poisson manifolds introduced in this paper is compatible with weak duality pairings between the tangent space and a subspace of the dual we investigate related notion like banach lie bialgebras and banach poisson lie groups suitably generalized to the non reflexive banach context the second part of the paper is devoted to the treatment of particular examples of banach poisson lie groups related to the restricted grassmannian and the kdv hierarchy more precisely we construct a banach poisson lie group structure on the unitary restricted banach lie group which acts transitively on the restricted grassmannian a dual banach lie group consisting of a class of upper triangular bounded operators admits also a banach poisson lie group structure of the same kind we show that the restricted grassmannian inherits a generalized banach poisson structure from the unitary banach lie group called bruhat poisson structure moreover the action of the triangular banach poisson lie group on it is a poisson map this action generates the kdv hierarchy and its orbits are the schubert cells of the restricted grassmannian

structural properties of the noncommutative kdv recursion operator the present work studies structural properties of the recursion operator of the noncommutative kdv equation as the main result it is proved that this operator is hereditary the notion of hereditary operators was introduced by fuchssteiner for infinite dimensional integrable systems building on classical concepts from differential topology as an illustration for the consequences of this property it is deduced that the flows of the noncommutative kdv hierarchy mutually commute

boundary value problems involving upper and lower solutions in reverse order in this paper we discuss the existence of extreme solutions of the boundary value problem for a class of first order functional equations with a nonlinear boundary condition in the presence of a lower solution a and an upper solution b with email protected email protected we establish existence results of extreme solutions by using the method of upper and lower solutions and a monotone iterative technique

monotone method for the neumann problem with lower and upper solutions in the reverse order this work is devoted to the study of the existence and approximation of solutions of the problem u f t u u u a u b 0 in the presence of lower and upper solutions in the reverse order to this end we prove anti maximum principles for the nonlinear operator u 2k u email protected

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

backlund transformations for the kp and mkp hierarchies with self consistent sources using gauge transformations for the corresponding generating pseudo differential operators ln in terms of eigenfunctions and adjoint eigenfunctions we construct several types of auto backlund transformations for the kp hierarchy with self consistent sources kphscs and mkp hierarchy with self consistent sources mkphscs respectively the backlund transformations from kphscs to mkphscs are also constructed in this way

invasion percolation a new form of percolation theory a new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system the model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance the name invasion percolation is proposed for this new process similarities to and differences from ordinary percolation theory are discussed

the monotone method for periodic differential equations with the non well ordered upper and lower solutions in this paper we consider the following periodic problem x t g t x t x t a e t a b x a x b x a x b the unique solution is obtained by constructing an auxiliary periodic system with bounded solutions and proving these bounds to be equal in which case they are the expected unique solution

novel position space renormalization group for bond directed percolation in two dimensions abstract a new position space renormalization group approach is investigated for bond directed percolation in two dimensions the threshold value for the bond occupation probabilities is found to be p c 0 6443 correlation length exponents on time parallel and space transverse directions are found to be 1 719 and 1 076 respectively which are in very good agreement with the best known series expansion results

percolation as a fractal growth problem abstract we consider percolation in two dimensions as a fractal growth problem and apply to it the theory of fractal growth based on the fixed scale transformation approach developed for diffusion limited aggregation and the dielectric breakdown model this represents an important test for this new theoretical method based on an additional invariance property with respect to the renormalization group we compute the fractal dimension of the percolating cluster including terms up to third order the result is d 1 8830 for the square lattice and d 1 8650 for the triangular lattice these values are in excellent agreement with the universal exact result d 91 48 1 8958 and also show the potential of this new method for standard problems

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

temporal development of invasion percolation the authors consider the dynamics of invasion percolation for different models this process exhibits scaling behaviours which in the simplest case can be related to some critical static properties of usual percolation in particular the concept of bursts is shown to be of fundamental importance the critical exponents they derive are checked numerically on a case of invasion percolation where growth is restricted to the hull of the existing cluster

local and non local multiplicative poisson vertex algebras and differential difference equations we develop the notions of multiplicative lie conformal and poisson vertex algebras local and non local and their connections to the theory of integrable differential difference hamiltonian equations we establish relations of these notions to q deformed w algebras and lattice poisson algebras we introduce the notion of adler type pseudodifference operators and apply them to integrability of differential difference hamiltonian equations

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

r structures yang baxter equations and related involution theorems there are several ways to construct functions in involution on a lie bi algebra a lie algebra equipped with a second lie bracket for the solvable systems associated to the casimir functions a second hamiltonian formulation can be constructed and a class of bi hamiltonian korteweg de vries like evolution equations with explicit space dependence is derived translating the casimir functions with the flow of a special vector field yields another set of functions in involution lenard relations are found for the corresponding hamiltonian systems finally solutions of the classical yang baxter equation lead to an analog of compatible hamiltonian pairs the invariants of the resulting hereditary operators are in involution

fixed scale transformation for ising and potts clusters the fractal dimension of ising and potts clusters are determined via the fixed scale transformation approach which exploits both the self similarity and the dynamical invariance of these systems at criticality the results are easily extended to droplets a discussion of the interrelationships between the present approach and renormalization group methods as well as glauber type dynamics is provided

comment on the run time statistics in models of growth in disordered media we point out an error in the equations for the evolution of the run time statistics given in a paper by marsili and give the correct expression moreover we discuss theannealing approximation which is implicit in the transformation

extremal solutions for third order nonlinear problems with upper and lower solutions in reversed order abstract this paper deals with the existence of extremal solutions for the third order nonlinear boundary value problem u t f t u t t a b u a a u a b u b c in the presence of a pair of lower and upper solutions in reversed order here r r is an increasing homeomorphism f a b r r is a carathedory function and a b c r the proof follows from monotone iterative techniques which are based on suitable anti maximum principles for adequate operators to deduce such results we study some related problems coupled with boundary value conditions of the form p 0 u a q 0 u a a p 1 u b q 1 u b b u a c and u a a u a b u b c

nonlocal quadratic poisson algebras monodromy map and bogoyavlensky lattices a new lax representation for the bogoyavlensky lattice is found and its r matrix interpretation is elaborated the r matrix structure turns out to be related to a highly nonlocal quadratic poisson structure on a direct sum of associative algebras the theory of such nonlocal structures is developed and the poisson property of the monodromy map is worked out in the most general situation some problems concerning the duality of lax representations are raised

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

non pfaffian quasi bi hamiltonian systems with two degrees of freedom in the case of two degrees of freedom a non pfaffian quasi bi hamiltonian system with a separable integrating factor is presented in terms of nijenhuis coordinates and its separability is proved indications are given in the case of a general integrating factor

the monotone method for neumann functional differential equations with upper and lower solutions in the reverse order abstract in this paper we show that the monotone technique produces two monotone sequences that converge uniformly to extremal solutions of second order functional differential equations and laplacian equations with neumann boundary value conditions moreover we obtain existence results assuming upper and lower solutions in the reverse order

fixed scale transformation applied to diffusion limited aggregation and dielectric breakdown model in three dimensions abstract we extend the method of the fixed scale transformation fst to the case of fractal growth in three dimensions and apply it to diffusion limited aggregation and to the dielectric breakdown model for different values of the parameter the scheme is formally similar to the two dimensional case with the following technical complications i the basis configurations for the fine graining process are five instead of two and consist of 2 2 cells ii the treatment of the fluctuations of boundary conditions is far more complex and requires new schemes of approximations in order to test the convergency of the theoretical results we consider three different schemes of increasing complexity for dbm in three dimensions the computed values of the fractal dimension for 1 2 and 3 result to be in very good agreement with corresponding values obtained by computer simulations these results provide an important test for the fst method as a new theoretical tool to study irreversible fractal growth

theory of laplacian fractals diffusion limited aggregation and dielectric breakdown model abstract we describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the laplace equation and a stochastic field this new theory provides a systematic method for the calculation of the fractal dimension d and of the multifractal spectrum of the growth probability a detailed application to the dielectric breakdown model and diffusion limited aggregation in two dimensions is presented our approach exploits the scale invariance of the laplace equation that implies that the structure is self similar both under growth and scale transformation this allows one to introduce a fixed scale transformation instead of coarse graining as in the renormalization group theory that defines a functional equation for the fixed point of the distribution of basic diagrams used in the coarse graining process for the calculation of the matrix elements of this transformation one has to consider an infinite but rapidly convergent number of processes that occurs outside a considered diagram

samples of noncommutative products in certain differential equations a set of associative noncommutative products is considered in different differential equations of the ordinary and partial types a method of separation of variables is considered for a large set of those systems the products involved include for example some products and some products based on nijenhuis tensors which are embedded in the differential equations of the laplace poisson lax and schr dinger styles a comment on the products of reshetikhin jambor sykora type is also given in relation to products of vey type

existence and comparison results for difference φ laplacian boundary value problems with lower and upper solutions in reverse order abstract this paper is devoted to the study of the existence and comparison results for nonlinear difference laplacian problems with mixed dirichlet neumann and periodic boundary value conditions we deduce existence of extremal solutions of periodic and neumann boundary value problems lying between a pair of lower and upper solutions given in reverse order we prove the optimality of some assumptions in