towards the classification of scalar nonpolynomial evolution equations quasilinearity we prove that for m 7 scalar evolution equations of the form u t f x t u u m which admit a nontrivial conserved density of order m 1 are linear in u m the existence of such conserved densities is a necessary condition for integrability in the sense of admitting a formal symmetry hence integrable scalar evolution equations of order m 7 admitting nontrivial conserved densities are quasilinear

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

a m isometric operators in semi hilbertian spaces abstract in this work the concept of m isometry on a hilbert space are generalized when an additional semi inner product is considered this new concept is described by means of oblique projections

tensor product of n isometries abstract a hilbert space operator a is an m isometry if k 0 m 1 k m k a m k a m k 0 we prove that the tensor product of an m isometry a with an n isometry b is an m n 1 isometry this proves a conjecture of botelho and jamison 5 remark 5 1

the evolution partial differential equation ut uxxx 3 uxxu2 3u2xu 3uxu4 the evolution equation ut uxxx 3 uxxu2 3u2xu 3uxu4 u u x t is integrable it can be exactly linearized by an appropriate change of dependent variable hence several explicit solutions of the partial differential equation pde can be exhibited some of them display a remarkable solitronic phenomenology

one symmetry does not imply integrability we show that bakirov s counter example which had been checked by computer algebra methods up to order 53 to the conjecture that one nontrivial symmetry of an evolution equation implies infinitely many is indeed a counter example to prove this we use thesymbolic methodof gel fand dikii andp adic analysis we also formulate a conjecture to the effect that almost all equations in the family considered by bakirov have at most finitely many symmetries this conjecture depends on the solution of a diophantine problem which we explicitly state

generalized schrodinger equations and jordan pairs the criteria of integrability for the nonlinear schrodinger type systems are obtained one to one correspondence between such integrable systems and the jordan paris is established it turns out that irreducible systems correspond to simple jordan pairs an infinite series of generalized symmetries and local conservation laws for such systems are completely described

the noncommutative akns system projection to matrix systems countable superposition and soliton like solutions starting from the recent work on noncommutative akns systems for functions with values in the bounded operators on a banach space it is shown how their formal 1 soliton solution depending on operator parameters can be mapped to solutions of matrix akns systems the main result is rather general solution formulas for matrix akns systems the most important applications are the countable superposition of matrix solitons and explicit expressions for the soliton like solutions of the classical akns system

on the integrability of systems of second order evolution equations with two components this paper is devoted to classifying second order evolution equations with two components combining the symbolic method and number theory we give the complete list of such homogeneous polynomial symmetry integrable systems with non zero diagonal linear terms the technique is applicable for more general systems

on recursion operators abstract we observe that application of a recursion operator of the burgers equation does not produce the expected symmetries this is explained by the incorrect assumption that d x 1 d x 1 we then proceed to give a method to compute the symmetries using the recursion operator as a first order approximation

symmetries and integrability bakirov system revisited all local generalized symmetries including x t dependent ones of the bakirov system are found in particular it is shown that its only non lie point local generalized symmetry is the sixth order one found by bakirov this result generalizes a similar result of beukers sanders and wang on x t independent symmetries and completes the refutation of the popular conjecture stating that the existence of one non lie point local generalized symmetry for a 1 1 dimensional system of pdes implies the existence of infinitely many such symmetries

on the structure of 2 1 dimensional commutative and noncommutative integrable equations we develop the symbolic representation method to derive the hierarchies of 2 1 dimensional integrable equations from the scalar lax operators and to study their properties globally the method applies to both commutative and noncommutative cases in the sense that the dependent variable takes its values in c or a noncommutative associative algebra we prove that these hierarchies are indeed quasi local in the commutative case as conjectured by mikhailov and yamilov j phys a 31 6707 1998 we propose a ring extension in the noncommutative case based on the symbolic representation as examples we give noncommutative versions of kadomtsev petviashvili kp modified kadomtsev petviashvili mkp and boussinesq equations

on a functional equation related to the intermediate long wave equation we resolve an open problem stated by ablowitz et al 1982 j phys a math gen 15 781 concerning the integral operator appearing in the intermediate long wave equation we explain how this is resolved using the perturbative symmetry approach introduced by one of us with mikhailov by solving a certain functional equation we prove that the intermediate long wave equation and the benjamin ono equation are the unique integrable cases within a particular class of integro differential equations furthermore we explain how the perturbative symmetry approach is naturally extended to treat equations on a periodic domain

classification of certain integrable coupled potential kdv and modified kdv type equations in this article we describe the classification of integrable symmetrically coupled potential kdv and modified kdv type equations that possess higher symmetries restricting our attention to the systems that cannot be decoupled by a change of dependent variables we obtain 11 previously unknown classes of integrable equations in some cases we present hamiltonian or bi hamiltonian formulations

on the integrability of homogeneous scalar evolution equations abstract we determine the existence of infinitely many symmetries for equations of the formut uk f u uk 1 when they are homogeneous with respect to the scalinguk k with 0 algorithms are given to determine whether a system has a symmetry also independent oftandx if it has one generalized symmetry we prove it has infinitely many and these can be found using recursion operators or master symmetries the method of proof uses the symbolic method and results from diophantine approximation theory we list the 10 integrable hierarchies the methods can in principle be applied to the 0 case as we illustrate for one example with 0 which seems to be new in principle they can also be used for systems of evolution equations but so far this has only been demonstrated for one class of examples

extension of the module of invertible transformations classification of integrable systems we demonstrate that for the systems of equations which are invariant under a point group or possess conservation laws of the zeroth or first order a nontrivial extension of the module of invertible transformations is possible that simplifies greatly a classification of the integrable systems of equations here we present an exhaustive list and a classification of the second order systems of the formu t u xx f u v u x v x v t v xx g u v u x v x which possess the conservation laws of higher order the reduction group approach allows us to define the lax type representations for some new equations of our list

factorization of operators ii we extend the methods of a previous paper j math phys 21 2508 1980 factorizing the general scalar third order differential operator and obtain a miura transformation for the boussinesq equation we give a general factorized eigenvalue problem we also give a hamiltonian structure associated with the factorized eigenvalue problem we derive several isospectral flows some of klein gordon type

decomposing algebraic m isometric tuples abstract we show that any m isometric tuple of commuting algebraic operators on a hilbert space can be decomposed as a sum of a spherical isometry and a commuting nilpotent tuple our approach applies as well to tuples of algebraic operators that are hereditary roots of polynomials in several variables

linearizability for third order evolution equations the problem of linearization for third order evolution equations is considered criteria for testing equations for linearity are presented a class of linearizable equations depending on arbitrary functions is obtained by requiring presence of an infinite dimensional symmetry group linearizing transformations for this class are found using symmetry structure and local conservation laws a number of special cases as examples are discussed their transformation to equations within the same class by differential substitutions and connection with kdv and mkdv equations is also reviewed in this framework

on homogeneous distributed parameter systems a geometric homogeneity is introduced for evolution equations in a banach space scalability property of solutions of homogeneous evolution equations is proven some qualitative characteristics of stability of trivial solution are also provided in particular finite time stability of homogeneous evolution equations is studied theoretical results are illustrated on important classes of partial differential equations

level grading a new graded algebra structure on differential polynomials application to the classification of scalar evolution equations we define a new grading which we call the level grading on the algebra of polynomials generated by the derivatives uk i over the ring of c functions of x t u u1 uk where this grading has the property that the total derivative and the integration by parts with respect to x are filtered algebra maps in addition if u satisfies the evolution equation ut f u where f is a polynomial of order m k p and of level p then the total derivative with respect to t dt is also a filtered algebra map furthermore if the separant belongs to then the canonical densities i are polynomials of level 2i 1 and dt i is of level 2i 1 m we define kdv like evolution equations as those equations for which all the odd canonical densities are non trivial we use the properties of level grading to obtain a preliminary classification of scalar evolution equations of orders m 7 9 11 13 up to their dependence on x t u u1 and u2 these equations have the property that the canonical density 1 is where and are functions of x t u u1 u2 this form of 1 is shared by the essentially nonlinear class of third order equations and a new class of fifth order equations

master symmetries for volterra equation belov chaltikian and blaszak marciniak lattice equations it is shown how to derive master symmetries for nonlinear lattice equations systematically using the basic principles but without using either their zero curvature equations or the bi hamiltonian structure this has been illustrated for volterra equation two coupled belov chaltikian bc and three coupled blaszak marciniak bm lattice equations the existence of a sequence of master symmetries is one of the characteristics of completely integrable nonlinear partial differential and differential difference equations admitting hamiltonian structure

vortex filament on symmetric lie algebras and generalized bi schrodinger flows in this article we display an evolving model on symmetric lie algebras from a purely geometric way by using the hamiltonian or para hamiltonian gradient flow of a fourth order functional called generalized bi schrodinger flows which corresponds to the fukumoto moffatt s model in the theory of moving curves or the vortex filament in physical words in mathbb r 3 the theory of vortex filament in mathbb r 3 or mathbb r 2 1 up to the third order approximation is shown to be generalized to symmetric lie algebras in a unified way

integrable systems and their recursion operators in this paper we discuss the structure of recursion operators we show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries this enables us to compute recursion operators more systematically under certain conditions which hold for all examples known to us nijenhuis operators are well defined i e they give rise to hierarchies of infinitely many commuting symmetries of the operator moreover the nonlocal part of a nijenhuis operator contains the candidates of roots and coroots

a m isometries on hilbert spaces abstract a bounded linear operator t on a hilbert space h is called an a m isometry for some positive operator a on h and integer m if k 0 m 1 m k m k t k a t k 0 we give some properties of a m isometries in particular we focus on spectral properties and the relation between a m isometries and m isometries also we obtain some dynamic properties of a m isometries as a negative answer to 22 question 1 with an example of an a isometric which is n supercyclic and sufficient conditions for an a m isometry to be not n supercyclic moreover we prove that the perturbation of a m isometry by a bigger class than nilpotent operators is not n supercyclic

on the integrability of non polynomial scalar evolution equations we show the existence of infinitely many symmetries for homogeneous equations when 0 if the equation has one generalized symmetry we prove that it has infinitely many and these can be produced by recursion operators identifying equations under homogeneous transformations we find that the only integrable equations in this class are the potential burgers potential modified korteweg de vries and potential kupershmidt equations we can draw some conclusions from these results for the case 1 which although theoretically incomplete seem to cover the known integrable systems for this case

an isometry plus a nilpotent operator is an m isometry applications abstract we prove that if an isometry a and a nilpotent operator q of order n commute then a q is a strict 2 n 1 isometry as an application of the main result we prove that a q cannot be n supercyclic for any n also we find an m isometry with prescribed spectrum k where k is the closed unit disk or a compact subset of the unit circle

stability of stationary fronts in a non linear wave equation with spatial inhomogeneity we consider inhomogeneous non linear wave equations of the type u u v u x u 0 the spatial real axis is divided in intervals i i 0 n 1 and on each individual interval the potential is homogeneous i e v u x v u for x i by varying the lengths of the middle intervals typically one can obtain large families of stationary front or solitary wave solutions in these families the lengths are functions of the energies associated with the potentials v in this paper we show that the existence of an eigenvalue zero of the linearisation operator about such a front or stationary wave is related to zeroes of the determinant of a jacobian associated to the length functions furthermore the methods by which the result is obtained is fully constructive and can subsequently be used to deduce the stability and instability of stationary fronts or solitary waves as will be illustrated in examples 2012 elsevier inc

a family of wave equations with some remarkable properties we consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions recursion operators are found for two members of the family investigated for one of them a lax pair is also obtained proving its complete integrability from the lax pair we construct a miura type transformation relating the original equation to the korteweg de vries kdv equation this transformation on the other hand enables us to obtain solutions of the equation from the kernel of a schrodinger operator with potential parametrized by the solutions of the kdv equation in particular this allows us to exhibit a kink solution to the completely integrable equation from the 1 soliton solution of the kdv equation finally peakon type solutions are also found for a certain choice of the parameters although for this particular case the equation is reduced to a homogeneous second order nonlinear evolution equation

matrix korteweg de vries and modified korteweg de vries hierarchies noncommutative soliton solutions the present work continues work on kdv type hierarchies presented by s carillo and c schiebold noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods j math phys 50 073510 2009 10 1063 1 3155080 general solution formulas for the kdv and mkdv hierarchies are derived by means of banach space techniques both in the scalar and matrix case a detailed analysis is given of solitons breathers their countable superpositions as well as of multisoliton solutions for the matrix hierarchies

classification of hereditary matrices abstract a classical approach used to obtain basic facts in the theory of square matrices involves an analysis of the relationship between polynomials p in one variable and square matrices t such that p t 0 we consider matrices and operators which satisfy a different type of polynomial constraint let h be a complex hilbert space t be a bounded linear transformation of h t be the adjoint of t and c x y be the algebra of polynomials in x and y with complex coefficients for a polynomial p c x y in two variables with complex coefficients define p t m n 0 p m n t n t m where p m n is the coefficient of ynxm in the expansion of p in a power series about the point 0 0 t is called a root of p if and only if p t 0 note that if p c x y is a polynomial in the single variable x then the definition of p t given here agrees with the classical definition in this paper we study the relationships which p t 0 forces between p and t when t is an algebraic operator i e there exists n 1 and complex numbers a0 an 1 such that 0 a0 a1t an 1tn 1 tn the classification starts with the following observation suppose p c x y and an algebraic operator t l h satisfy p t 0 then certain subspaces of h which are invariant for t must be orthogonal or certain coefficients of p must vanish this leads to the notions of a graph attached to each p c x y and a graph attached to each square matrix t for diagonalizable t a necessary and sufficient graph theoretic condition for solving p t 0 is given for nondiagonalizable t this condition is necessary but not sufficient the use of these graphs does however reduce the problem to the problem of solving the equation p t 0 for t with exactly one or two eigenvalues for t with one eigenvalue we give a necessary and sufficient condition for solving p t 0 this leaves the case of solving p t 0 when t has exactly two eigenvalues this problem mixes algebra involving polynomials with matrix theory we show that it is equivalent to the purely algebraic problem of determining if equations of the form i j eci jxi r j s 0 have solutions of finite support with certain nonvanishing properties we call these equations bi hankel equations subordinate to a given subset e of the lattice of integer pairs i j 0 i m 1 0 j n 1 it turns out that there is an algorithm which uses grobner bases for determining if the type of solution we seek exists and for computing it

algebraic properties of operator roots of polynomials abstract properties of m selfadjoint and m isometric operators have been investigated by several researchers particularly interesting to us are algebraic properties of nilpotent perturbations of such operators mccullough and rodman showed in the nineties that if q n 0 and a is a selfadjoint operator commuting with q then the sum a q is a 2 n 1 selfadjoint operator very recently bermudez martinon and noda proved a similar result for nilpotent perturbations of isometries via a new approach we obtain simple proofs of these results and other generalizations to operator roots of polynomials

derivative nonlinear schrodinger equations and hermitian symmetric spaces it is shown that to each hermitian symmetric space there corresponds an integrable system of generalised derivative nonlinear schrodinger equations nls the nonlinear terms are related to the curvature tensor of the associated symmetric space the hamiltonian form of the equations is presented the results are an extension of those presented in an earlier paper 1983 on generalised nls equations associated with symmetric and reductive homogeneous spaces

integrable evolution equations on associative algebras this paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra which includes matrix clifford and group algebra valued systems a variety of new examples of integrable systems possessing higher order symmetries are presented symmetry reductions lead to an associative algebra valued version of the painleve transcendent equations the basic theory of hamiltonian structures for associative algebra valued systems is developed and the bihamiltonian structures for several examples are found

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

some results on higher order isometries and symmetries products and sums with a nilpotent operator abstract we study the sum of an m isometry or an m symmetric operator with a nilpotent operator we also study the product of two m isometries or two n symmetries we obtain several theorems generalizing previous work our method is straightforward and thus offers direct and simple insights into why these theorems are true

on tuples of commuting operators in positive semidefinite inner product spaces abstract this paper is concerned with the study of certain properties of operator tuples on a complex hilbert space h when a semi inner product induced by a positive operator a on h is considered in particular we show that r a t a t for every commuting operator tuple t t 1 t d such that each t k admits an a adjoint operator where r a t and a t denote respectively the a joint spectral radius and the a joint numerical radius of t this study allows to establish that r a t a t t a for every a normal commuting tuple of operators t where t a is denoted to be the a joint operator seminorm of t in addition the a joint spectral radius of an a m isometric tuple of commuting operators is studied

towards the complete classification of homogeneous two component integrable equations in this article we suggest an improved method for classifying general two component integrable evolution equations homogeneous in a given weighting scheme this method relies on linear changes of variables and on an appropriate splitting of the solution space to illustrate the method we implement the classification of coupled kdv type equations we show that there are five nontriangular systems possessing higher order generalized symmetries one of these systems is previously unknown this seems to be the first classification of coupled integrable equations homogeneous in a given weighting scheme without any restrictions on the form of the main matrix

classification of integrable b equations abstract we classify integrable equations which have the form u t a 1 u n k v 0 v 1 v t a 2 v n where a 1 a 2 c n n and k a quadratic polynomial in derivatives of v this is done using the symbolic calculus biunit coordinates and the lech mahler theorem furthermore we present a method based on resultants to determine whether an equation is in a hierarchy of lower order

classification of polynomial integrable systems of mixed scalar and vector evolution equations i we perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries we consider polynomial systems that are homogeneous under a suitable weighting of variables this paper deals with the kdv weighting the burgers or potential kdv or modified kdv weighting the ibragimov shabat weighting and two unfamiliar weightings the case of other weightings will be studied in a subsequent paper making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems we give the complete lists of second order systems with a third order or a fourth order symmetry and third order systems with a fifth order symmetry for all but a few systems in the lists we show that the system or at least a subsystem of it admits either a lax representation or a linearizing transformation a thorough comparison with recent work of foursov and olver is made

on the equivalence of linearization and formal symmetries as integrability tests for evolution equations gurses integrability test 1991 consists of the compatibility of the linearized equation with an eigenvalue equation and leads to the recursion operator this test is applied to quasilinear fifth order equations and the same classification as the formal symmetry method of mikhailov et al 1990 is obtained the same classification for polynomial equations is obtained using fokas test 1987 i e the existence of one higher order symmetry it is shown that the recursion operators of a specific form can be constructed using symmetries and conserved covariants and the recursion operators for polynomial equations are obtained with this method

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

classification of integrable one component systems on associative algebras this paper is devoted to the complete classification of integrable one component evolution equations whose field variable takes its values in an associative algebra the proof that the list of non commutative integrable homogeneous evolution equations is complete relies on the symbolic method each equation in the list has infinitely many local symmetries and these can be generated by its recursion recursive operator or master symmetry

on the classification of scalar evolution equations with non constant separant the separant of the evolution equation u t f where f is some differentiable function of the derivatives of u up to order m is the partial derivative where as an integrability test we use the formal symmetry method of mikhailov shabat sokolov which is based on the existence of a recursion operator as a formal series the solvability of its coefficients in the class of local functions gives a sequence of conservation laws called the conserved densities we apply this method to the classification of scalar evolution equations of orders for which and are non trivial i e they are not total derivatives and is not linear in its highest order derivative we obtain the top level parts of these equations and their top dependencies with respect to the level grading that we defined in a previous paper as a grading on the algebra of polynomials generated by the derivatives u b i over the ring of functions of in this setting b and i are called base and level respectively we solve the conserved density conditions to show that if depends on then these equations are level homogeneous polynomials in furthermore we prove that if is non trivial then with while if is trivial then where and and are functions of we show that the equations that we obtain form commuting flows and we construct their recursion operators that are respectively of orders 2 and 6 for non trivial and trivial respectively omitting lower order dependencies we show that equations with non trivial and b 3 are symmetries of the essentially non linear third order equation for trivial the equations with b 5 are symmetries of a non quasilinear fifth order equation obtained in previous work while for b 3 4 they are symmetries of quasilinear fifth order equations

partially integrable nonlinear equations with one higher symmetry in this letter we present a family of second order in time nonlinear partial differential equations which have only one higher symmetry these equations are not integrable but have a solution depending on one arbitrary function

new reductions of integrable matrix partial differential equations sp m invariant systems we propose a new type of reduction for integrable systems of coupled matrix partial differential equations this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix via this reduction we obtain a new integrable system of coupled derivative modified kdv equations and a new integrable variant of the massive thirring model in addition to the already known systems we also discuss integrable semidiscretizations of the obtained systems and present new soliton solutions to both continuous and semidiscrete systems as a by product a new integrable semidiscretization of the manakov model self focusing vector nonlinear schrodinger equation is obtained

classification of integrable polynomial vector evolution equations several classes of systems of evolution equations with one or two vector unknowns are considered we also investigate systems with one vector and one scalar unknown for these classes all equations having the simplest higher symmetry are listed

functional representations of integrable hierarchies we consider a general framework for integrable hierarchies in lax form and derive certain universal equations from which functional representations of particular hierarchies such as kp discrete kp mkp akns i e formulations in terms of functional equations are systematically and quite easily obtained the formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative typically matrix algebra the obtained functional representations can be understood as noncommutative analogues of fay identities for the kp hierarchy

a stability criterion for the non linear wave equation with spatial inhomogeneity abstract in this paper the non linear wave equation with a spatial inhomogeneity is considered the inhomogeneity splits the unbounded spatial domain into three or more intervals on each of which the non linear wave equation is homogeneous in such setting there often exist multiple stationary fronts in this paper we present a necessary and sufficient stability criterion in terms of the length of the middle interval s and the energy associated with the front in these interval s to prove this criterion it is shown that critical points of the length function and zeros of the linearisation have the same order furthermore the evans function is used to identify the stable branch the criterion is illustrated with an example which shows the existence of bi stability two stable fronts one of which is non monotonic the evans function also gives a sufficient instability criterion in terms of the derivative of the length function

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

joint numerical ranges of operators in semi hilbertian spaces abstract in this paper we aim to investigate the concept of numerical range and maximal numerical range relative to a positive operator of a d tuple of bounded linear operators on a hilbert space some properties and applications of these sets are studied mainly it is proved that they are convex for d 1 this generalizes the well known toeplitz hausdorff theorem 24 16 and stampfli s result 23 moreover under additional hypotheses we show that these sets are convex for d 2

on integrability of systems of evolution equations we prove the conjecture formulated in bsw98 that almost all systems in the family formula have at most finitely many symmetries by using number theory we list the nine exceptional cases when the systems do have infinitively many symmetries for such systems we give the recursive operators to generate their symmetries we treat both 1the commutative and the noncommutative or quantum cases this is the first example of a class of equations where such a classification has been possible

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

perturbative symmetry approach the aim of our paper is to formulate a perturbative version of the symmetry approach in the symbolic representation and to generalize it in order to make it suitable for the study of nonlocal and non evolution equations our formalism is the development and incorporation of the perturbative approach of zakharov and schulman the symbolic method of sanders and wang and the standard symmetry approach of shabat et al we apply our theory to describe integrable generalizations of the benjamin ono type equations and to isolate integrable cases of the camassa holm type equations