relativistic chern simons higgs vortex equations an existence theorem is established for the solutions to the non abelian relativistic chern simons higgs vortex equations over a doubly periodic domain when the gauge group g assumes the most general and important prototype form g su n mathematics subject classification 2010 35c08 35j50 35q70 81e13

harmonic maps and periodic toda systems harmonic maps from the point of view of periodic toda systems are considered the correspondence between certain harmonic immersions of surfaces and solutions to periodic toda systems related to nonexceptional simple lie algebras is constructed

periodic solutions with prescribed energy for singular conservative systems involving strong forces this paper establishes under appropriate conditions which are stated below the existence of periodic solutions of the equation ii t vv q t 0 1 1 where q t e rn n 1 2 and v i is the gradient of a real function i which has a singularity at the origin in the literature the problem of the existence of periodic solutions of 1 1 has been addressed following two different points of view i given t 0 find periodic solutions of 1 1 having t as period fixed period problem ii given h e r find periodic solutions of 1 1 such that for all t e r

multiple brake orbits for some classes of singular hamiltonian systems from which one can easily construct a periodic solution of period 2t by reflecting q around t 0 t if q x e e 1 v x hj is homeomorphic to a ball then the existence of one brake orbit is a classical result of seifert 11 generalizations of that result as well as alternative proofs were given in 2 6 141 in the case in which the topology of sz is richer then the existence of multiple solutions has been proved by bolotin and kozlov in 7 see also 8 in this paper we are mainly concerned with the case in which the potential v has singularities inside 2 in the sense that v x co as x x e ai and the topology of q is nontrivial more precisely in section 2 we shall assume that sz has a certain number of holes k and that v behaves like lxi o1 with 0 cx 2 near x 0 and we shall show that 1 3 has at least k classical brake orbits let us point out that the main interest of this result lies in the fact that we are able to prove that the solutions we find are classical ones that is they do not collide with the singularity while most of the papers dealing with existence of fixed energy 9 or fixed period lo solutions for singular potentials only prove existence of generalized solutions that is solutions which can collide with the singularity existence of a noncollision periodic solution for a singular potential

the relativistic non abelian chern simons equations we study ther xr system of nonlinear elliptic equations delta u a lambda sum limits b 1 r k ab e u b lambda sum limits b 1 r sum limits c 1 r e u b k ab e u c k bc 4 pi sum limits j 1 n a delta p aj a 1 2 r x r 2 where 0 is a constant parameter k kab is the cartan matrix of a semi simple lie algebra and p is the dirac measure concentrated atp r 2 this system of equations arises in the relativistic non abelian chern simons theory and may be viewed as a nonintegrable deformation of the integrable toda system we establish the existence of a class of solutions known as topological multivortices the crucial step in our method is the use of the decomposition theorem of cholesky for positive definite matrices so that a variational principle can be formulated

generalized drinfeld sokolov reductions and kdv type hierarchies generalized drinfeld sokolov ds hierarchies are constructed through local reductions of hamiltonian flows generated by monodromy invariants on the dual of a loop algebra following earlier work of de groot et al reductions based upon graded regular elements of arbitrary heisenberg subalgebras are considered we show that in the case of the nontwisted loop algebra l gl n graded regular elements exist only in those heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn pr or to equal numbers plus onen pr 1 we prove that the reduction belonging to the grade 1 regular elements in the casen pr yields thep p matrix version of the gelfand dickeyr kdv hierarchy generalizing the scalar casep 1 considered by ds the methods of ds are utilized throughout the analysis but formulating the reduction entirely within the hamiltonian framework provided by the classical r matrix approach leads to some simplifications even forp 1

on the τ functions of the degasperis procesi equation the degasperis procesi dp equation is investigated from the point of view of determinant pfaffian identities the reciprocal link between the dp equation and the pseudo 3 reduction of the c two dimensional toda system is used to construct the n soliton solution of the dp equation the n soliton solution of the dp equation is presented in the form of pfaffian through a hodograph reciprocal transformation the bilinear equations the identities between determinants and pfaffians and the functions of the dp equation are obtained from the pseudo 3 reduction of the c two dimensional toda system

the unified approach to integrable relativistic equations soliton solutions over nonvanishing backgrounds ii in part i of this work the n soliton solution has been constructed for the generic system associated with the sl 2 c case of the scheme for the unified description of integrable relativistic massive fields here solutions are isolated for reductions of this system including the conventional complex sine gordon equation the massive thirring model and another complexification of the sine gordon equation defined by the lagrangian l 2 1 2 2 j 2 2 2 1 2 j i the latter model is shown to exhibit decays and fusion of subluminal solitons the reduction to the conventional complex sine gordon appears to be even more interesting as it cannot be defined by simply restricting the linear problem to some real form of sl 2 c algebra and the relevant involution turns out to be quite nontrivial when the background is flat this involution degenerates and so the n kink solution for the reduction cannot be extracted from the generic n kink solution directly this difficulty is

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

quantum line operators from lax pairs motivated by the realization of the yang baxter equation of 2d integrable models in the 4d gauge theory of costello witten yamazaki cwy we study the embedding of integrable 2d toda field models inside this construction this is done by using the lax formulation of 2d integrable systems and by thinking of the standard lax pair l in terms of components of cwy gauge connection propagating along particular directions in the gauge bundle we also use the results of the cwy theory to build quantum line operators for 2d toda models and compute the one loop contribution of two intersecting lines exchanging one gluon other features such as local symmetries and comments on extension of our method to other 2d integrable models are also discussed we also comment on some basic points that still need a refinement before discussing about a fully consistent embedding of lax pairs into cwy theory

stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks in this paper we consider in a bounded and smooth domain the wave equation with a delay term in the boundary condition we also consider the wave equation with a delayed velocity term and mixed dirichlet neumann boundary condition in both cases under suitable assumptions we prove exponential stability of the solution these results are obtained by introducing suitable energies and by using some observability inequalities if one of the above assumptions is not satisfied some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero

classical and quantum integrable systems with boundaries we study two dimensional classically integrable field theory with independent boundary conditions at each end and obtain three possible generating functions for integrals of motion when this model is an ultralocal one classically integrable boundary conditions can be found when solving boundary k equations in the quantum case we also find that the unitarity condition of the quantum r matrix is sufficient to construct commutative quantities with boundaries and its reflection equations are obtained

prolongation structures and lie algebra real forms a technique useful as an aid for finding finite dimensional representations of a wahlquist estabrook prolongation algebra is introduced the technique is illustrated with an examination of equations of the form 2za x1 x2 fa zb

the generalised zakharov shabat system and the gauge group action the generalized zakharov shabat systems with complex valued non regular cartan elements and the systems studied by caudrey beals and coifman cbc systems and their gauge equivalent are studied this study includes the properties of fundamental analytical solutions for the gauge equivalent to cbc systems and the minimal set of scattering data the description of the class of nonlinear evolutionary equations solvable by the inverse scattering method and the recursion operator related to such systems the hierarchies of hamiltonian structures the results are illustrated in the example of the multi component nonlinear schrodinger equations and the corresponding gauge equivalent multi component heisenberg ferromagnetic type models related to so 5 c algebra

stabilization of wave and plate like equations with nonlinear dissipation on the boundary abstract in this paper we consider second order in time equations of petrovsky type wave and plate like with nonlinear dissipative boundary conditions under certain geometric conditions imposed on the domain the results on asymptotic behavior of the solutions are established

periodic solutions of a class of singular hamiltonian systems soit r n 0 n n n 3 et soit v c 1 rx r on etudie l existence des solutions periodiques t u t avec une periode prescrite du systeme suivant de n equations differentielles d ordre 2 u v t u

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

a note on affine toda couplings the author proves the clebsch gordan property for the couplings of an untwisted affine toda field theory utilizing dorey s 1991 conjecture

discretization of hyperbolic type darboux integrable equations preserving integrability a method of integrable discretization of the liouville type nonlinear partial differential equations based on integrals is suggested new examples of the discrete liouville type models are presented

the variational contribution of the periodic orbits provided by the birkhoff lewis theorem with applications to convex hamiltonians and to three body type problems

uniform stabilization of the wave equation by nonlinear boundary feedback the question of uniformly stabilizing the solution of the wave equation y delta y 0 in omega times 0 infty omega is a bounded domain of bf r n by means of a nonlinear feedback law of the following form is studied partial y partial v k x g y on gamma 0 times 0 infty y 0 on gamma 1 times 0 infty gamma 0 gamma 1 being a suitable partition of the boundary of omega and g a continuous nondecreasing function such that g 0 0 we choose k x in l infty k x geqq 0 such that k x vanishes linearly at the interface points x in bar gamma 0 cap bar gamma 1 then if g s behaves like s p 1 s as s to 0 with p 1 and linearly as s to infty it is proved that the energy of every solution decays like t 2 p 1 as t to infty in the case where p 1 the exponential decay rate is proved

a re examination of the basic postulates of thermomechanics this paper is mainly concerned with a re examination of the basic postulates and the consequent procedure for the construction of the constitutive equations of material behaviour in thermomechanics however the implication of the basic postulates and the significance of the related procedure for the development of the constitutive equations is also illustrated in some detail in the context of flow of heat in a rigid solid with particular reference to the propagation of thermal waves at finite speed more specifically after briefly examining the nature of the basic equations of motion for a system of particles within the scope of the classical newtonian mechanics the basic postulates of the purely mechanical theory for a continuum including its specialization for a rigid body is re examined this includes some differences from the usual procedure on the subject next thermal variables are introduced and after observing a useful analogy between the thermal and mechanical variables a discussion of a theory of heat or a purely thermal theory is provided which differs from the usual development in the classical thermodynamics a detailed application of the latter development is then made to the problem of heat flow in a stationary rigid solid using several different and well motivated constitutive equations special cases of these include linearized theories of the classical heat flow by conduction and of heat flow transmitted as thermal waves the remainder of the paper is concerned with thermal mechanical theory of deformable media along with discussions of a number of related issues on the subject

more non abelian loop toda solitons we find new solutions including soliton like ones for a special case of non abelian loop toda equations associated with complex general linear groups we use the method of rational dressing based on an appropriate block matrix representation suggested by the gradation under consideration we present solutions in the form of a direct matrix generalization of hirota s soliton solution already well known in the case of abelian loop toda systems

conservation laws and symmetries of generalized sine gordon equations we study some systems of non linear pde s eqs 1 1 below which can be regarded either as generalizations of the sine gordon equation or as two dimensional versions of the toda lattice equations we show that these systems have an infinite number of non trivial conservation laws and an infinite number of symmetries the second result is deduced from the first by a variant of the hamiltonian formalism for evolution equations we also consider some specializations of the systems

a minimax method for a class of hamiltonian systems with singular potentials abstract this paper presents a minimax method which gives existence and multiplicity results for time periodic solutions of a class of hamiltonian systems when a singular potential is present the singularity satisfies the strong force condition of gordon when milder singularities are permitted a notion of generalized t periodic solution is introduced and we get existence and multiplicity results for such solutions

comments on defects in the ar toda field theories a simple basic argument is given based solely on energy momentum considerations to recover conditions under which ar affine or conformal toda field theories can support defects of integrable type associated triangle relations are solved to provide expressions for transmission matrices that generalize previously known examples calculated for the sine gordon model and the a2 affine toda model

kac moody groups and integrability of soliton equations a new approach to integrability of affine toda field theories and closely related to them kdv hierarchies is proposed the flows of a hierarchy are explicity identified with infinitesimal action of the principal abelian subalgebra of the corresponding affine kac moody algebra on a homogeneous space

the fixed energy problem for a class of nonconvex singular hamiltonian systems we consider a noncompact hypersurface h in r2n which is the energy level of a singular hamiltonian of strong force type under global geometric assumptions on h we prove that it carries a closed characteristic as a consequence of a result by hofer and viterbo on the weinstein conjecture in cotangent bundles of compact manifolds our theorem contains as particular cases earlier results on the fixed energy problem for singular lagrangian systems of strong force type

the generalized additional symmetries of the two toda lattice hierarchy the generalized additional symmetries of the two toda lattice hierarchy are investigated in this paper the algebraic structure of this generalized additional symmetry is showed as w w and the actions of the generalized additional symmetries on function are also discussed by restricting the two toda lattice hierarchy to the semi infinite case

dynamical systems with effective like potentials on etudie l existence des solutions t periodiques pour des systemes hamiltoniens d ordre 2 de la forme x v x ou v r n 0 r est de classe c 2 et v x quand x 0

topologically distinct collision free periodic solutions for the n center problem this work concerns the planar n center problem with homogeneous potential of degree alpha alpha in 1 2 the existence of infinitely many topologically distinct non collision periodic solutions with a prescribed energy is proved a notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive h the existence of a classical periodic solution with energy h for the n center problem with alpha in 1 2 is proven in case alpha 1 a slightly different result is shown it is the case that there is either a non collision periodic solution or a collision reflection solution the results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions the proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments

a uniform stability result for thermoelasticity of type iii with boundary distributed delay abstract in this paper we consider a thermoelastic system of type iii with boundary distributed delay under suitable assumption on the weight of the delay we prove using the energy method that the damping effect through heat conduction given by green and naghdi s theory is still strong enough to uniformly stabilize the system even in the presence of time delay

periodic solutions of prescribed energy for a class of hamiltonian systems with singular potentials abstract we study the existence of periodic solutions of prescribed energy for a class of hamiltonian systems with singular potentials we found periodic solutions lying between two prescribed spheres

local conservation laws for the two dimensional periodic su n 1 toda lattices an infinite number of local conservation laws are derived for the two dimensional periodic su n 1 toda lattice equations following the well known method for the sine gordon equation

hidden quantum group symmetry and integrable perturbations of conformal field theories the hidden quantum group symmetry in the quantum sine gordon model is found this symmetry provides the possibility to restrict the operator algebra of the model to subalgebras it is shown that these subalgebras are massive deformations of minimal conformal field theories

on the squared eigenfunction symmetry of the toda lattice hierarchy the squared eigenfunction symmetry for the toda lattice hierarchy is explicitly constructed in the form of the kronecker product of the vector eigenfunction and the vector adjoint eigenfunction which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively then after the fay like identities and some important relations about the wave functions are investigated the action of the squared eigenfunction related to the additional symmetry on the tau function is derived which is equivalent to the adler shiota van moerbeke formulas

non crystallographic reduction of generalized calogero moser models we apply a recently introduced reduction procedure based on the embedding of non crystallographic coxeter groups into crystallographic ones to calogero moser systems for rational potentials the familiar generalized calogero hamiltonian is recovered for the hamiltonians of trigonometric hyperbolic and elliptic types we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio we explicitly show for the simplest of these non crystallographic models how the corresponding classical equations of motion can be derived from a lie algebraic lax pair based on the larger crystallographic coxeter group

lie superalgebraic approach to super toda lattice and generalized super kdv equations we propose a super lax type equation based on a certain class of lie superalgebra as a supersymmetric extension of generalized modified kdv hierarchy we are able to construct an infinite set of conservation laws and the consistent time evolution generators for generalized modified super kdv equations thefirst few of the conserved currents the modified super kdv equation and the super miura transformation are worked out explicitly in the case of twisted affine lie superalgebraosp 2 2 2

method of lyapunov functionals construction in stability of delay evolution equations the investigation of stability for hereditary systems is often related to the construction of lyapunov functionals the general method of lyapunov functionals construction which was proposed by v kolmanovskii and l shaikhet and successfully used already for functional differential equations for difference equations with discrete time for difference equations with continuous time is used here to investigate the stability of delay evolution equations in particular partial differential equations

toda lattice mass transport in lagrangian mechanics and in a two dimensional system abstract this paper explores the connection between the hydrodynamic mass transport description and the thermodynamic description for a nonlinear range of the toda lattices particular attention is paid to the broken isotropy in the kdv and burgers equations the flow variable representation is established from the lagrangian mechanics for hydrodynamic mass transport based on the inverse scattering transform the gel fand levitan marchenko glm equation is formulated from the kdv equation expressed by the flow variable representation we found that a kernel of the glm equation is given by the concentration variable q x t a lagrangian is formulated for the kdv equation in state space q x t k x t next an extension of the flow variable representation is sought in a two dimensional system the lhs of the kadomtsev petviashvili kp equation takes the same form as in the second formalism of the kdv equation by setting up the flow variable representation of the kp equation the burgers equation in two dimensions is formulated these results contribute to an understanding of the broken isotropy for the nonlinear mass transport equations these results provide physical insight into various consequences of the generalized form of the kawasaki ohta equation from the viewpoint of mass transport

linearizing integral transform for the multicomponent lattice kp abstract we present a generalization of the direct linearization method for the kadomtsev petviashvili kp equation and its lattice analogues such as the two dimensional toda equation the generalization consists of a riemann hilbert type of integral transform which relates solutions of the associated spectral problem to any given solution of the spectral problem by dimensional reduction we obtain the integral transform for the korteweg de vries kdv equation and its lattice analogues the integral transform is well suited for the investigation of the finite matrix generalizations of the above mentioned equations such finite matrix generalizations may be of use in connection with partial difference equations for spin correlation functions in the two dimensional ising model

supersymmetric two dimensional toda lattice the two dimensional toda lattice connected with contragradient lie superalgebras is studied the systems of linear equations associated with the models for which the inverse scattering method is applicable are written down the reduction group is calculated

boundary form factors of the sinh gordon model with dirichlet boundary conditions at the self dual point in this paper we present a detailed investigation of the form factors of boundary fields of the sinh gordon model with a particular type of dirichlet boundary condition corresponding to zero value of the sinh gordon field at the boundary at the self dual point we follow for this the boundary form factor program recently proposed by z bajnok l palla and g takacs 2006 on the boundary form factor program preprint hep th 0603171 extending the analysis of the boundary sinh gordon model initiated there the main result of the paper is a conjecture for the structure of all n particle form factors of the boundary operators x and x 2 in terms of elementary symmetric polynomials in certain functions of the rapidity variables in addition form factors of boundary descendant fields have been constructed

toda lattice hierarchy and conservation laws time evolutions of the toda lattice hierarchies of ueno and takasaki are induced by hamiltonians which are conservation laws for the original one and two dimensional toda lattice obtained by olive and turok moreover these hamiltonians for two dimensional toda lattice hierarchy are also conserved quantities of the two component kp hierarchy in which that system is embedded the one dimensional toda lattice hierarchy is characterized by the bilinear relations and a new version of the one dimensional toda lattice hierarchy is constructed generalized toda lattice hierarchies associated to all affine lie algebras are presented

the determinant representation for quantum correlation functions of the sinh gordon model we consider the quantum sinh gordon model in this paper form factors in this model were calculated by mussardo and colleagues we sum up all contributions of form factors and obtain a closed expression for a correlation function this expression is a determinant of an integral operator similar determinant representations have been proven to be useful not only in the theory of correlation functions but also in matrix models

reductions of integrable equations on a iii type symmetric spaces we study a class of integrable nonlinear differential equations related to the a iii type symmetric spaces these spaces are realized as factor groups of the form su n s u n k u k we use the cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic lax operators to this symmetric space the symmetries of the lax operator are inherited by the fundamental analytic solutions and give a characterization of the corresponding riemann hilbert data

closed orbits of fixed energy for singular hamiltonian systems this paper deals with the existence of periodic solutions of q v q 0 such that 1 2 q 2 v q h where q r n h is a given number v c 2 r n 0 r has a singularity at x 0 and v denotes the gradient of v

specializations of integrable systems and affine lie algebras we analyze a number of restrictions on the evolution systems associated with the zero curvature equations corresponding to the extended dynkin diagram a 1 n 1 the resulting specialized evolution systems contain exponential terms like a third order differential equation previously derived in ref 5

bethe ansatz equations for the classical a 1 _ n affine toda field theories we establish a correspondence between classical affine toda field theories and an bethe ansatz systems we show that the connection coefficients relating specific solutions of the associated classical linear problem satisfy functional relations of the type that appear in the context of the massive quantum integrable model

multiple closed orbits of fixed energy forn body type problems with gravitational potentials abstract using the equivariant ljusternik schnirelmann theory and the estimates of the upper bound and lower bound of the critical values for the variational functional we obtain results in general concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planarn body type problems

spatial chaotic structure of attractors of reaction diffusion systems the dynamics described by a system of reaction diffusion equations with a nonlinear potential exhibits complicated spatial patterns these patterns emerge from preservation of homotopy classes of solutions with bounded energies chaotically arranged stable patterns exist because of realizability of all elements of a fundamental homotopy group of a fixed degree this group corresponds to level sets of the potential the estimates of homotopy complexity of attractors are obtained in terms of geometric characteristics of the potential and other data of the problem

a note on periodic solutions of prescribed energy for singular hamiltonian systems abstract in this note we show under a less restrictive setting than previously that there exists a collision brake orbit of prescribed energy to a certain second order hamiltonian system

constrained reductions of two dimensional dispersionless toda hierarchy hamiltonian structure and interface dynamics finite dimensional reductions of the two dimensional dispersionless toda hierarchy constrained by the string equation are studied these include solutions determined by polynomial rational or logarithmic functions which are of interest in relation to the laplacian growth or hele shaw problem governing interface dynamics the consistency of such reductions is proved and the hamiltonian structure of the reduced dynamics is derived the poisson structure of the rationally reduced dispersionless toda hierarchies is also derived

form factors of boundary fields for the a2 affine toda field theory in this paper we carry out the boundary form factor program for the a2 affine toda field theory at the self dual point the latter is an integrable model consisting of a pair of particles which are conjugated to each other that is and possessing two bound states resulting from the scattering processes 1 1 2 and 2 2 1 we obtain solutions up to four particle form factors for two families of fields which can be identified with spinless and spin 1 fields of the bulk theory previously known as well as new bulk form factor solutions are obtained as a particular limit of ours minimal solutions of the boundary form factor equations for all an affine toda field theories are given which will serve as the starting point for a generalization of our results to higher rank algebras

on the liouville integrability of the periodic kostant toda flow on matrix loops of level k in this work we consider the periodic kostant toda flow on matrix loops in mathfrak s mathfrak l n mathbb c of level k which correspond to periodic infinite band matrices with period n with lower bandwidth equal to k and fixed upper bandwidth equal to 1 with 1 s on the first superdiagonal we show that the coadjoint orbits through the submanifold of such matrix loops can be identified with those of a finite dimensional lie group which appears in the form of a semi direct product we then characterize the generic coadjoint orbits and obtain an explicit global cross section for such orbits we also establish the liouville integrability of the periodic kostant toda flow on such orbits via the construction of action angle variables

point symmetries of generalized toda field theories a class of two dimensional field theories with exponential interactions is introduced the interaction depends on two coupling matrices and is sufficiently general to include all toda field theories existing in the literature lie point symmetries of these theories are found for an infinite semi infinite and finite number of fields special attention is accorded to conformal invariance and its breaking

periodic solutions of singular hamiltonian systems with prescribed energy abstract we study the existence of non collision periodic solutions for some class of singular hamiltonian systems in particular in the case that the weak force potential has a set of singular points which is a compact subset of r n we prove the existence of a non collision periodic solution for all positive prescribed energy

analytical mini max methods for hamiltonian brake orbits of prescribed energy on considere des caracterisations explicites de solutions periodiques de systemes hamiltoniens autonomes d ordre 2 de la forme q v q q t r n ou le potentiel v satisfait les conditions suivantes v c 2 r n v x v o 0 v est strictement convexe sur r n

exponential stability and inequalities of solutions of abstract functional differential equations abstract with the lyapunov second method we study the abstract functional differential equation d u d t f t u t we obtain inequalities of solutions and exponential stability with conditions like i w 1 u t x v t u t w 2 d t u t t h t l s w 1 u s x d s ii v 1 t u t t w 2 d t u t p t applications in ordinary and partial functional differential equations are given

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

an example on the effect of time delays in boundary feedback stabilization of wave equations this note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations the question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays

an analysis of delay dependent stability for ordinary and partial differential equations with fixed and distributed delays this paper is concerned with the study of the stability of ordinary and partial differential equations with both fixed and distributed delays and with the study of the stability of discretizations of such differential equations we start with a delay dependent asymptotic stability analysis of scalar ordinary differential equations with real coefficients we study the exact stability region of the continuous problem as a function of the parameters of the model next it is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems in the second part of the paper we study delay partial differential equations the stability region of the fully continuous problem is analyzed first then a semidiscretization in space is applied it is shown that the spatial discretization leads to a reduction of the stability region when the standard second order central difference operator is employed to approximate the diffusion operator finally we consider the delay dependent stability of the fully discrete problem where the partial differential equation is discretized both in space and in time some numerical examples and further discussions are given