three component nonlinear dynamical system generated by the new third order discrete spectral problem we propose a nonlinear model on a regular infinite one dimensional lattice it describes the three component dynamical system with modulated on site masses and is shown to admit a zero curvature representation the associated auxiliary spectral problem is basically of third order and gives rise to fairly complicated subdivision into domains of regularity of jost functions in the plane of complex spectral parameter as a result both the direct and the inverse scattering problems turn out to be substantially nontrivial the caudrey version of the direct and inverse scattering technique for the needs of model integration is adapted the simplest soliton solution is found

nonisospectral negative volterra flows and mixed volterra flows lax pairs infinitely many conservation laws and integrable time discretization in this paper by means of the discrete zero curvature representation nonisospectral negative volterra flows and mixed volterra flows are proposed by means of solving corresponding discrete spectral equations we demonstrate the existence of infinitely many conservation laws for the two nonisospectral flows and obtain the formulae of the corresponding conserved densities and associated fluxes integrable time discretizations for several isospectral equations of the two flows are also presented

curvature diameter homotopy groups and cohomology rings we establish two topological results a if m is a 1 connected compact n manifold and q 2 then the minimal number of generators for the qth homotopy group q m mng q m is bounded above by a number depending only on mng h m and q where h m is the homology group c let mathscr m n y be the collection of compact orientable n manifolds whose oriented bundles admit so n invariant fibrations over y with fiber compact nilpotent manifolds such that the induced so n actions on y are equivalent then q m finitely generated m mathscr m n y contains only finite isomorphism classes depending only on n y q together with the results of cg and gr1 from a we conclude that i if m is a complete n manifold of nonnegative curvature then mng q m is bounded above by a number depending only on n and q 2 together with the results of ch and cfg from c we conclude that ii if m is a compact n manifold whose sectional curvature and diameter satisfy secm and diamm d then q m has a finite number of possible isomorphism classes depending on n d q 2 provided q m is finitely generated we also show that b if m is a compact n manifold with secm and diam m d then the cohomology ring h m may have infinitely many isomorphism classes in particular b answers some questions raised by k grove gro

a controlled plus construction for crumpled laminations given a closed n manifold m n 4 and a finitely generated perfect subgroup p of it1 m we previously developed a controlled version of quillen s plus construction namely a cobordism w m n with the inclusion j n f w a homotopy equivalence and kernel of i i1 m f it1 w equalling the smallest normal subgroup of xl1 m containing p together with a closed map p w 0 1 such that p l t is a closed n manifold for every t e 0 1 and in particular m p 1 o and n p l l we accomplished this by constructing an acyclic map of manifolds f m f n having the right fundamental groups and w arose as the mapping cylinder of f with a collar attached along n the main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when it1 m contains no finitely generated perfect subgroups by products of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups

minimal entropy and collapsing with curvature bounded from below we show that if a closed manifold m admits an structure not necessarily polarized possibly of rank zero then its minimal entropy vanishes in particular this is the case if m admits a non trivial s1 action as a corollary we obtain that the simplicial volume of a manifold admitting an structure is zero we also show that if m admits an structure then it collapses with curvature bounded from below this in turn implies that m collapses with bounded scalar curvature or equivalently its yamabe invariant is non negative we show that structures of rank zero appear rather frequently every compact complex elliptic surface admits one as well as any simply connected closed 5 manifold we use these results to study the minimal entropy problem we show the following two theorems suppose that m is a closed manifold obtained by taking connected sums of copies of s4 p2 overline mathbb c p 2 s2 s2and the k3 surface then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s4 p2 s2 s2 p2 overline mathbb c p 2 or p2 p2 finally suppose that m is a closed simply connected 5 manifold then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s5 s3 s2 then on trivial s3 bundle over s2 or the wu manifold su 3 so 3

discrete nonlinear schrodinger equations with arbitrarily high order nonlinearities a class of discrete nonlinear schrodinger equations with arbitrarily high order nonlinearities is introduced these equations are derived from the same hamiltonian using different poisson brackets and include as particular cases the saturable discrete nonlinear schrodinger equation and the ablowitz ladik equation as a common property these equations possess three kinds of exact analytical stationary solutions for which the peierls nabarro barrier is zero several properties of these solutions including stability discrete breathers and moving solutions are investigated

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

expansions over the squared solutions and difference evolution equations the completeness relation for the system of squared solutions of the discrete analog of the zakharov shabat problem is derived it allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results these include i the expansion of the potential and its variations over the system of squared solutions the expansion coefficients being the scattering data and their variations respectively thus the interpretation of the inverse scattering transform ist as a generalized fourier transform becomes obvious ii compact expressions for the trace identities through the operator for which the squared solutions are eigenfunctions iii brief exposition of the spectral theory of the operator iv direct calculation of the action angle variables based on the symplectic form of the completeness relation v the generating functional of the m operators in the lax representation vi the quantum ve

absence of singular spectrum for a perturbation of a two dimensional laplace beltrami operator with periodic electromagnetic potential let be a lattice on we consider a metric g a one form a and a real function v on all periodic we prove that the spectrum of the schrodinger operator on is absolutely continuous

integrable three dimensional lattices following the work of zakharov and shabat 1974 the authors derive nonlinear equations which are integrable through a discrete gel fand levitan integral equation in two resp one resp zero continuous and one resp two resp three discrete variables

supersymmetric extension of the integrable box ball system we introduce the supersymmetric extension of the box ball system by use of the isomorphism of the crystal base for the super lie algebra we define the time evolution operator and give the evolution equation explicitly we also construct the soliton solutions

new integrable systems related to the relativistic toda lattice the bi hamiltonian structure of the relativistic toda lattice is exploited to introduce some new integrable lattice systems their integrable discretizations are obtained by means of the general procedure proposed recently by the author backlund transformations between the new systems and the relativistic toda lattice in both the continuous and discrete time formulations are established

on realizations of solutions of the kdv equation by determinants on operator ideals using new developments in the theory of traces determinants and elementary operators on quasi banach operator ideals we clarify and extend marchenko s method for realizing solutions of the kdv equation moreover we point out why abstract traces and determinants on quasi banach operator ideals are appropriate tools for obtaining solutions of the kdv equation the method we present can also be applied to other nonlinear equations in soliton physics

solitons in parametrically driven discrete nonlinear schrodinger systems with the exploding range of intersite interactions we present the sequence of parametrically driven discrete nonlinear schrodinger systems with the progressively extending range of intersite couplings in the case of time independent coupling parameters the sequence is reduced to the ablowitz ladik hierarchy which is known to be integrable by the inverse scattering transform however the models with the time dependent intersite interactions are shown to be integrable too irrespective of a particular form of time dependencies of coupling parameters any of such parametrically driven systems might exhibit rather complex soliton dynamics and is described by the unconserved hamiltonian function we reveal an important subclass of parametrically driven systems demonstrating the parametrical localization of soliton dynamics on a confined domain of space meanwhile an appropriate choice of time dependencies in intersite interactions allow us to transform the original parametrically driven system into another one but subjected to the linear external potential as a result the latter system can be readily integrated as well in particular the peculiarities of bloch oscillations in the systems with time independent long range intersite interactions and linear external potential of constant strength are analyzed in general regulating the range of intersite couplings the strengths and time dependencies of coupling parameters we are able to model a number of physically important quasi one dimensional systems we develop an alternative approach to solve the marchenko equations permitting one to obtain the multisoliton solutions in the most simple and natural way finally we point out how to reformulate any model in row in terms of corrected amplitudes with the standard poisson brackets

simply connected 4 manifolds with a given boundary let m be a closed oriented connected 3 manifold for each bilinear symmetric pairing zn l our goal is to calculate the set vl m of all oriented homeomorphism types of compact 1 connected oriented 4manifolds with boundary m and intersection pairing isomorphic to zn l for each pair zn l which presents h m we construct a double coset space bl m and a function ct vl m bl m the set bt m is the quotient of the group of all link pairing preserving isomorphisms of the torsion subgroup of h1 m by two naturally occuring subgroups when zn l is an even pairing we construct another double coset space bl m a function 6l vl m b fl m and a projection p2 bl m bl m such that p2 c l ct our main result states that when zn l is even the function c l is injective as is the function c x a vl m bl m x z 2 when z l is odd here a is a kirby siebenmann obstruction to smoothing it follows that the sets vl m are finite and of an order bounded above by a constant depending only on h1 m we also show that when hi m q 0 and z l is even ct p2 c 6l is injective it seems likely that via the functions cl x a and 6l the sets bl m x z 2 and bl m calculate vl m when zn l is respectively odd and even we verify this in several cases most notably when h1 m is free abelian the results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two 1connected 4 manifolds extending a given homeomorphism of their boundaries the theory developed is then applied to show that there is an m 0 depending only on h1 m such that for any self homeomorphism f of m fm extends to a self homeomorphism of any 1 connected compact 4 manifold with boundary m introduction in m freedman s fundamental paper fr he classified closed simply connected oriented 4 manifolds up to orientation preserving homeomorphism he showed that modulo the kirby siebenmann invariant these manifolds are in bijective correspondence through their intersection pairings with the set of unimodular bilinear symmetric pairings over z see theorem 1 5 of fr and corollary 2 2 3 of q in this paper we begin a classification of simply connected 4 manifolds with connected boundary our results with regard to this problem are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between 1 connected 4 manifolds extending a given homeomorphism of their boundaries received by the editors september 19 1985 1980 mathematics subject classification 1985 revision primary 57n15 research supported by an nserc canada post doctoral fellowship 1986 american mathematical society 0002 9947 86 1 00 25 per page

algebra geometric constructions of a 2 1 dimensional discrete integrable model a new 2 1 dimensional differential difference equation is considered with the aid of the nonlinearization of the lax pair the 2 1 dimensional differential difference equation is decomposed into a new integrable symplectic map and a class of finite dimensional integrable hamiltonian systems based on the theory of algebra curve the continuous flow and discrete flow are straightened out in view of the introduced abel jacobi coordinates

enigma of probability amplitudes in hamiltonian formulation of integrable semidiscrete nonlinear schrodinger systems an attempt to find a probability amplitude based hamiltonian representation of the symmetrical version of integrable semidiscrete multicomponent nonlinear schrodinger systems is made thus the on cell locality of the general point transformation in combination with the model multicomponentness is shown to contradict the concept of canonical hamiltonian representation in terms of probability amplitudes nevertheless the above concept can be realized in a slightly adjusted semidiscrete multicomponent nonlinear schrodinger system that preserves some physically valuable solutions of the original either symmetric or asymmetric integrable model the advantages of the adjusted hamiltonian model for the analysis of real physical systems are formulated examples of longitudinal and lateral soliton dynamics on multichain tubular lattices subjected to uniform electric and magnetic fields are given

nonlinear differential difference equations associated backlund transformations and lax technique the lax technique can be successfully employed to derive the class of nonlinear differential difference equations associated with the discrete analogue of the matrix schrodinger spectral problem and solvable by the spectral transform

integral action in output feedback for multi input multi output nonlinear systems we address a particular problem of output regulation for multi input multi output nonlinear systems specifically we are interested in making the stability of an equilibrium point and the regulation to zero of an output robust to small unmodelled discrepancies between design model and actual system in particular those introducing an offset we propose a novel procedure which is intended to be relevant to realistic scenarios as illustrated by a non academic example

lie algebraic approach to the construction of 2 1 dimensional lattice field and field integrable hamiltonian equations two different methods for the construction of 2 1 dimensional integrable lattice field and field hamiltonian dynamical systems are presented the first method is based on the so called central extension procedure applied to the lie algebra of shift operators and the lie algebra of pseudodifferential operators the second method is the so called operand formalism both methods allow a construction of some new integrable nonlinear hamiltonian lattice field and field equations in 2 1 dimensional space

finite genus solution of a new 2 1 burgers equation with a discrete variable a new 2 1 burgers equation with a discrete variable is given as a compatible condition of the lax pair which are transformed into two hamilton flows and a discrete symplectic flow these flows are integrable in liouville sense and share the same integrals by the abel jacobi coordinates the flows are straightened based on the riemann jacobi inverse theorem a finite genus solution is obtained

heegaard floer homology of matsumoto s manifolds abstract we consider a homology sphere m n k 1 k 2 presented by two knots k 1 k 2 with linking number 1 and framing 0 n we call the manifold matsumoto s manifold we show that m n t 2 3 k 2 never bounds any contractible 4 manifold if n 2 k 2 holds we also give a formula of ozsvath szabo s invariant as the total sum of the euler numbers of the reduced filtration we compute the invariants of the twisted whitehead doubles of torus knots and correction terms of the branched covers of the whitehead doubles by using owens and strle s obstruction we show that the 12 twisted whitehead double of the 2 7 torus knot and the 20 twisted whitehead double of the 3 7 torus knot are not slice but the double branched covers bound rational homology 4 balls these are new examples having a gap between what a knot is slice and what a double branched cover bounds a rational homology 4 ball

the two dimensional periodic box ball system and its fundamental cycle we study a two dimensional box ball system which is an ultradiscrete analogue of the discrete kp equation we construct an algorithm to calculate the fundamental cycle which is an important conserved quantity of the two dimensional box ball system with periodic boundary condition by using the tropical curve theory

absolutely continuous spectrum for the isotropic maxwell operator with coefficients that are periodic in some directions and decay in others the purpose of this paper is to prove that the spectrum of an isotropic maxwell operator with electric permittivity and magnetic permeability that are periodic along certain directions and tending to a constant super exponentially fast in the remaining directions is purely absolutely continuous the basic technical tools is a new operatorial identity relating the maxwell operator to a vector valued schrodinger operator the analysis of the spectrum of that operator is then handled as in 3 4

nonlinear integrable model of frenkel like excitations on a ribbon of triangular lattice following the considerable progress in nanoribbon technology we propose to model the nonlinear frenkel like excitations on a triangular lattice ribbon by the integrable nonlinear ladder system with the background controlled intersite resonant coupling the system of interest arises as a proper reduction of first general semidiscrete integrable system from an infinite hierarchy the most significant local conservation laws related to the first general integrable system are found explicitly in the framework of generalized recursive approach the obtained general local densities are equally applicable to any general semidiscrete integrable system from the respective infinite hierarchy using the recovered second densities the hamiltonian formulation of integrable nonlinear ladder system with background controlled intersite resonant coupling is presented in doing so the relevant poisson structure turns out to be essentially nontrivial the darboux transformation scheme as applied to the first general semidiscrete system is developed and the key role of backlund transformation in justification of its self consistency is pointed out the spectral properties of darboux matrix allow to restore the whole darboux matrix thus ensuring generation one more soliton as compared with a priori known seed solution of integrable nonlinear system the power of darboux dressing method is explicitly demonstrated in generating the multicomponent one soliton solution to the integrable nonlinear ladder system with background controlled intersite resonant coupling

method of descent for integrable lattices a method of descent for constructing integrable hamiltonian systems is introduced the derived periodic and nonperiodic lattices possess lax representations with spectral parameter and have plenty of first integrals examples of liouville integrable four dimensional hamiltonian lotka volterra systems are presented

integrable nonlinear ladder system with background controlled intersite resonant coupling a new spectral problem on one dimensional lattices is found allowing consistently to support the zero curvature representation for a wide class of integrable nonlinear ladder systems the modified recurrence technique for obtaining an infinite set of conservation laws is developed and some basic conserved quantities are explicitly derived the eigenvalue problems associated with the limiting spectral operator for the special case of rapidly vanishing boundary conditions on schrodinger type fields and finite background condition on a concomitant field are solved and the domains of analyticity of jost functions are presented both analytically and graphically this particular example shows that the original auxiliary spectral problem is basically of fourth order and must generate a set of four distinct jost functions that have to be involved in the procedure of inverse scattering transform moreover there exists a critical background value of accompanying field which separates two principally different possibilities in the organization of analyticity domains of jost functions this crossover should inevitably lead to qualitative rearrangements in the structure of model solutions thus already in the limit of low amplitude excitations we strictly observe the loss of stability regarding the linear spectrum of schrodinger subsystem just above the critical background value of practically unexcited concomitant field whereas in the stability region the structure of linear spectrum is essentially controlled by the magnitude of background level via effective modification of both intersite resonant coupling and self site coupling

infinitely many conservation laws and integrable discretizations for some lattice soliton equations in this paper by means of the lax representations we demonstrate the existence of infinitely many conservation laws for the general toda type lattice equation the relativistic volterra lattice equation the suris lattice equation and some other lattice equations the conserved density and the associated flux are given formulaically we also give an integrable discretization for a lattice equation with n dependent coefficients

an integrable symmetric 2 1 dimensional lotka volterra equation and a family of its solutions a symmetric 2 1 dimensional lotka volterra equation is proposed by means of a dependent variable transformation the equation is firstly transformed into multilinear form and further decoupled into bilinear form by introducing auxiliary independent variables a bilinear backlund transformation is found and then the corresponding lax pair is derived explicit solutions expressed in terms of pfaffian solutions of the bilinear form of the symmetric 2 1 dimensional lotka volterra equation are given as a special case of the pfaffian solutions we obtain soliton solutions and dromions

bilinear discrete painleve equations based on the results of singularity confinement we derive bilinear expressions for the discrete painleve equations in these cases where a bilinear expression is not sufficient we obtain trilinear or higher multilinear expressions we show that the bilinear approach provides a natural framework for the derivation of backlund and schlesinger transforms for the discrete painleve equations

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

4 manifold topology ii dwyer s filtration and surgery kernels even when the fundamental group is intractable i e not good many interesting 4 dimensional surgery problems have topological solutions we unify and extend the known examples and show how they compare to the presumed counterexamples by reference to dwyer s filtration on second homology the development brings together many basic results on the nilpotent theory of links as a special case a class of links only slightly smaller than homotopically trivial links is shown to have free slices on their whitehead doubles

integrable semi discretization of the coupled modified kdv equations the discrete version of the inverse scattering method proposed by ablowitz and ladik is extended to solve multi component systems the extension enables one to solve the initial value problem which proves directly the complete integrability of a semi discrete version of the coupled modified korteweg de vries kdv equations and their hierarchy it also provides a procedure to obtain conservation laws and multi soliton solutions of the hierarchy

affine toda field theory as a 3 dimensional integrable system the affine toda field theory is studied as a 2 1 dimensional system the third dimension appears as the discrete space dimension corresponding to the simple roots in the an affine root system enumerated according to the cyclic order on the an affine dynkin diagram we show that there exists a natural discretization of the affine toda theory the quantum analog of the variables is found the thermodynamic bethe ansatz of the affine toda system is studied in the limit l n it is shown that the free energy of the systems grows proportionally to the volume

r matrix approach to lattice integrable systems an r matrix formalism is applied to the construction of the integrable lattice systems and their bi hamiltonian structure miura like gauge transformations between the hierarchies are also investigated in the end the ladder of linear maps between generated hierarchies is established and described

surgery in dimension four and noncompact 5 manifolds this paper describes a precise relationship between the problems of completing surgery in dimension four and finding boundaries for noncompact 5 manifolds two outstanding problems of low dimensional topology are completing 4 dimensional surgery and finding boundaries for noncompact 5 dimensional manifolds although no complete theory is known for either problem some interesting results have been found see 1 4 5 10 and addendum in this paper we describe a complete relationship between these two problems consequently knowledge of 4 dimensional surgery yields information about noncompact 5 manifolds and conversely all manifolds and maps in this paper are assumed to be smooth each manifold is provided with a basepoint and when appropriate basepaths will be chosen to given 2 spheres without explicit mention i wish to thank gerald a anderson and frank quinn for helpful discussions also i wish to thank patrick gilmer for his advice and encouragement during this project 1 statement of results let f b m am y x be a degree one normal map in the sense of wall 11 where m is a compact connected 4 manifold and y x is a poincare pair not necessarily simple if x 0 then we require that f jam am x induces an isomorphism of homology groups with a z vi y coefficients also we assume that f is 2 connected k2 m is a free a module and the surgery obstruction a f b e lh 7 c is zero if h c k2 m is a subkernel we will say that h is representable in m if we can find disjoint framed embedded 2 spheres si sr c int m such that the inclusion m ui si m induces an isomorphism of fundamental groups and the classes of sl sr in h2 m a 7t2 m form a basis over a for h the subkernel h is stably representable if for some integer k 0 h ed kei ek is representable in m k s2 x s2 here ei denotes the class of s2 x 1 c ith summand s2 x s2 and e1 ek is the submodule generated by el ek we regard h ed ke ek as received by the editors august 10 1982 1980 mathematics subject classification primary 57r65 secondary 57m99 1985 american mathematical society 0002 9947 85 1 00 25 per page

integrable discretizations for toda type lattice soliton equations from a proper 2 2 discrete isospectral problem a new integrable lattice soliton system is proposed integrable discretizations of a general toda type lattice soliton equation associated with the discrete isospectral problem are established the lagrangian and newtonian forms of integrable discretizations of toda type lattice equations which occur in the literature are given uniformly and some new integrable discretizations of the toda type lattice are obtained

localization of classical waves ii electromagnetic waves we consider electromagnetic waves in a medium described by a position dependent dielectric constant we assume that is a random perturbation of a periodic function and that the periodic maxwell operator has a gap in the spectrum where we prove the existence of localized waves i e finite energy solutions of maxwell s equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times localization of electromagnetic waves is a consequence of anderson localization for the self adjoint operators we prove that in the random medium described by the random operator exhibits anderson localization inside the gap in the spectrum of this is shown even in situations when the gap is totally filled by the spectrum of the random operator we can prescribe random environments that ensure localization in almost the whole gap

lax pairs for integrable lattice systems this paper studies the structure of lax pairs associated with integrable lattice systems where space is a one dimensional lattice and time is continuous it describes a procedure for generating examples of such systems and emphasizes the features that are needed to obtain equations which are local on the spatial lattice

smooth s cobordisms of elliptic 3 manifolds the main result of this paper states that a symplectic s cobordism of elliptic 3 manifolds is diffeomorphic to a product assuming a canonical contact structure on the boundary based on this theorem we conjecture that a smooth s cobordism of elliptic 3 manifolds is smoothly a product if its universal cover is smoothly a product we explain how the conjecture fits naturally into the program of taubes of constructing symplectic structures on an oriented smooth 4 manifold with b 1 from generic self dual harmonic forms the paper also contains an auxiliary result of independent interest which generalizes taubes theorem sw gr to the case of symplectic 4 orbifolds

the global structure of simple space times according to a standard definition of penrose a space time admitting well defined future and past null infinitiesi andi is asymptotically simple if it has no closed timelike curves and all its endless null geodesics originate fromi and terminate ati the global structure of such space times has previously been successfully investigated only in the presence of additional constraints the present paper deals with the general case it is shown thati is diffeomorphic to the complement of a point in some contractible open 3 manifold the strongly causal regioni0 ofi is diffeomorphic to mathbb s 2 times mathbb r and every compact connected spacelike 2 surface ini is contained ini0 and is a strong deformation retract of bothi0 andi moreover the space time must be globally hyperbolic with cauchy surfaces which subject to the truth of the poincare conjecture are diffeomorphic to 3

fundamental groups of positively curved manifolds with symmetry we obtain a classification for the fundamental groups of closed n manifolds of positive sectional curvature which admit an isometric t k action with k ge frac n 6 1 n ne 11 15

integrability of the semi discrete toda equation with self consistent sources a new procedure to construct and solve soliton equations with self consistent sources sescss is applied to the semi discrete toda equation based on its bilinear from bilinear backlund transformation bt for the semi discrete toda escs is presented starting from the bt a lax pair is derived for the semi discrete toda escs

a discrete time relativistic toda lattice for each of the two simplest hamiltonian flows from the relativistic toda hierarchy we introduce two integrable symplectic discretizations all four discrete time systems are demonstrated to belong to the same hierarchy and to exemplify the general scheme for symplectic maps on groups equipped with quadratic poisson brackets the initial value problem for the difference equations is solved in terms of a factorization problem in a group interpolating hamiltonian flows are found for all maps

a family of freely slice good boundary links we show that every good boundary link with a pair of derivative links on a seifert surface satisfying a homotopically trivial plus assumption is freely slice this subsumes all previously known methods for freely slicing good boundary links with two or more components and provides new freely slice links

an overview of periodic elliptic operators the article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas close attention is paid to studying analytic properties of bloch and fermi varieties which influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

the heat flow in nonlinear hodge theory we study the nonlinear hodge systemd 0 and 2 0 for an exterior form on a compact oriented riemannian manifold m where q is a given positive function the solutions are called harmonic forms they are the stationary points on cohomology classes of the functional e me 2 dm with e q q 2 the codifferential of a form is defined as 1 with 2 we evolve a given closed form 0 by the nonlinear heat flow system d for a time dependent exterior form x t on m this system is the differential of the normalized gradient flow u for e with 0 du under a technical assumption on the function 2 q q q we show that the nonlinear heat flow system d with initial condition 0 0 has a unique solution for all times which converges to a harmonic form in the cohomology class of 0 this yields a nonlinear hodge theorem that every cohomology class of m has a unique harmonic representative

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

symmetry broken canonizations of the semi discrete integrable nonlinear schrodinger system with background controlled inter site coupling the integrable system of nonlinear schrodinger type consisting of two mutually symmetric basic subsystems and one concomitant subsystem given on a ribbon of triangular lattice is characterized by the highly nonstandard form of poisson structure the fact of system criticality against the positive semi definite background parameter provides us the clue how to organize the nonlinear point transformation ensuring the canonization of field variables precisely it allows to introduce two subsets of mutually symmetric intermediate auxiliary field variables and to involve one or another of them into a rational procedure of canonization realizable already in terms of two mutually asymmetric subsets of field amplitudes there are two variants of system asymmetric standardization namely the minus asymmetric and plus asymmetric ones emanated from the two leg structure of underlying space lattice of the primary noncanonical nonlinear system either of the two asymmetrically standardized nonlinear systems consi

construction of soliton cellular automaton from the vertex model the discrete 2d toda equation and the bogoyavlensky lattice we study the soliton cellular automaton sca in 2 1 dimensions from the viewpoint of the integrable vertex model as in our previous paper we relate the sca the so called box ball system to an integrable vertex model associated with the bogoyavlensky lattice we extend this framework and introduce the 2 1 dimensional sca which can be interpreted as the ultradiscretization of the 2d toda equation we also construct the n soliton solutions for this system

convergence of the supercell method for defect modes calculations in photonic crystals we present a rigorous study of the convergence of the supercell method used for determing defect modes in photonic crystals with compactly supported perturbations transverse electric and transverse magnetic polarized waves are investigated in 2 d structures we prove an exponential convergence of the defect frequencies with the supercell size and give a justification of the quasi independence of the corresponding eigenfunctions on the wave vector we also give a characterization of the supercell eigenvalues corresponding to the background photonic crystal

on the continuous limits and integrability of a new coupled semidiscrete mkdv system in this paper we aim to get more insight on the relation between semidiscrete coupled mkdv system where semidiscrete means that the system is discrete in the space variable and continuous in time and the coupled mkdv equations to this purpose we propose a new coupled semidiscrete mkdv system the lax pairs the darboux transformation soliton solutions and conservation laws for the coupled semidiscrete mkdv system are given the coupled mkdv theory including the lax pairs the darboux transformation soliton solutions and conservation laws is recovered through the continuous limits of corresponding theory for the new semidiscrete mkdv system

stable mapping class groups of 4 manifolds with boundary we give a complete algebraic description of the mapping class groups of compact simply connected 4 manifolds with boundary up to connected sum with copies of s 2 s 2

matrix semidiscrete ablowitz ladik equation hierarchy and a matrix discrete second painleve equation in this article we first show that the semidiscrete ablowitz ladik equation hierarchy can be extended to the matrix case by considering a matrix semidiscrete ablowitz ladik isospectral linear problem we then extend this linear problem to the nonisospectral case and a new coupled matrix semidiscrete integrable equation is constructed it is shown that the continuum limit of this new matrix semidiscrete integrable equation is the coupled matrix variable coefficient modified korteweg de vries equation and that a matrix discrete second painleve equation can be obtained as a stationary reduction in this new matrix semidiscrete equation

the absolute continuity of the spectrum of maxwell operator in a periodic media we prove that the spectrum of maxwell operator b e rot e a x rot b x b with non negative and periodic functions a x and b x is absolutely continuous

solution of a discrete inverse scattering problem and of the cauchy problem of a class of discrete evolution equations the cauchy problem of a class of nonlinear evolution equations is solved by finding explicit solutions of a discrete inverse scattering problem that are not restricted to the pure soliton case and implementing appropriate time evolution of the scattering data this yields operator valued functions which are shown to solve a hierarchy of operator evolution equations by applying methods similar to those in marchenko s work in addition the relation to canonical lax constructions is investigated using methods introduced by aden and carl and schiebold one obtains scalar solutions to corresponding scalar equations sometimes representable by determinants on operator ideals

integrable discretizations of derivative nonlinear schrodinger equations we propose integrable discretizations of derivative nonlinear schrodinger dnls equations such as the kaup newell equation the chen lee liu equation and the gerdjikov ivanov equation by constructing lax pairs the discrete dnls systems admit the reduction of complex conjugation between two dependent variables and possess bi hamiltonian structure through transformations of variables and reductions we obtain novel integrable discretizations of the nonlinear schrodinger nls modified kdv mkdv mixed nls matrix nls matrix kdv matrix mkdv coupled nls coupled hirota coupled sasa satsuma and burgers equations we also discuss integrable discretizations of the sine gordon equation the massive thirring model and their generalizations

discrete third order spectral problems and a new toda type equation in this paper we construct the class of equations associated with a discrete spectral problem of third order among the equations of this class we find an integrable differential equation on the lattice with exponential nonlinearity by considering its associated darboux transformation we construct an explicit solution

symplectic methods for the ablowitz ladik discrete nonlinear schrodinger equation using several kinds of coordinate transformations we standardize the noncanonical symplectic structure of the ablowitz ladik model a l model of nonlinear schrodinger equation nlse then we employ some symplectic scheme to simulate the solitons motion and test the evolution of the discrete invariants of the a l model and also the conserved quantities of the original nlse in comparison with a higher order non symplectic scheme applied directly to the a l model we show the overwhelming superiorities of the symplectic method we also compare the implementation of the same symplectic scheme to different standardized hamiltonian systems resulting from different coordinate transformations and show that the symmetric coordinate transformation improves the numerical results obtained via the asymmetric one in preserving the invariants of the a l model and the original nlse

rigidity theorems for right angled reflection groups let t be a right angled reflection group let m and m be coxeter manifolds then any t map f m fm is t homotopic to a homeo morphism introduction this paper contains the results given in the author s ph d the sis davis in 9 showed that in every dimension 4 there exists a cocompact reflection group on a contractible manifold not homeomorphic to an euclidean space also he gave the first example of a closed aspherical manifold not covered by euclidean space davis construction provides an infinite coxeter group acting locally smoothly effectively and properly discontinuously on a contractible manifold with compact quotient

hierarchies integrable decompositions and conservation laws of geng equation abstract in this paper two hierarchies of the geng equations are presented including positive non isospectral hierarchy and negative non isospectral hierarchy moreover integrable couplings of the corresponding geng hierarchies are also constructed by enlarging the associated matrix spectral problem three new integrable decompositions and conservation laws of the isospectral geng equation are also obtained the gauge transformations are used to obtain the associated binary symmetry constraints of the geng equation at the first time

construction of group actions on four manifolds it is shown that any cyclic group of odd prime order acts on any closed simply connected topological 4 manifold inducing the identity on integral homology the action is locally linear except perhaps at one isolated fixed point in the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear introduction in this paper we shall show that every closed simply connected topological 4 manifold m4 admits an action of any cyclic group zp of odd prime order p the action will be homologically trivial that is induce the identity on integral homology be pseudofree that is have only isolated fixed points except in certain cases when p 3 and be locally linear except perhaps at one isolated fixed point when p 2 and the intersection form of m4 has even type then the same conclusion holds this is also due to s kwasik and p vogel 1985 by a somewhat different proof if p 2 and the intersection form of m has odd type however then an action of z2 which is homologically trivial must have a fixed point set containing two dimensional components see 7 further it is harder to control the kirby siebenmann triangulation obstruction which is then not determined by the intersection form it is interesting to ask whether these actions can be locally linear or smooth when m4 is smooth the existence of a locally linear z2 action implies the vanishing of the kirby siebenmann obstruction see kwasik and vogel 1984 we shall show however that for p 3 in general these actions can be constructed to be locally linear in a recent preprint kwasik has shown that this is the case for the fake cp2 when p is odd by a rather different proof these actions are homologically trivial and have only isolated fixed points except sometimes when p 3 the g signature theorem shows that not every simply connected 4 manifold admits pseudofree locally linear actions when p 3 we shall see for example that neither the e8 manifold the kummer surface nor a nontrivial connected sum of copies of cp2 admits such an action for p 3 in broad outline our construction goes as follows let a closed oriented simply connected 4 manifold m be given by studying equivariant framed links we construct a compact smooth 4 manifold with boundary having the intersection form of m and admitting a smooth zp action which is pseudofree and homologically trivial the action is free on the boundary homology sphere e in a paper primarily focused on equivariant plumbing diagrams for high dimensional highly connected manifolds with even intersection forms weintraub 1975 carried out most of this we give an independent development because we need to be able to better control received by the editors october 16 1985 and in revised form january 7 1986 1980 mathematics subject classification 1985 revision primary 57s17 57s25 57n15 research supported in part by a grant from the national science foundation 1987 american mathematical society 0002 9947 87 1 00 25 per page

symmetry of integrable cellular automaton we study an integrable cellular automaton which is called the box ball system bbs the bbs can be derived directly from the integrable differential difference equation by either ultradiscretization or crystallization we clarify the integrable structure and the hidden symmetry of the bbs

a surgery sequence in dimension four the relations with knot concordance we present a systematic treatment of the classification problem for compact smooth 4 manifolds m it is modeled on the surgery exact sequence the central theorem in the classification of n manifolds n 5 the price for the extension to dimension 4 is a hole in m where a homotopy 1 skeleton should be there is no homotopy theoretic or surgical obstruction to complet ing m with a wedge of circles so that the completion has the topology of a compact smooth manifold this point set problem is all that stands between 4 manifolds and the tranquility that prevails in higher dimensions when m is simply connected only a point is missing from the model the applications of this are discussed in f and fq 1 the general theory has application to knot and link theory in particular knots with alexander polynomial 1 are characterized geometrically as knots admitting a certain

linking and causality in globally hyperbolic space times the classical linking number lk is defined when link components are zero homologous in 15 we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes here we apply alk to the study of causality in lorentzian manifolds

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

on liouville integrability of zero curvature equations and the yang hierarchy sufficient conditions for a zero curvature equation ut vx u v 0 being liouville integrable are investigated in the case that the equation is integrable an explicit formula of the poisson bracket h lambda h mu for hamiltonians h is proposed the yang hierarchy is derived and shown to be liouville integrable

a diffeomorphism classification of manifolds which are like projective planes we give a complete diffeomorphism classification of 1 connected closed manifolds m with integral homology h m z z z provided that dim m 6 4

relativistic toda systems we present and study poincare invariant generalizations of the galilei invariant toda systems the classical nonperiodic systems are solved by means of an explicit action angle transformation

gauge theory on asymptotically periodic 4 manifolds il existe une famille non denombrable de classes de diffeomorphismes des 4 varietes orientees homeomorphes a r 4