diagonalization of the quadratic boson hamiltonian with zero modes i mathematical the necessary mathematical equipment is developed for an analysis of the zero modes connected with hamiltonians quadratic in boson creation and annihilation operators a cross matrix being a matrix of a certain partitioned form we consider the group g of conjuctivity transformations t where t is a 2m square cross matrix subjected to the restriction t it i i diag i i i is the m square unit matrix with respect to this group the concept of standard form is introduced for the set of positive semidefinite hermitian cross matrices d it is proved that any such matrix d can be transformed into a matrix e of the standard form according to t dt e t g an algorithm suitable as a basis for computer programs is given for finding standardized and standardizing matrices e and t for any given d

multiplicity of solutions for a class of elliptic systems with critical sobolev exponent abstract in this paper we consider systems of p q laplacian elliptic equations with critical sobolev exponent the existence and multiplicity results of solutions are obtained by a limit index method

existence of multiple weak solutions for asymptotically linear wave equations abstract we consider the problem of multiple existence of 2 periodic weak solutions to wave equations u x t h x t u x t f x t of space dimension 1 where h x t is asymptotically linear in both as 0 and as it is shown by variational methods that there exist at least three solutions under several conditions on h x t if f is sufficiently small one of the results reads as follows let b lim h x t and assume that the convergence is uniform with respect to x t and that b non resonant case then the following conditions guarantee the existence of at least three solutions for sufficiently small f a h x t h x t 0 is non decreasing resp non increasing in and sup x t h x t min b resp inf x t h x t max b to obtain these results we prove that a c 1 class functional which is bounded from above and satisfies a condition similar to the linking condition has at least three critical points under wps condition

an infinite dimensional morse theory with applications in this paper we construct an infinite dimensional extraordinary cohomology theory and a morse theory corresponding to it these theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension several applications are given to hamiltonian systems the onedimensional wave equation of vibrating string type and systems of elliptic partial differential equations 0 introduction let e be a real hilbert space with an inner product and let be a twice continuously differentiable functional denote the frechet derivative and the gradient of at x by x and x respectively where as usual x y x y y e recall that a point x0 e is said to be critical if x0 0 or equivalently if x0 0 the level c r will be called regular if 1 c contains no critical points and critical if x0 0 for some x0 1 c let a b a b be two regular levels of denote m 1 a b and consider the restriction of to m in morse theory one is interested in the local topological structure of the level sets of m near a critical point and in the relation between this local structure and the topological structure of the set m to be more specific suppose that x0 m is an isolated critical point of then one defines a sequence of critical groups of at x0 by setting cq x0 hq c u u x0 q 0 1 2 0 1 where c x0 c x e x c hq is the q th singular homology group with coefficients in some field f and u is a neighbourhood of x0 define the morse index of x0 to be the maximal dimension of a subspace of e on which the quadratic form x0 y y is negative definite one shows that if x0 is a nondegenerate critical point i e if x0 e e is invertible then cq x0 f received by the editors march 20 1995 1991 mathematics subject classification primary 58e05 secondary 34c25 35j65 35l05 55n20 58f05

a superquadratic indefinite elliptic system and its morse conley floer homology we study critical points of the indefinite functional f h u v int nabla u cdot nabla v h x u v mathrm d x by applying floer s homology construction to the ordinary gradient flow of the functional f on a suitable sobolev space one of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the floer homology for f h

ground states for diffusion system with periodic and asymptotically periodic nonlinearity in this paper we study the following diffusion system t u x u b t x x u v x u g t x v t v x v b t x x v v x v f t x u where b c 1 r r n r n and v x c r n r by using non nehari manifold method developed in tang 2014 we investigate the existence of the ground state solutions of nehari type for the above problem with periodic and asymptotically periodic nonlinearity

multiple solutions for asymptotically linear elliptic systems in this article we consider cooperative and noncooperative elliptic systems that are asymptotically linear at infinity we obtain infinitely many solutions with small energy if the potential is even if the noncooperative system is resonant both at zero and at infinity then the number of nontrivial solutions depends on the dimension of the eigenspaces between resonant values

procedure of quasi averaging for heterophase mixtures abstract for a continuous heterophase system of arbitrary type the procedure of averaging over heterophase fluctuations is defined the consideration is based on the use of the gibbs quasi equilibrium ensembles in the method of separating surfaces averaging over phase configurations is given by means of a functional integration over characteristic functions of phase submanifolds as an illustration a renormalized hamiltonian for a system with a two particle interaction is found convenient methods of separating phases and of calculating the corresponding quasi averages are formulated

forced vibrations of superquadratic hamiltonian systems on etudie l existence des solutions t periodiques ter t 0 du systeme hamiltonien suivant z gh z f t ou g est la matrice asymetrique standard

infinite dimensional cohomology groups and periodic solutions of asymptotically linear hamiltonian systems abstract in this paper we study the existence of nontrivial 2 periodic solutions of asymptotically linear hamiltonian systems we consider the case of resonance both at zero and at infinity and we permit time dependent asymptotic matrices our main tools are an infinite dimensional cohomology theory and a corresponding morse theory recently constructed by w kryszewski and the first author we develop a method to compute the new critical groups

generalized fountain theorem and applications to strongly indefinite semilinear problems abstract by using the degree theory and the topology of kryszewski and szulkin we establish a version of the fountain theorem for strongly indefinite functionals the abstract result will be applied for studying the existence of infinitely many solutions of two strongly indefinite semilinear problems including the semilinear schrodinger equation

infinitely many solutions for non cooperative elliptic systems in this paper we consider the existence of infinitely many solutions of noncooperative elliptic systems perturbed from odd cases

morse theory and asymptotic linear hamiltonian system on considere l existence de solutions 2 periodiques des systemes hamiltoniens asymptotiquement quadratiques jz h z t ou j 0 i i 0 i in la matrice identite de n n

effective hamiltonians for systems with mixed symmetry abstract a rigorous definition for the averaging over phase configurations by means of functional sums over characteristic functions is given for an arbitrary lattice system being the mixture of several phases the general form for the effective hamiltonian of the heterophase system on a lattice is found

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

remarks on approximation methods in degree theory an approximative approach to a generalized theory of the topological mapping degree is presented some new wide classes of operators acting in banach spaces which include a proper mappings of petryshyn are introduced and studied from the viewpoint of the homotopic properties of the topological degree the results are applied in some existence aspects of abstract nonlinear equations

morse index of some min max critical points i application to multiplicity results on donne un resultat general sur des bornes inferieures pour des indices de morse de points critiques obtenus par des principes de min max en combinant cette information avec une inegalite semiclassique on obtient des estimations pointues sur la croissance de certaines valeurs critiques a partir desquelles on deduit de nouveaux resultats de multiplicite pour des solutions d equations elliptiques semi lineaires d ordre 2

existence and multiplicity of solutions for asymptotically hamiltonian elliptic systems in rn abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h u x u v in r n v v x v h v x u v in r n u x 0 and v x 0 as x where the potential v is bounded below and h is asymptotically linear in z as z with z u v by applying a generalized linking theorem of strongly indefinite functionals we obtain solutions for the above system

on hamiltonian elliptic systems with periodic or non periodic potentials abstract this paper is concerned with the following hamiltonian elliptic system u b x u v x u h v x u v v b x v v x v h u x u v for x r n existence and multiplicity of solutions are obtained for the systems with periodic or non periodic potentials v via variational methods

periodic solutions of autonomous newtonian systems partially asymptotically linear at infinity abstract the goal of this article is to study the existence continuation and global bifurcation of autonomous newtonian systems partially asymptotically linear at infinity the main idea is to prove a kind of splitting lemma and apply the degree for s 1 equivariant gradient maps defined by rybicki 1994 in 16

a note on superlinear hamiltonian elliptic systems this paper is concerned with the superlinear periodic elliptic systems of hamiltonian type in the whole space the existence of a ground state solution as well as an infinite number of geometrically distinct solutions is obtained

multiple solutions for discrete periodic nonlinear schrodinger equations in this paper we study the existence of multibump solutions for discrete nonlinear schrodinger equations with periodic potentials we first reduce the existence of multibump homoclinic solutions to the existence of an isolated homoclinic solution with a nontrivial critical group then we study the existence of homoclinics with nontrivial critical groups for both superlinear and asymptotically linear discrete periodic nonlinear schrodinger equations and we provide simple sufficient conditions for the existence of homoclinics with nontrivial critical groups in the positive definite case as an application we get without any symmetry assumptions infinitely many geometrically distinct homoclinic solutions with exponential decay at infinity

simplified bifurcation analysis of the sixteen vertex model based on the standard representation of equivalence classes of models the search for bifurcation points in the parameter space of sixteen vertex models as described in three papers by gaaff and hijmans1 and hijmans and schram2 3 canbe considerably simplified and clarified by representing the equivalence classes of models having the same partition function by their standard models instead of by their normal models the preference for the standard model arises from the fact that it is characterized by parameters belonging to the irreducible representations of the symmetry group of the partition function whereas the parameters defining the normal model belong to a mixed representation the standard model defined in terms of 10 characteristic parameters is unique apart from a finite stargroup of 96 operations arising from the freedom to permute the eigenvalues and eigenvectors of a characteristic 4 x 4 matrix qss bifurcations occur whenever the set of 10 characteristic parameters is invariant with respect to a non trivial little group of operations from the stargroup

infinitely many solutions for diffusion equations without symmetry abstract we consider the following diffusion system t u x u b t x x u v x u h v t x u v t v x v b t x x v v x v h u t x u v t x r r n which is an unbounded hamiltonian system in l 2 r r n r 2 m z u v r r n r m r m b c r r n r n v c r n r and h c 1 r r n r 2 m r suppose that h b and v depend periodically on t and x and that h t x z is superquadratic in z as z without a symmetry assumption on h we establish the existence of infinitely many geometrically distinct solutions via a variational approach

on the bifurcations occurring in the parameter space of the sixteen vertex model ii bifurcations arising from degeneracies in the n matrix the analysis of the bifurcation structure in the 16 dimensional parameter space of homogeneous sixteen vertex models started in part i is completed in this paper as before equivalence classes of models having the same partition function are identified by means of a 4 4 diagonal matrix n and a pair of characteristic 2 2 matrices a and b the bifurcation classes of models studied in this part include also classes for which the n matrix shows degeneracies the four primary bifurcation classes uncovered in part i i e the general and complementary eight vertex models and the one sided and doubly cyclic models each give rise to a hierarchy of two kinds of subbifurcations on the one hand the models corresponding to the different patterns of degeneracies of the n matrix and on the other hand the models for which the rotation angles and characterizing the matrices a and b have values 2 na and 2 nb respectively with na and nb integers

a new morse index theory for strongly indefinite functionals abstract in this paper a new morse index theory for strongly indefinite functionals was developed via g lerkin approximation in particular the abstract theory is valid for those kinds of strongly indefinite functionals corresponding to wave equation and beam equation

on the bifurcations occuring in the parameter space of the sixteen vertex model i discrete bifurcations abstract the 16 dimensional parameter space of homogeneous sixteen vertex models is scanned for bifurcation points i e points corresponding to models which possess extra symmetries not existing in nearby points equivalence classes of models having the same partition function are identified by means of a characteristic normal model represented by a 4 x 4 diagonal matrix n and a pair of 2 x 2 matrices a and b in this paper the matrix n is assumed to be non degenerate and the only bifurcations found are those associated with special types of matrices a and b i e matrices whose decomposition in terms of pauli matrices corresponds to a vector a a 1 a 2 a 3 or b b 1 b 2 b 3 that is invariant with respect to one or more elements of the cubic symmetry group the various bifurcation classes of models found include a the families of general and complementary eight vertex models b discrete sets of doubly and one sided cyclic models and c a number of secondary bifurcation classes within the eight vertex families among which is baxters symmetric eight vertex model

the cohomological span of ls conley index abstract in this paper we introduce a new homotopy invariant the cohomological span of ls conley index we prove the theorems on the existence of critical points for a class of strongly indefinite functionals with the gradient of the form l x k x where l is bounded linear and k is completely continuous we give examples of hamiltonian systems for which our methods give better results than the morse inequalities we also give a formula for the ls index of an isolated critical point which is an extension of the classical dancer theorem for the case of ls index

conley type index applied to hamiltonian inclusions in the paper we develop the theory of a cohomological index of the conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional hilbert space an application to the existence of periodic orbits to asymptotically linear hamiltonian inclusions is presented

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

the o 3 gauge transformation and 3 state vertex models the authors consider the o 3 gauge transformation for three state vertex models on lattices of coordination number three using an explicit mapping between o 3 and sl 2 they establish that there exist exactly six polynomials of the vertex weights which are fundamentally invariant under the o 3 transformation explicit expressions of these fundamental invariants are obtained in the case of symmetric vertex weights

statistical theory of heterophase fluctuations abstract we present the physical basis and mathematical foundations for constructing a microscopic statistical theory of systems with heterophase fluctuations averaging over these fluctuations allows us to find an effective hamiltonian corresponding to a quasi equilibrium situation phase probabilities are defined and new classifications of phase transitions are suggested

little groups of irreps of o 3 so 3 and the infinite axial subgroups a new chain criterion for the determination of little groups is presented this result distinguishes massive rotationally inequivalent irrep basis functions and allows for multiple branching paths consequently it is suitable for application to lie groups applied to the groups o 3 so 3 d h d c v c h and c the result enables the enumeration of all possible little groups together with their associated basis functions these results are relevant to the determination of the symmetry of a material from its linear and nonlinear optical properties and to the choices of order parameters for symmetry breaking in liquid crystals

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

critical point theory for indefinite functionals with symmetries letxbe a hilbert space and c1 x r be strongly indefinite assume in addition that a compact lie groupgacts orthogonally onxand that is invariant in order to find critical points of we develop the limit relative category of fournieret al in the equivariant context we use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem in the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension also other conditions of the classical symmetric mountain pass theorem forg z 2 due to ambrosetti and rabinowitz can be weakened considerably for example we are able to deal with infinite dimensional fixed point spaces the proofs consist of a direct reduction to a relative borsuk ulam type theorem this provides a new proof even for the classical mountain pass theorem the abstract critical point theorems are applied to an elliptic system with dirichlet boundary conditions we only need a weak version of the usual superquadraticity condition the linking theorem can be applied to asymptotically linear hamiltonian systems which are symmetric with respect to a generalized symplectic group action

a new index theory for linear self adjoint operator equations and its applications abstract we develop an index theory for linear self adjoint operator equation with the linear self adjoint operator having no compact resolvent as applications we consider the existence and multiplicity of periodic solutions of wave equations and beam equations

infinitely many periodic solutions for the equation u tt u xx u p 1 u f x t ii on considere l existence de vibrations forcees pour l equation d onde non lineaire u tt u xx u p 1 u f x t x t 0 r u 0 t u t 0 pour t r u x t 2 u x t x t 0 r pour tout p 1 et f x t l p 1 p on demontre l existence d une infinite de solutions periodiques

multiple solutions for a system of equations with p laplacian abstract in the paper by using of the limit index we prove a theorem applying to get multiple critical values of some strongly indefinite nonsmooth functionals and then we apply it to a system of equations involving the p laplacian

the 32 minimal general generator sets of 230 double space groups the 32 minimal general generator sets mggs for the three dimensional double space groups are provided each mggs has a very limited number of parameters that describe the translational parts of the space groups belonging to a crystal class the method of construction is based on the algebraic defining relations of the abstract generators of the point groups the equivalence or inequivalence criteria for the space groups with respect to lattice transformations are established by introducing the definite set u of the unimodular matrices that leaves a lattice type l of each crystal class g based on the set u it will be shown that mere shifts of the lattice origin are necessary and sufficient to determine the inequivalence of almost all the space groups belonging to a crystal class of high symmetry it is simpler to construct mggs of higher symmetry than those of lower symmetry this contrasts with the existing methods which are based on the solvability of the space groups

on the classification of the homogeneous 16 vertex models on a square lattice ii connection with a new type of phase transition abstract in the previous paper we have shown that the classification problem of the homogeneous real 16 vertex models on a square lattice as proposed by gaaff and hijmans is closely connected with the similar problem corresponding to the boson bogoliubov transformation in quantum mechanics this insight leads us in the present paper to use arguments and examples from papers dealing with the latter problem in a further study of the classification problem of the real 16 vertex models we also use a somewhat modified concept of standard or normal 16 vertex model which was introduced by hijmans and schram in a similar study concerning the complex 16 vertex models we show that rather than the single family found by hijmans and schram for the complex variant of the problem the set of equivalence classes of real 16 vertex models can be divided up into seven families this strongly suggests the existence of a new type of phase transitions which manifest themselves when the physical system in the 16 parameter space moves from one family to another

global bifurcations of critical orbits of g invariant strongly indefinite functionals abstract let h be a separable hilbert space which is an orthogonal representation of a compact lie group g and let h r be a g invariant strongly indefinite functional of the class c 1 to study critical orbits of we have defined the degree for g invariant strongly indefinite functionals which is an element of the euler ring u g using this degree we have formulated the rabinowitz alternative for g invariant strongly indefinite functionals the abstract results are applied to the study of global bifurcations of weak solutions of non cooperative systems of elliptic differential equations

ground state solutions for a diffusion system in this paper we study the following diffusion system t u x u v x u g t x v t v x v v x v f t x u where z u v r r n r 2 v x c r n r is a general periodic function under weaker conditions on nonlinearity we establish the existence of ground state solutions via variational methods

existence and continuation of periodic solutions of autonomous newtonian systems abstract in this article we study the existence and the continuation of periodic solutions of autonomous newtonian systems to prove the results we apply the infinite dimensional version of the degree for so 2 equivariant gradient operators defined by the third author in nonlinear anal theory methods appl 23 1 1994 83 102 and developed in topol meth nonlinear anal 9 2 1997 383 417 using the results due to rabier symmetries topological degree and a theorem of z q wang j math 24 3 1994 1087 1115 and wang symmetries and calculation of the degree chinese ann math 10 1989 520 536 we show that the leray schauder degree is not applicable in the proofs of our theorems because it vanishes

a variational approach to subquadratic perturbations of elliptic systems abstract we consider subquadratic perturbations of semilinear elliptic systems which are in variational form a unified approach to both the cooperative and noncooperative problems is presented and a condition of nonquadraticity at infinity is introduced to treat the resonant cases through minimax techniques

a generalized saddle point theorem abstract the well known saddle point theorem is extended to the case of functions defined on a product space x v where x is a banach space and v is a compact manifold under some linking conditions the existence of at least cuplength v 1 critical points is proved the abstract theorems are applied to the existence problems of periodic solutions of hamiltonian systems with periodic nonlinearity and or resonance

periodic solutions of asymptotically linear autonomous hamiltonian systems abstract in this paper we study the existence and the continuation of periodic solutions of autonomous hamiltonian systems we apply the degree for g invariant strongly indefinite functionals defined in gol biewska and rybicki 2011 2 we consider resonant as well as nonresonant case at stationary solutions

existence of forced oscillations for some nonlinear differential equations abstract this article studies the existence of t periodic solutions for systems of nonlinear second order ordinary differential equations assuming function v to be superquadratic it is shown that this system possesses infinitely many t periodic solutions the proof of this result rests on showing that certain homotopy groups of level sets of the functional associated with the system are not trivial some more general results concerning systems of another type are also presented

a twist condition and periodic solutions of hamiltonian systems abstract in this paper we investigate existence of nontrivial periodic solutions to the hamiltonian system hs j z h t z z r 2 n under a general twist condition for the hamiltonian function in terms of the difference of the conley zehnder index at the origin and at infinity we establish existence of nontrivial periodic solutions compared with the existing work in the literature our results do not require the hamiltonian function to have linearization at infinity our results allow interactions at infinity between the hamiltonian and the linear spectra the general twist condition raised here seems to resemble more the spirit of poincare s last geometric theorem

existence and multiplicity of solutions for asymptotically linear nonperiodic hamiltonian elliptic system abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h v x u v v v x v h u x u v x r n u x v x 0 as x where z u v r n r r n 3 assuming the potential v is nonperiodic and sign changing h x z is nonperiodic in x and asymptotically quadratic in z u v existence and multiplicity of solutions are obtained via a variational approach

multibump solutions and critical groups we consider the newtonian system ddot q b t q w q q t with b w periodic in t b positive definite and show that for each isolated homoclinic solution q 0 having a nontrivial critical group in the sense of morse theory multibump solutions with 2 le k le iy bumps can be constructed by gluing translates of q 0 further we show that the collection of multibumps is semiconjugate to the bernoulli shift next we consider the schr odinger equation delta u v x u g x u in rn where v g are periodic in x 1 ldots x n sigma delta v subset 0 iy and we show that similar results hold in this case as well in particular if g x u u 2 2 u n ge 4 and v changes sign then there exists a solution minimizing the associated functional on the nehari manifold this solution gives rise to multibumps if it is isolated

spaces of states for heterophase systems abstract by constructing the spaces of physical states and realizing their decomposition into mutually orthogonal subspaces we show that our statistical theory of heterophase fluctuations 1 is applicable to a large class of systems exhibiting configurational and magnetic transitions we demonstrate that this theory can describe not only transitions between a phase with a discrete symmetry and another phase with a continuous summetry as in the case of a crystal liquid transition of a ferromagnet paramagnet one but also transitions between phases corresponding to point symmetry groups such as polymorphic transitions or reorientational magnetic transitions

on the classification of the homogeneous 16 vertex models on a square lattice i connection with the boson bogoliubov transformation abstract we consider the classification of all homogeneous 16 vertex models on a square lattice which was proposed by gaaff and hijmans we show that this classification can be translated into that is reformulated in terms of a classification of matrices which closely resembles one already investigated with some success in the literature viz the classification connected with the boson bogoliubov transformation in quantum mechanics the main difference between our approach and the one of gaaff and hijmans who also made a reformulation is that we can keep our considerations in conformity with the actual physical problem within the field of real rather than complex numbers in the accompanying paper this leads to results with features not present in the analysis over c which point to a possible connection with a new kind of phase transitions

computations of critical groups and applications to asymptotically linear wave equation and beam equation in this paper by using the morse index theory for strongly indefinite functionals developed in nonlinear anal tma in press we compute precisely the critical groups at the origin and at infinity respectively the abstract theorems are used to study the existence multiplicity of nontrivial periodical solutions for asymptotically wave equation and beam equation with resonance both at infinity and at zero

lattice mixtures of fluctuating phases abstract a method of constructing a renormalized hamiltonian for a heterophase mixture on a lattice is formulated two ways of separating thermodynamic phases are discussed a direct construction of spaces of states having desirable properties and by introducing into the hamiltonian the terms explicitly containing order parameters characterizing the needed phases the methods are illustrated by heterophase models of the ising and heisenberg types

periodic solutions for an asymptotically linear wave equation with resonance abstract in this paper we study the existence and multiplicity of nontrivial periodic solutions for an asymptotically linear wave equation with resonance both at infinity and at zero the main features are using morse theory for the strongly indefinite functional and the precise computation of critical groups under conditions which are more general

computations of cohomology groups and nontrivial periodic solutions of hamiltonian systems by computing the e critical groups at and infinity of the corresponding functional of hamiltonian systems we proved the existence of nontrivial periodic solutions for the systems which may be resonant at and infinity under some new conditions some results in the literature are extended and some new type of theorems are proved the main tool is the e morse theory developed by kryszewski and szulkin

the existence of solutions for superquadratic hamiltonian elliptic systems on rn abstract this paper is concerned with the following hamiltonian elliptic system u v x u h u x u v in r n v v x v h v x u v in r n u x 0 and v x 0 as x where v x c r n r and h x z is superquadratic in z as z with z u v by applying a critical point theorem for strongly indefinite functionals we obtain infinitely many solutions for the above system

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians

existence and multiplicity of semiclassical solutions for asymptotically hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system e 2 u v x u h u x u v in r n e 2 v v x v h v x u v in r n u x 0 and v x 0 as x where e 0 is a small parameter and the potential v is bounded below and h is asymptotically linear in z as z with z u v by applying a generalized linking theorem of strongly indefinite functionals we obtain the existence of multiple semiclassical solutions for the above system

ground state solutions of hamiltonian elliptic systems in dimension two the aim of this paper is to study hamiltonian elliptic system of the form 0 1 where 2 is a bounded domain in the second place we present existence results for the following stationary schrodinger systems defined in the whole plane 0 2 we assume that the nonlinearities f g have critical growth in the sense of trudinger moser by using a suitable variational framework based on the generalized nehari manifold method we obtain the existence of ground state solutions of both systems 0 1 and 0 2

multiple solutions for asymptotically quadratic and superquadratic elliptic system of hamiltonian type this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h v x u v x r n v v x v h u x u v x r n u x 0 v x 0 as x where z u v r n r i r n 3 and the potential v x is nonperiodic and sign changing by applying a generalized linking theorem for strongly indefinite functionals we establish the existence of multiple solutions for asymptotically quadratic nonlinearity as well as the existence of infinitely many solutions for superquadratic nonlinearity

existence and multiplicity of rotating periodic solutions for resonant hamiltonian systems abstract in the present paper we consider a class of resonant hamiltonian systems x j h x t x in r 2 n we will use saddle point reduction morse theory combining the technique of penalized functionals to obtain the existence of nontrivial rotating periodic solutions i e x t t q x t for any t r with t 0 and q an symplectic orthogonal matrix in the case q k i 2 n for any positive integer k such a rotating periodic solution is just a quasi periodic solution moreover if h is even in x we will give the multiplicity of nontrivial rotating periodic solutions by using two abstract critical theorems and previous techniques

existence of solutions for nonperiodic superquadratic hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u r v x u v in r n v v x v r u x u v in r n u x 0 and v x 0 as x where r x z is superquadratic in z as z with z u v by applying a critical point theorem for strongly indefinite functionals we prove the existence of solution for the above system

selection rules for symmetry breaking and symmetry restoration in continuous transitions abstract using landau s phenomenological theory as a framework subduction and chain subduction criteria are discussed as methods for obtaining directly selection rules for symmetry breaking for symmetry restoration induction and chain induction criteria give selection rules minimal symmetry breaking and minimal symmetry restoration are related to satisfaction of chain subduction and induction criteria

differential systems with strongly indefinite variational structure abstract the euler lagrange equations of lagrangians with a strongly indefinite quadratic part are studied by means of the direct min max method of benci and rabinowitz the simplest case is a system of two coupled semilinear poisson equations using an analytic framework of a suitable family of products of fractional sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates

strongly indefinite functionals and multiple solutions of elliptic systems we study existence and multiplicity of solutions of the elliptic system left begin array 20 l updelta u h u x u v hfill text in upomega hfill updelta v h v x u v hfill text in upomega quad u x v x 0 quad text on partial upomega hfill end array right where upomega subset mathbb r n n ge 3 is a smooth bounded domain and h in mathcal c 1 overline upomega times mathbb r 2 mathbb r we assume that the nonlinear term h x u v sim left u right p left v right q r x u v text with mathop lim limits left u v right to infty frac r x u v left u right p left v right q 0 where p in 1 2 2 2n n 2 text and q in 1 infty so some supercritical systems are included nontrivial solutions are obtained when h x u v is even in u v we show that the system possesses a sequence of solutions associated with a sequence of positive energies resp negative energies going toward infinity resp zero if p 2 resp p 2 all results are proved using variational methods some new critical point theorems for strongly indefinite functionals are proved