variational properties and orbital stability of standing waves for nls equation on a star graph we study standing waves for a model of nonlinear schr odinger equation on a graph the graph is obtained joining n halflines at a vertex i e it is a star graph at the vertex an interaction occurs due to a boundary condition of delta type with strength alpha leqslant 0 which includes the free or kirchhoff case alpha 0 there imposed the nonlinearity is of focusing power type the dynamics is given by an equation of the form i frac d dt psi t h psi t psi t 2 mu psi t where h is the s a operator which generates the linear schr odinger dynamics on the graph in the hilbert space l 2 gg we show the existence of several families of standing waves solutions of the form phi t e i omega t psi omega where psi omega is a vector amplitude parametrized by a frequency omega they exist for every sign of the coupling at the vertex attractive alpha 0 and for every omega frac alpha 2 n 2 the number of the families depends on the number n of the edges of the graph we moreover determine the ground state which is the standing wave the amplitude of which minimizes the action on the natural nehari constraint and order the various families of standing waves according to their increasing action thus determining a nonlinear spectrum of the problem finally we show that the ground state is orbitally stable for every allowed omega if the nonlinearity is subcritical or critical for supercritical nonlinearities there exists an omega frac alpha 2 n 2 such that the ground state is orbitally stable for frac alpha 2 n 2 omega omega action minimization requires technical subtleties in particular an adaptation of decreasing rearrangements on a star graph is developed with a generalization of the p olya szeg h o inequality

instability of standing waves for klein gordon zakharov equations with different propagation speeds in three space dimensions abstract this paper is concerned with the standing wave for klein gordon zakharov equations with different propagation speeds in three space dimensions the existence of standing wave with the ground state is established by applying an intricate variational argument and the instability of the standing wave is shown by applying pagne and sattinger s potential well argument and levine s concavity method

on transitions to stationary states in one dimensional nonlinear wave equations we consider the long time asymptotics of solutions to one dimensional nonlinear wave equations which are infinite dimensional hamiltonian systems we assume that the nonlinear term is concentrated at a finite segment of the line we prove long time convergence to stationary states for all finite energy solutions in the frechet topology defined by local energy seminorms this means that the set of stationary states is a point attractor for the systems in the frechet topology the investigation is inspired by n bohr s postulate on the transitions between stationary states in quantum systems

analysis of the linearization around a critical point of an infinite dimensional hamiltonian system partial differential equations that conserve energy can often be written as infinite dimensional hamiltonian systems of the following general form du dt je u where j x x is a symplectic matrix i e jj 1 and e x r is a c 2 functional defined on some hilbert space x

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

representation theory of finite semigroups semigroup radicals and formal language theory in this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field generalizing results of rhodes for the field of complex numbers applications are given to obtain many new results as well as easier proofs of several results in the literature involving triangularizability of finite semigroups which semigroups have split basic semigroup algebras two sided semidirect product decompositions of finite monoids unambiguous products of rational languages products of rational languages with counter and cerny s conjecture for an important class of automata

solitary wave dynamics in an external potential we study the behavior of solitary wave solutions of some generalized nonlinear schrodinger equations with an external potential the equations have the feature that in the absence of the external potential they have solutions describing inertial motions of stable solitary waves we consider solutions of the equations with a non vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that over a large interval of time they describe a solitary wave whose center of mass motion is a solution of newton s equations of motion for a point particle in the given external potential up to small corrections corresponding to radiation damping

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

synchronizing automata with a letter of deficiency 2 we present two infinite series of synchronizing automata with a letter of deficiency 2 whose shortest reset words are longer than those for synchronizing automata obtained by a straightforward modification of cerny s construction

actions and irreducible representations of the mapping class group let g be a countable discrete group call two subgroups h 1 and h 2 of g commensurable if h 1 cap h 2 has finite index in both h 1 and h 2 we say that an action of g on a discrete set x has noncommensurable stabilizers if the stabilizers of any two distinct points of x are not commensurable we prove in this paper that the action of the map ping class group on the complex of curves has noncommensurable stabilizers following a method due to burger and de la harpe this action leads to constructions of irreducible unitary representations of the mapping class group

on asymptotic stability of stationary solutions to nonlinear wave and klein gordon equations we consider nonlinear wave and klein gordon equations with general nonlinear terms localized in space conditions are found which provide asymptotic stability of stationary solutions in local energy norms these conditions are formulated in terms of spectral properties of the schrodinger operator corresponding to the linearized problem they are natural extensions to partial differential equations of the known lyapunov condition for the nonlinear wave equation in three dimensional space we find asymptotic expansions as t of the solutions which are close enough to a stationary asymptotically stable solution

an instability mechanism for radially symmetric standing waves of a nonlinear schrodinger equation abstract a condition is proved for the spectrum of nonlinear schrodinger equations linearised at a standing wave to have a positive eigenvalue the standing waves considered are radially symmetric ones in higher space dimensions the instability result is applied to show that if there are multiple asymptotically positive nondegenerate waves with a fixed number of zeroes then there is an unstable one the techniques used are dynamical systems arguments and involve a shooting argument in the space of lagrangian planes

stable standing waves of nonlinear klein gordon equations in this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear klein gordon equations

braid groups and the co hopfian property abstract let b n be the braid group on n 4 strands we prove that b n modulo its center is co hopfian we then show that any injective endomorphism of b n is geometric in the sense that it is induced by a homeomorphism of a punctured disk we further prove that any injection from b n to b n 1 is geometric for n 7 additionally we obtain analogous results for mapping class groups of punctured spheres the methods use thurston s theory of surface homeomorphisms and build upon work of ivanov mccarthy

on the dynamics of solitons in the nonlinear schrodinger equation we study the behavior of the soliton solutions of the equation i frac partial psi partial t frac 1 2m delta psi frac 1 2 w varepsilon prime psi v x psi where w varepsilon prime is a suitable nonlinear term which is singular for varepsilon 0 we use the strong nonlinearity to obtain results on existence shape stability and dynamics of the soliton the main result of this paper theorem 1 shows that for varepsilon rightarrow 0 the orbit of our soliton approaches the orbit of a classical particle in a potential v x

stability and folds it is known that when one branch of a simple fold in a bifurcation diagram represents linearly stable solutions the other branch represents unstable solutions the theory developed here can predict instability of some branches close to folds without knowledge of stability of the adjacent branch provided that the underlying problem has a variational structure first one particular bifurcation diagram is identified as playing a special role the relevant diagram being specified by the choice of functional plotted as ordinate the results are then stated in terms of the shape of the solution branch in this distinguished bifurcation diagram in many problems arising in elasticity the preferred bifurcation diagram is the loaddisplacement graph the theory is particularly useful in applications where a solution branch has a succession of folds

an algebraic characterization of the point pushing subgroup abstract the point pushing subgroup p g of the mapping class group mod g 1 of a surface with marked point is an embedding of 1 g given by pushing the marked point around loops we prove that for g 3 the subgroup p g is the unique normal genus g surface subgroup of mod g 1 as a corollary to this uniqueness result we give a new proof that out mod g 1 1 where out denotes the outer automorphism group a proof which does not use automorphisms of complexes of curves ingredients in our proof of this characterization theorem include combinatorial group theory representation theory the johnson theory of the torelli group surface topology and the theory of lie algebras

curve complexes are rigid any quasi isometry of the curve complex is bounded distance from a simplicial automorphism as a consequence the quasi isometry type of the curve complex determines the homeomorphism type of the surface

slowly synchronizing automata with fixed alphabet size abstract it was conjectured by cern in 1964 that a synchronizing dfa on n states always has a shortest synchronizing word of length at most n 1 2 and he gave a sequence of dfas reaching this bound in this paper we investigate the role of the alphabet size for each possible alphabet size we count dfas on n 6 states which synchronize in n 1 2 e steps for all e 2 n 2 furthermore we give constructions of automata with any number of states and 3 4 or 5 symbols which synchronize slowly namely in n 2 3 n o 1 steps in addition our results prove cern s conjecture for n 6 our computations lead to 31 dfas on 3 4 5 or 6 states which synchronize in n 1 2 steps of these dfa s 19 are new the remaining 12 which were already known are exactly the minimal ones the 19 new dfas are extensions of automata which were already known but for n 3 we prove that the cern automaton on n states does not admit non trivial extensions with the same smallest synchronizing word length n 1 2

local well posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude abstract we study the local well posedness of a periodic nonlinear equation for surface waves of moderate amplitude in shallow water we use an approach proposed by kato which is based on semigroup theory for quasi linear equations we also show that singularities for the model equation can occur only in the form of wave breaking in particular surging breakers

relaxation of solitons in nonlinear schrodinger equations with potential abstract in this paper we study dynamics of solitons in the generalized nonlinear schrodinger equation nls with an external potential in all dimensions except for 2 for a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space centered near different critical points of the potential we call those solutions which are centered near the minima of the potential and which minimize energy restricted to l 2 unit sphere trapped solitons or just solitons in this paper we prove under certain conditions on the potentials and initial conditions that trapped solitons are asymptotically stable moreover if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position the dynamical law of motion of the soliton i e effective equations of motion for the soliton s center and momentum is close to newton s equation but with a dissipative term due to radiation of the energy to infinity

global attractor for a nonlinear oscillator coupled to the klein gordon field the long time asymptotics is analyzed for all finite energy solutions to a model mathbf u 1 invariant nonlinear klein gordon equation in one dimension with the nonlinearity concentrated at a single point each finite energy solution converges as t to the set of all nonlinear eigenfunctions of the form x e i t the global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

orbital stability of double solitons for the benjamin ono equation in this paper we prove the orbital stability of double solitons for the benjamin ono equation in the case of the kdv equation this stability has been proved in 17 parts of the proof given there rely on the fact that the euler lagrange equations for the conserved quantities of the kdv equation are ordinary differential equations since this is not the case for the benjamin ono equation new methods are required our approach consists in using a new invariant for multi solitons and certain new identities motivated by the sylvester law of inertia

on stabilization of string nonlinear oscillator interaction abstract in the present paper we consider a system of equations that describes the interaction of a nonlinear oscillator with an infinite string the main result is the stabilization roughly speaking each finite energy solution to the system tends to a stationary solution as t and similarly as t the proof uses the description of a reversible system by an irreversible the limit stationary solutions corresponding to t may be different and arbitrary the result gives a mathematical model of transitions to stationary states in reversible systems these transitions are similar to bohr ones such transitions are impossible for finite dimensional hamiltonian systems and for linear autonomous shrodinger equations the paper contains the complete exposition and an extension of the author s recent results

unstable ground state of nonlinear klein gordon equations on demontre l instabilite de l etat fondamental des equations de klein gordon non lineaires a n dimensions d espace n 3

stability and instability of the kdv solitary wave under the kp i flow we consider the kp i and gkp i equations in mathbb r times mathbb r 2 pi mathbb z we prove that the kdv soliton with subcritical speed 0 c in the spirit of the work by duyckaerts and merle gafa 18 1787 1840 2009 we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity this last result also holds for the gkp i equation

a centre stable manifold for the focussing cubic nls in mathbb r 1 3 consider the focussing cubic nonlinear schrodinger equation in mathbb r 3 i psi t delta psi psi 2 psi quad 0 1 it admits special solutions of the form eit where phi in mathcal s mathbb r 3 is a positive 0 solution of delta phi alpha phi phi 3 quad 0 2 the space of all such solutions together with those obtained from them by rescaling and applying phase and galilean coordinate changes called standing waves is the 8 dimensional manifold that consists of functions of the form e i v cdot gamma phi cdot y alpha we prove that any solution starting sufficiently close to a standing wave in the sigma w 1 2 mathbb r 3 cap x 1 l 2 mathbb r 3 norm and situated on a certain codimension one local lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves furthermore we show that mathcal n is invariant under the hamiltonian flow locally in time and is a centre stable manifold in the sense of bates jones batjon the proof is based on the modulation method introduced by soffer and weinstein for the l2 subcritical case and adapted by schlag to the l2 supercritical case an important part of the proof is the keel tao endpoint strichartz estimate in mathbb r 3 for the nonselfadjoint schrodinger operator obtained by linearizing 0 1 around a standing wave solution all results in this paper depend on the standard spectral assumption that the hamiltonian mathcal h left begin array cc delta 2 phi cdot alpha 2 alpha quad phi cdot alpha 2 phi cdot alpha 2 quad delta 2 phi cdot alpha 2 alpha end array right quad 0 3 has no embedded eigenvalues in the interior of its essential spectrum infty alpha cup alpha infty

the cerný conjecture for one cluster automata with prime length cycle abstract we prove the cern conjecture for one cluster automata with prime length cycle consequences are given for the hybrid road coloring cern conjecture for strongly connected aperiodic digraphs with a proper cycle of prime length

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

on injective homomorphisms between teichmuller modular groups i in this paper and its sequel we prove that injective homomorphisms between teichmuller modular groups of compact orientable surfaces are necessarily isomorphisms if an appropriately measured size of the surfaces in question differs by at most one in particular we establish the co hopfian property for modular groups of surfaces of positive genus

the structure of the torelli group i a finite set of generators for j this is the first of three papers concerning the so called torelli group let m mg be a compact orientable surface of genus g having n boundary components and let 9 xg be its mapping class group that is the group of orientation preserving diffeomorphisms of m which are 1 on the boundary am modulo isotopies which fix 3m pointwise this group is also known to the complex analysts as the teichmuller group or modular group if n 0 or 1 let further 4 jg be the subgroup of d1 which acts trivially on h1 m z the topologists have no traditional name for a but the analysts tell me it was known classically and is called the torelli group several interesting problems and conjectures exist concerning f the principal one can be found in kirby s problem list k and asks if gg is finitely generated in this first paper we shall answer the question affirmatively for both gg 0 and fgg when g 3 and shall give a fairly simple set of generators two other conjectures were made by the author the first involves the subgroup j of f which is generated by twists on nulhomologous simple closed curves ji produces a surjective homomorphism t fgg 1 a3h1 m z which kills c and it is conjectured there that in fact 5y ker t the proof of this is the content of the second paper in the third paper we use the results of the second to compute the abelianization f f explicitly thereby verifying another conjecture in ji the first reasonably simple but infinite set of generators for fg0 was produced by powell in p his generators consist of two types a twists on bounding simple closed curves b opposite twists on a bounding pair of disjoint homologous simple closed curves each of which are nonbounding using his result the author showed in j2 that the maps of the second type which we call bp maps for bounding pair are actually sufficient to generate both kg 0 and fg 1 for g 3 and in fact that we need only those whose two curves bound a genus one subsurface of m note that for g 2 all bp maps are trivial and hence the result fails in this case in the finite set of generators produced in this paper only

on the asymptotic stability of solitary waves in the discrete schrodinger equation coupled to a nonlinear oscillator abstract the long time asymptotics is analyzed for finite energy solutions of the 1d discrete schrodinger equation coupled to a nonlinear oscillator the coupled system is invariant with respect to the phase rotation group u 1 for initial states close to a solitary wave the solution converges to a sum of another solitary wave and a dispersive wave which is a solution to the free schrodinger equation the proofs use the strategy of buslaev perelman v s buslaev g s perelman on the stability of solitary waves for nonlinear schrodinger equations amer math soc trans 2 164 1995 75 98 the linearization of the dynamics on the solitary manifold the symplectic orthogonal projection method of majorants etc

the regularized benjamin ono and bbm equations well posedness and nonlinear stability abstract nonlinear stability of nonlinear periodic solutions of the regularized benjamin ono equation and the benjamin bona mahony equation with respect to perturbations of the same wavelength is analytically studied these perturbations are shown to be stable we also develop a global well posedness theory for the regularized benjamin ono equation in the periodic and in the line setting in particular we show that the cauchy problem in both periodic and nonperiodic case cannot be solved by an iteration scheme based on the duhamel formula for negative sobolev indices

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

sharp threshold of global existence and instability of standing wave for a davey stewartson system this paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave phi t x e i omega t u x for the system i phi t delta phi a phi p 1 phi b e 1 phi 2 phi 0 quad t geq 0 quad x in bf r n quad quad quad rm ds where a 0 b 0 1 0 by combining the former results

normal subgroups of mapping class groups and the metaconjecture of ivanov we prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group the proof relies on another theorem we prove which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group these results resolve the metaconjecture of n v ivanov which asserts that any sufficiently rich object associated to a surface has automorphism group isomorphic to the extended mapping class group for a broad class of such objects as applications we show 1 right angled artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support 2 normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support and 3 distinct normal subgroups of the mapping class group with elements of small support are not isomorphic our results also suggest a new framework for the classification of normal subgroups of the mapping class group

long time oscillation of solutions of nonlinear schrodinger equations near minimal mass ground state abstract in this paper we consider the long time dynamics of radially symmetric solutions of nonlinear schrodinger equations nls having a minimal mass ground state in particular we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time more precisely we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator then we show that the finite dimensional 2 d part approximately obeys newton s equation of motion for a particle in an anharmonic potential well showing that the infinite dimensional part is well separated from the finite dimensional part we will have long time oscillation

on the cubical geometry of higman s group we investigate the cocompact action of higman s group on a cat 0 square complex associated to its standard presentation we show that this action is in a sense intrinsic which allows for the use of geometric techniques to study the endomorphisms of the group and show striking similarities with mapping class groups of hyperbolic surfaces outer automorphism groups of free groups and linear groups over the integers we compute explicitly the automorphism group and outer automorphism group of higman s group and show that the group is both hop an and co hop an we actually prove a stronger rigidity result about the endomorphisms of higman s group every non trivial morphism from the group to itself is an automorphism we also study the geometry of the action and prove a surprising result although the cat 0 square complex acted upon contains uncountably many ats the higman group does not contain subgroups isomorphic to z finally we show that this action possesses features reminiscent of negative curvature which we use to prove a re ned version of the tits alternative for higman s group higman s group was constructed in 15 as the rst example of a nitely presented in nite group without non trivial nite quotients it is given by the following presentation h ai i z 4z aiai 1a i a 2 i 1 i z 4z if some partial results about higman s group are known the structure of the group and its geometry have remained mostly elusive the group is known to split as an amalgamated product over several non abelian free subgroups and such splittings can be used to derive various properties of the group for instance the group has nite asymptotic dimension by work of bell dranishnikov 1 a property that implies the novikov conjecture and the coarse baum connes conjecture for the group 36 37 such splittings have also been used to show that the higman group admits in a sense many quotients schupp developed small cancellation tools to show that the group is sq universal 30 that is every countable group embeds in a quotient of h minasyan osin used these splittings to show that the group is 2010 mathematics subject classi cation primary 20f65 secondary 20f28

periodic finite genus solutions of the kdv equation are orbitally stable abstract the stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade the kdv equation is known to have large families of periodic solutions that are parameterized by hyperelliptic riemann surfaces they are generalizations of the famous multi soliton solutions we show that all such periodic solutions are orbitally stable with respect to subharmonic perturbations perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution

automorphisms of surface mapping class groups a recent theorem of n ivanov ivanov of the leningrad branch of the steklov mathematical institute has shown that the outer automorphism groups of surface mapping class groups are finite in this report we shall give explicit descriptions of the outer automorphism groups of both the mapping class groups and extended mapping class groups of closed connected orientable surfaces of negative euler characteristic the extended mapping class groups are the extensions of the mapping class groups by the orientation reversing isotopy classes for surfaces of genus greater than two these are respectively a cyclic group of order two and the trivial group for a surface of genus two these are both noncyclic groups of order four once again the hypergeometric involution in genus two plays a unique role

reset sequences for monotonic automata natarajan reduced the problem of designing a certain type of mechanical parts orienter to that of finding reset sequences for monotonic deterministic finite automata he gave algorithms that in polynomial time either find such sequences or prove that no such sequence exists in this paper a new algorithm based on breadth first search is presented that runs in faster asymptotic time than natarajan s algorithms and in addition finds the shortest possible reset sequence if such a sequence exists tight bounds on the length of the minimum reset sequence are given the time and space bounds of another algorithm given by natarajan are further improved that algorithm finds reset sequences for arbitrary deterministicfinite automata when all states are initially possible

on the solitary wave dynamics under slowly varying medium for nonlinear schrodinger equations we consider the problem of a solitary wave propagation in a slowly varying medium for a variable coefficients nonlinear schrodinger equation we prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media moreover we describe for all time the behavior of these solutions which include refracted and reflected solitary waves depending on the initial energy

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples