the twisting trick for double well hamiltonians we show that the use of a twisting trick allows a transparent geometrical analysis of the spectral properties of double well hamiltonians in particular one can provenorm resolvent convergence of the relevant hamiltonians whenever one has two centres of force whose separationr diverges to infinity

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

nonlinear schrodinger equation with growing potential on infinite metric graphs abstract the paper deals with nonlinear schrodinger equations on infinite metric graphs we assume that the linear potential is infinitely growing we prove an existence and multiplicity result that covers both self focusing and defocusing cases furthermore under some additional assumptions we show that solutions obtained bifurcate from trivial ones we prove that these solutions are superexponentially localized our approach is variational and based on generalized nehari manifold

potential approximations to δ an inverse klauder phenomenon with norm resolvent convergence we show that there is a family schrodinger operators with scaled potentials which approximates the interaction hamiltonian in the norm resolvent sense this approximation based on a formal scheme proposed by cheon and shigehara has nontrivial convergence properties which are in several respects opposite to those of the klauder phenomenon

existence of infinitely many stationary solutions of the l 2 subcritical and critical nlse on compact metric graphs abstract we investigate the existence of stationary solutions for the nonlinear schrodinger equation on compact metric graphs in the l 2 subcritical setting we prove the existence of an infinite number of such solutions for every value of the mass in the critical regime the existence of infinitely many solutions is established if the mass is lower than a threshold value while global minimizers of the nls energy exist if and only if the mass is lower or equal to the threshold moreover the relation between this threshold and the topology of the graph is characterized the investigation is based on variational techniques and some new versions of gagliardo nirenberg inequalities

blow up and strong instability of standing waves for the nls δ equation on a star graph abstract we study strong instability by blow up of the standing waves for the nonlinear schrodinger equation with interaction on a star graph the key ingredient is a novel variational technique applied to the standing wave solutions being minimizers of a specific variational problem we also show well posedness of the corresponding cauchy problem in the domain of the self adjoint operator which defines interaction this permits to prove virial identity for the h 1 solutions to the cauchy problem we also prove certain strong instability results for the standing waves of the nls equation on the line

fundamental solution of the heat and schrodinger equations with point interaction abstract we find explicit formulae for the fundamental solution for the heat and time dependent schrodinger equations with point interaction in dimension n 3 we discuss in detail the small time asymptotics of these fundamental solutions the case n 2 has the peculiarity of involving logarithmic terms in the expansion

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

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

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

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

on uncontrollable discrete time bilinear systems which are nearly controllable in this note for a class of uncontrollable discrete time bilinear systems it is shown that the controllable region nearly covers the whole space while the uncontrollable region is only a hypersurface as a result for almost any initial state and any terminal state of the system the former can be transferred to the latter in addition the two dimensional controllability counterexamples in are generalized to arbitrary finite dimensional cases

stability and symmetry breaking bifurcation for the ground states of a nls with a δ interaction we determine and study the ground states of a focusing schrodinger equation in dimension one with a power nonlinearity 2 and a strong inhomogeneity represented by a singular point perturbation the so called attractive interaction located at the origin the time dependent problem turns out to be globally well posed in the subcritical regime and locally well posed in the supercritical and critical regime in the appropriate energy space the set of the nonlinear ground states is completely determined for any value of the nonlinearity power it exhibits a symmetry breaking bifurcation structure as a function of the frequency i e the nonlinear eigenvalue more precisely there exists a critical value of the nonlinear eigenvalue such that if 0 then there exist two non symmetric ground states we prove that before bifurcation i e for finally for 2 and omega gg omega all ground states are unstable the branch of odd ground states for obtaining a family of orbitally unstable stationary states existence of ground states is proved by variational techniques and the stability properties of stationary states are investigated by means of the grillakis shatah strauss framework where some non standard techniques have to be used to establish the needed properties of linearization operators

on near controllability and stabilizability of a class of discrete time bilinear systems in this paper the near controllability and stabilizability of a class of discrete time bilinear systems are studied a necessary and sufficient condition for the near controllability of the systems is first derived then on the basis of the corresponding near controllability criterion the stabilizability problem of the systems is solved in turn

stability of solitary waves for nonlinear schrodinger equations with inhomogeneous nonlinearities we show that the ground state solitary waves of the critical nonlinear schrodinger equation i t t r v 4 d 0 in dimension d 2 are orbitally stable as 0i f v 0 v 4 0 g d v 0 2 where gd is a constant that depends only on d 2002 elsevier science b v all rights reserved

spectral properties of metastable markov semigroups abstract it is shown that if a symmetric markov semigroup e ht on the hilbert space l 2 x is hypercontractive then the approximate degeneracy of the ground state has several consequences concerning other parts of the spectrum of h and concerning the unitary group e iht in particular in the presence of a space inversion symmetry all the eigenvalues occur in pairs with gaps comparable to the gap between the bottom two eigenvalues

locally linearized runge kutta method of dormand and prince in this paper the effect that produces the local linearization of the embedded runge kutta formulas of dormand and prince for initial value problems is studied for this embedded locally linearized runge kutta formulas are defined and their performance is analyzed by means of exhaustive numerical simulations for a variety of well known test equations with different dynamics the simulation results show that the locally linearized formulas exhibit significant higher accuracy than the original ones which implies a substantial reduction of the number of time steps and consequently a sensitive reduction of the overall computational cost of their adaptive implementation

nonlinear instability of half solitons on star graphs abstract we consider a half soliton stationary state of the nonlinear schrodinger equation with the power nonlinearity on a star graph consisting of n edges and a single vertex for the subcritical power nonlinearity the half soliton state is a degenerate critical point of the action functional under the mass constraint such that the second variation is nonnegative by using normal forms we prove that the degenerate critical point is a saddle point for which the small perturbations to the half soliton state grow slowly in time resulting in the nonlinear instability of the half soliton state the result holds for any n 3 and arbitrary subcritical power nonlinearity it gives a precise dynamical characterization of the previous result of adami et al 2012 2 where the half soliton state was shown to be a saddle point of the action functional under the mass constraint for n 3 and for cubic nonlinearity

negative energy ground states for the l2 critical nlse on metric graphs we investigate the existence of ground states with prescribed mass for the focusing nonlinear schrodinger equation with l2 critical power nonlinearity on noncompact quantum graphs we prove that unlike the case of the real line for certain classes of graphs there exist ground states with negative energy for a whole interval of masses a key role is played by a thorough analysis of gagliardo nirenberg inequalities and by estimates of the optimal constants most of the techniques are new and suited to the investigation of variational problems on metric graphs

asymptotic stability of small gap solitons in nonlinear dirac equations we prove dispersive decay estimates for the one dimensional dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear dirac equations with quintic and higher order nonlinear terms

a higher order local linearization method for solving ordinary differential equations the local linearization ll method for the integration of ordinary differential equations is an explicit one step method that has a number of suitable dynamical properties however a major drawback of the ll integrator is that its order of convergence is only two the present paper overcomes this limitation by introducing a new class of numerical integrators called the llt method that is based on the addition of a correction term to the ll approximation in this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the ll method in particular it is proved that the llt method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence the performance of the introduced method is further illustrated through computer simulations

on the lack of bound states for certain nls equations on metric graphs abstract the purpose of this paper is to prove some results on the absence of bound states for certain nonlinear schrodinger equations on noncompact metric graphs with localized nonlinearity in particular we show how the topological and metric properties of graphs affect the existence nonexistence of bound states this work completes the discussion initiated in serra and tentarelli 2016 and tentarelli 2016

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

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

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

dispersion for the schrodinger equation on networks in this paper we consider the schrodinger equation on a network formed by a tree with the last generation of edges formed by infinite strips we give an explicit description of the solution of the linear schrodinger equation with constant coefficients this allows us to prove dispersive estimates which in turn are useful for solving the nonlinear schrodinger equation the proof extends also to the laminar case of positive step function coefficients having a finite number of discontinuities

stable solutions of the logarithmic schrodinger equation on considere l equation de schrodinger suivante u t u ulog u 2 0 dans r r n on donne des resultats sur le comportement asymptotique des solutions

a solution to the focusing 3d nls that blows up on a contracting sphere we rigorously construct radial h 1 solutions to the 3d cubic focusing nls equation i t 2j j 2 0 that blow up along a contracting sphere with blow up time set to t 0 the solutions concentrate on a sphere at radius t 1 3 but focus towards this sphere at the faster rate t 2 3 such dynamics were originally proposed heuristically by degtyarev zakharov rudakov 2 in 1975 and independently later in holmer roudenko 5 in 2007 where it was demonstrated to be consistent with all conservation laws of this equation in the latter paper it was proposed as a solution that would yield divergence of the l 3 norm within the wide radius kr u t k 1 2 l2 but not within the tight radius kr u t k 2 l2

efficient computation of phi functions in exponential integrators abstract a crucial point for the computational efficiency of the exponential type integrators for differential equations is the effective evaluation of the integrals involving matrix exponential called phi functions focusing on exponential integrators for differential equations of small dimensions for which the krylov based approximations for the phi functions are not viable in this article refined algorithms are introduced for speeding up both the evaluation of several phi functions simultaneously and the computation of a linear combination of phi functions times matrices the algorithms are derived from a meticulous approximation to the exponential of certain partitioned matrix by adapting the conventional pade method to the special structure of this matrix numerical simulations are provided to evaluate the performance of the proposed algorithms and compare them with others in the literature

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

rate of convergence of local linearization schemes for initial value problems there is a large variety of local linearization ll schemes for the numerical integration of initial value problems which differ with respect to the algorithm that is used in the numerical implementation of the local linear discretization however in contrast with the ll discretization the order of convergence of the ll schemes have not been studied so far in this paper a general result about that matter is given in addition a brief survey of the main implementations of the ll method is also presented

stable standing waves for a nls on star graphs as local minimizers of the constrained energy abstract on a star graph made of n 3 halflines edges we consider a schrodinger equation with a subcritical power type nonlinearity and an attractive delta interaction located at the vertex from previous works it is known that there exists a family of standing waves symmetric with respect to the exchange of edges that can be parametrized by the mass or l 2 norm of its elements furthermore if the mass is small enough then the corresponding symmetric standing wave is a ground state and consequently it is orbitally stable on the other hand if the mass is above a threshold value then the system has no ground state here we prove that orbital stability holds for every value of the mass even if the corresponding symmetric standing wave is not a ground state since it is anyway a local minimizer of the energy among functions with the same mass the proof is based on a new technique that allows to restrict the analysis to functions made of pieces of soliton reducing the problem to a finite dimensional one in such a way we do not need to use direct methods of calculus of variations nor linearization procedures

recent progress in extrapolation methods for ordinary differential equations the paper is intended to give a survey on the state of the art of extrapolation methods for initial value problems in ordinary differential equation systems all basic discretizations which are su

analysis still matters a surprising instance of failure of runge kutta felberg ode solvers this paper provides a nice example to illustrate that without supporting analysis a numerical simulation may lead to incorrect conclusions we explore a pedagogical example of failure of runge kutta felberg rkf algorithms for a simple dynamical system that models the coupling of two oscillators although the system appears to be well behaved the explicit rkf solvers provide erratic numerical solutions the mode of failure is based in a period doubling route to chaos due to the existence of stable linear solutions in the problem

dynamical stability of metastable states abstract we continue our study of symmetric markov semigroups for which the zero eigenvalue is almost but not quite degenerate we show that the metastable states of the semigroup are dynamically stable under small perturbations without any of the technical restrictions or approximations of our earlier papers

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

symmetry breaking bifurcation in the nonlinear schrodinger equation with symmetric potentials we consider the focusing attractive nonlinear schrodinger nls equation with an external symmetric potential which vanishes at infinity and supports a linear bound state we prove that the symmetric nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non degenerate local maxima at zero under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork in the particular case of double well potentials with large separation the power of nonlinearity determines the subcritical or supercritical character of the bifurcation the results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter

self focusing on bounded domains the critical nonlinear schrodinger equation nls on bounded domains models the propagation of cw laser beams in hollow core fibers unlike the nls on unbounded domains which models propagation in bulk media the ground state waveguide solutions are stable and the condition of critical power for singularity formation is generically sharp

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

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

solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities we study the nonlinear schrodinger nls generalized korteweg de vries gkdv and generalized benjamin ono equations gbo with critical power nonlinearities the ground state solitary wave is conditionally stable modulo the group of translations in space and phase and dilations this is a natural extension of the result for subcritical nonlinearities where solitary waves are stable modulo translations alone to a case where solutions may blow up we also present results on global existence for gbo and the structure of singularities for certain solutions of gbo which are analogues of previous results by the author for nls and gkdv

metastability and low lying spectra in reversible markov chains we study a large class of reversible markov chains with discrete state space and transition matrix pn we define the notion of a set of metastable points as a subset of the state space n such that i this set is reached from any point x n without return to x with probability at least bn while ii for any two points x y in the metastable set the probability t 1x y to reach y from x without return to x is smaller than an 1 bn under some additional non degeneracy assumption we show that in such a situation

purely nonlinear instability of standing waves with minimal energy we consider hamiltonian systems with u 1 symmetry we prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable in spite of the absence of linear instability essentially the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system we apply our theory to the non

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

dynamic properties of the local linearization method for initial value problems some dynamic properties of the local linearization ll scheme for the numerical integration of initial value problems in ordinary differential equations odes are investigated specifically the general conditions under which this scheme preserves the stationary points and periodic orbits of the odes and the local stability at these steady states are studied these dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes in addition a brief review of the computational implementations of the ll scheme is also presented

comments on a controllability counterexample and the continuation lemma a technical note in this journal vol 50 no 6 pp 840 841 june 2005 by elliott gives a bilinear example showing that the euler discretization of a noncontrollable continuous time system can be controllable the example is correct but there was a flaw in a result of the tn lemma 1 for discrete time systems local controllability implies controllability that has independent interest in this note the lemma is reformulated as a conjecture for continuous in state systems and it is also proved under additional conditions for a class of two dimensional bilinear systems the euler discretization is shown directly to be small controllable a fortiori controllable

normal form for the symmetry breaking bifurcation in the nonlinear schrodinger equation abstract we derive and justify a normal form reduction of the nonlinear schrodinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double well symmetric potential we prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time dependent solutions of the nonlinear schrodinger equation over long but finite time intervals

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

existence of dynamics for a 1d nls equation perturbed with a generalized point defect in the present paper we study the well posedness for the one dimensional cubic nls perturbed by a generic point interaction point interactions are described as the 4 parameter family of self adjoint extensions of the symmetric 1d laplacian defined on the regular functions vanishing at a point and in the present context can be interpreted as localized defects interacting with the nls field a previously treated special case is given by an nls equation with a defect which we generalize and extend as far as well posedness is concerned to the whole family of point interactions we prove existence and uniqueness of the local cauchy problem in strong form initial data and evolution in the operator domain of point interactions weak form initial data and evolution in the form domain of point interactions and conservation laws of mass and energy are proved for finite energy weak solutions of the problem which imply global existence of the dynamics a technical difficulty arises due to the fact that a power nonlinearity does not preserve the form domain for a subclass of point interactions to overcome it a technique based on the extension of resolvents of the linear part of the generator to maps between a suitable hilbert space and the energy space is devised and estimates are given which show the needed regularization properties of the nonlinear flow

instability of bound states of a nonlinear schrodinger equation with a dirac potential abstract we study analytically and numerically the stability of the standing waves for a nonlinear schrodinger equation with a point defect and a power type nonlinearity a major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves this is overcome by a perturbation method and continuation arguments among others in the case of a repulsive defect we show that the standing wave solution is stable in h rad 1 r and unstable in h 1 r under subcritical nonlinearity further we investigate the nature of instability under critical or supercritical nonlinear interaction we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints in the subcritical radial case unstable bound states cannot collapse but rather narrow down until they reach the stable regime a finite width instability in the nonradial repulsive case all bound states are unstable and the instability is manifested by a lateral drift away from the defect sometimes in combination with a finite width instability or a blowup instability

a controllability counterexample simulation by time discretizations can be qualitatively misleading as counterexamples a class of noncontrollable single input two dimensional bilinear control systems is presented whose euler discretizations are controllable on the punctured plane

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

orbital stability of solitary waves of a 3 coupled nonlinear schrodinger system abstract in this paper consideration is given to the 3 coupled nonlinear schrodinger system i t u j 2 x 2 u j i 1 3 b i j u i 2 u j 0 where u j are complex valued functions of x t r 2 j 1 2 3 and b i j are positive constants satisfying b i j b j i it will be shown first that if the symmetric matrix b b i j satisfies certain conditions then ground state solutions of the 3 coupled nonlinear schrodinger system exist and moreover they are orbitally stable the theory is then extended to include solitary waves as well in particular it will be shown that when a solitary wave is perturbed the perturbed solution must stay close to a solitary wave profile in which the translation and phase parameters are prescribed functions of time properties of these functions are then studied this is a continuous work of our previous paper where the 2 coupled nonlinear schrodinger system was considered

stability of standing waves for a nonlinear klein gordon equation with delta potentials abstract in this paper we study local well posedness and orbital stability of standing waves for a singularly perturbed one dimensional nonlinear klein gordon equation we first establish local well posedness of the cauchy problem by a fixed point argument unlike the unperturbed case a noteworthy difficulty here arises from the possible non unitarity of the semigroup generating the corresponding linear evolution we then show that the equation is hamiltonian and we establish several stability instability results for its standing waves our analysis relies on a detailed study of the spectral properties of the linearization of the equation and on the well known slope condition for orbital stability

on the structure of critical energy levels for the cubic focusing nls on star graphs we provide information on a non trivial structure of phase space of the cubic nonlinear schrodinger nls on a three edge star graph we prove that in contrast to the case of the standard nls on the line the energy associated with the cubic focusing schrodinger equation on the three edge star graph with a free kirchhoff vertex does not attain a minimum value on any sphere of constant l2 norm we moreover show that the only stationary state with prescribed l2 norm is indeed a saddle point

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

bifurcation and stability for nonlinear schrodinger equations with double well potential in the semiclassical limit we consider the stationary solutions for a class of schrodinger equations with a symmetric double well potential and a nonlinear perturbation here in the semiclassical limit we prove that the reduction to a finite mode approximation give the stationary solutions up to an exponentially small term and that symmetry breaking bifurcation occurs at a given value for the strength of the nonlinear term the kind of bifurcation picture only depends on the nonlinearity power we then discuss the stability instability properties of each branch of the stationary solutions finally we consider an explicit one dimensional toy model where the double well potential is given by means of a couple of attractive dirac s delta pointwise interactions

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

exponential rosenbrock type methods we introduce a new class of exponential integrators for the numerical integration of large scale systems of stiff differential equations these so called rosenbrock type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the jacobian in contrast to standard integrators the methods are fully explicit and do not require the numerical solution of linear systems we analyze the convergence properties of these integrators in a semigroup framework of semilinear evolution equations in banach spaces in particular we derive an abstract stability and convergence result for variable step sizes this analysis further provides the required order conditions and thus allows us to construct pairs of embedded methods we present a third order method with two stages and a fourth order method with three stages respectively the application of the required matrix functions to vectors are computed by krylov subspace approximations we briefly discuss these implementation issues and we give numerical examples that demonstrate the efficiency of the new integrators

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

some new results on the two dimensional kinetic ising model in the phase coexistence region we consider a glauber dynamics reversible with respect to the two dimensional ising model in a finite square of sidel with open boundary conditions in the absence of an external field and at large inverse temperature we prove that the gap in the spectrum of the generator restricted to the invariant subspace of functions which are even under global spin flip is much larger than the true gap as a consequence we are able to show that there exists a new time scaleteven much smaller than the global relaxation timetrel such that with large probability any initial configuration first relaxes to one of the two phases in a time scale of orderteven and only after a time scale of the order oftrel does it reach the final equilibrium by jumping via a large deviation to the opposite phase it also follows that with large probability the time spent by the system during the first jump from one phase to the opposite one is much shorter than the relaxation time

precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime we investigate the close connection between metastability of the reversible diffusion process x defined by the stochastic differential equation d xt f xt d t 2e d wt e 0 and the spectrum near zero of its generator l f where f d and w denotes brownian motion on d for generic f to each local minimum of f there corresponds a metastable state we prove that the distribution of its rescaled relaxation time converges to the exponential distribution as 0 with optimal and uniform error estimates each metastable state can be viewed as an eigenstate of l with eigenvalue which converges to zero exponentially fast in 1 modulo errors of exponentially small order in 1 this eigenvalue is given as the inverse of the expected metastable relaxation time the eigenstate is highly concentrated in the basin of attraction of the corresponding trap

on the standing waves of the nls log equation with a point interaction on a star graph abstract we study the nonlinear schrodinger equation with logarithmic nonlinearity on a star graph g at the vertex an interaction occurs described by a boundary condition of delta type with strength r we investigate the orbital stability and the spectral instability of the standing wave solutions e i t x to the equation when the profile x has mixed structure i e has bumps and tails in our approach we essentially use the extension theory of symmetric operators by krein von neumann and the analytic perturbations theory

bounds on the spectrum for markov chains and markov processes a generalization of cheeger s inequality we prove a general version of cheeger s inequality for discrete time markov chains and continuous time markovian jump processes both reversible and nonreversible with general state space we also prove a version of cheeger s inequality for markov chains and processes with killing as an application we prove l 2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs

instability of bound states for abstract nonlinear schrodinger equations abstract we study the instability of bound states for abstract nonlinear schrodinger equations we prove a new instability result for a borderline case between stability and instability we also reprove some known results in a unified way

avoiding the exactness of the jacobian matrix in rosenbrock formulae abstract a new class of methods for solving stiff systems which avoids the exactness of the jacobian matrix is introduced the order conditions for methods of order p 5 are given the linear stability properties for such methods are analysed numerical testing are also included