the hamiltonian structure for dynamic free boundary problems hamiltonian structures for 2 or 3 dimensional incompressible flows with a free boundary are determined which generalize a previous structure of zakharov for irrotational flow our poisson bracket is determined using the method of arnold namely reduction from canonical variables in the lagrangian material description using this bracket the hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated the structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained in the case of two dimensional flows the vorticity bracket is determined and the generalized enstrophy is shown to be a casimir function this investigation also clears up some confusion in the literature concerning the vorticity bracket even for fixed boundary flows

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

kinematical symmetries of three dimensional incompressible flows abstract the motion of an incompressible fluid in lagrangian coordinates involves infinitely many symmetries generated by the left lie algebra of group of volume preserving diffeomorphisms of the three dimensional domain occupied by the fluid utilizing a 1 3 dimensional hamiltonian setting an explicit realization of this symmetry algebra is constructed recursively a dynamical connection is used to split the symmetries into reparametrization of trajectories and one parameter family of volume preserving diffeomorphisms of the fluid domain algebraic structures of symmetries and hamiltonian structures of their generators are inherited from the same construction a comparison with the properties of two dimensional flows is included

the small dispersion limit of the korteweg de vries equation iii in parts i and ii we have derived explicit formulas for the distribution limit u of the solution of the kdv equation as the coefficient of uxxx tends to zero this formula contains n parameters 1 n whose values as well as whose number depends on x and t in section 4 we have shown that for t tb n 1 and the value of was determined in section 5 we have shown that the parameters i satisfy a nonlinear system of partial differential equations in part iii section 6 we show that for t large n 3 and we determine the asymptotic behavior of 1 2 3 and of u and u2 for t large the explicit formulas show that u and u2 are o t 1 and o t 2 respectively see formulas 6 2 and 6 24 in section 7 we study initial data whose value tends to zero as x and to 1 as x if one accepts some plausible guesses about the behavior of solutions with such initial data we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution the function s is expressible in terms of complete elliptic integrals a similar formula is derived for u2 in section 8 we indicate how to extend the treatment of this series of papers to multihumped but still negative initial data

local well posedness for the maxwell schrodinger equation time local well posedness for the maxwell schrodinger equation in the coulomb gauge is studied in sobolev spaces by the contraction mapping principle the lorentz gauge and the temporal gauge cases are also treated by the gauge transform

averaged lagrangians and the mean effects of fluctuations in ideal fluid dynamics we begin by placing the generalized lagrangian mean glm equations for a compressible adiabatic fluid into the euler poincare ep variational framework of fluid dynamics for an averaged lagrangian we then state the ep averag ing result that glm equations arise from glm hamilton s principles in the ep framework next we derive a new set of approximate small amplitude glm equations gm equations at second order in the fluctuating displacement of a lagrangian trajectory from its mean position these equations express the linear and nonlinear back reaction effects on the eulerian mean fluid quantities by the fluctuating displacements of the lagrangian trajectories in terms of their eulerian second moments the derivation of the gm equations uses the linearized relations between eulerian and lagrangian fluctuations in the tradition of lagrangian stability analysis for fluids the gm derivation also uses the method of averaged lagrangians in the tradition of wave mean flow interaction wmfi the gm ep motion equations for compressible and incompressible ideal fluids are compared with the euler alpha turbulence closure equations an alpha model is a glm or gm fluid theory with a taylor hypothesis closure thc such closures are based on the linearized fluctuation relations that determine the dynamics of the lagrangian statistical quantities in the euler alpha closure equations we use the ep averaging result to bridge between the glm equations and the euler alpha closure equations hence combining the small amplitude approximation with thc yields in new gm turbulence closure equations for compressible fluids in the ep variational framework 2002 elsevier science b v all rights reserved

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

the cauchy problem for coupled yang mills and scalar fields in the temporal gauge we study the cauchy problem for minimally coupled classical yang mills and scalar fields inn 1 dimensional space time in the temporal gauge we prove the existence and uniqueness of solutions for small time intervals and for anyn we then develop a general theory of solutions in local spaces and extend the previous local in time results to this more general setting in space time dimensions two and three we prove the existence of global in time solutions by the method of a priori estimates both in global and local spaces in space time dimension four our estimates yield only partial results on the global existence problem

existence and uniqueness of energy solution to klein gordon schrodinger equations abstract this paper is concerned with the initial value problem for the nonlinear klein gordon schrodinger kgs equations in r 3 1 time space by using viscous approach the existence of the global finite energy solution is established for the nonlinear kgs equations by compactness argument in addition the uniqueness of the solution is proved by introducing a function with integral form

integrable and chaotic motions of four vortices i the case of identical vortices it is shown that the three vortex problem in two dimensional hydrodynamics is integrable whereas the motion of four identical vortices is not a sequence of canonical transformations is obtained that reduces the n degree of freedom hamiltonian which describes the interaction of n identical vortices to one with n 2 degrees of freedom for n 3 a reduction to a single degree of freedom is obtained and this problem can be solved in terms of elliptic functions for n 4 the reduction procedure leads to an effective hamiltonian with two degrees of freedom of the form found in problems with coupled nonlinear oscillators resonant interaction terms in this hamiltonian suggest non integrable behaviour and this is verified by numerical experiments explicit construction of a solution that corresponds to a heteroclinic orbit in phase space is possible the relevance of the results obtained to fundamental problems in hydrodynamics such as the question of integrability of euler s equation in two dimensions is discussed the paper also contains a general exposition of the hamiltonian and poisson bracket formalism for point vortices

blowup results for the kgs system with higher order yukawa coupling in this paper we investigate the klein gordon schrodinger kgs system with higher order yukawa coupling in spatial dimensions n 3 we establish a perturbed virial type identity and prove blowup results relied on lyapunov functionals for kgs system with a negative energy level additionally we give a result with respect to the blowup rate in finite time for the radial solution in 3 spatial dimensions

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

a rigorous derivation of the hamiltonian structure for the nonlinear schrodinger equation abstract we consider the cubic nonlinear schrodinger equation nls in any spatial dimension which is a well known example of an infinite dimensional hamiltonian system inspired by the knowledge that the nls is an effective equation for a system of interacting bosons as the particle number tends to infinity we provide a derivation of the hamiltonian structure which is comprised of both a hamiltonian functional and a weak symplectic structure for the nonlinear schrodinger equation from quantum many body systems our geometric constructions are based on a quantized version of the poisson structure introduced by marsden morrison and weinstein 24 for a system describing the evolution of finitely many indistinguishable classical particles

applications of a further extended tanh method by using a further extended tanh method chaos solitons fract 17 2003 669 and symbolic computation system maple we obtain abundant new explicit exact solutions of the 2 1 dimensional burgers equation and the 2 1 dimensional boussinesq equation the new solutions including soliton like solutions period form solutions and some other solutions

ideal hydrodynamics on lie groups abstract in this paper we consider the reduction to a lie group of geodesic flows of left or right invariant metrics for a fixed value of momentum we describe the reduced vector field on the lie group as well as its right or left invariant symmetry fields the reduced vector field has an important hydrodynamic interpretation it is a stationary flow of an ideal fluid with a constant pressure we use the explicit expressions for the reduced vector field and its symmetry fields to define secondary hydrodynamics i e we study the reduction for the euler equations of an ideal fluid that describe the geodesics of a right invariant metric on a lie group s diff m of the volume preserving diffeomorphisms of a riemannian manifold m to the group s diff m for a typical coadjoint orbit we find all symmetry fields of a reduced flow and as a corollary we get a simple proof for nonexistence of new invariants of coadjoint orbits which are the integrals of local densities over the flow domain

smoothing effects for schrodinger equations with electro magnetic potentials and applications to the maxwell schrodinger equations abstract we consider schrodinger equations in r 1 2 with electro magnetic potentials the potentials belong to h 1 and typically they are time independent or determined as solutions to inhomogeneous wave equations we prove kato type smoothing estimates for solutions we also apply this result to the maxwell schrodinger equations in the lorentz gauge and prove unique solvability of this system in the energy space

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

nonlinear stability of circular vortex patches this paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the euler equations of incompressible fluids this is achieved by establishing a relative variational principle in terms of either energy or angular momentum thus we exploit and extend arnold s idea in 1965 1969 to a nonsmooth setting as well

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

symmetry transformations of an ideal steady fluid flow determined by a potential function first we consider a transformation of 3d trajectories fluid particle paths of inviscid steady flows using the dual stream function approach for the local representation of velocity fields u x y z this enables to derive the equation governing the deformation of trajectories by the gradient field along the surface x y z 0 in fact is a symmetry transformation and it looks formally like the filament motion which preserves the curvature then we investigate in detail a fine structure of a lie algebra associated with an extension of the transformation which creates a visual appearance of sliding stream surfaces x y z 0 along itself the minimal set of generating differential invariants is found this set consists of a single invariant which coincides with a hamiltonian function

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

well posedness for the nonlinear klein gordon schrodinger equations with heterointeractions in this paper we study the well posedness of solution to a certain system of nonlinear klein gordon schrodinger equations in three space dimensions basing on the strichartz estimates we obtain the global existence uniqueness of the solutions and continuous dependence with respect to initial data in the sobolev spaces of low regularities by setting appropriate contraction and taking difference estimates

on the coupling between an ideal fluid and immersed particles in this paper we present finite dimensional particle based models for fluids which respect a number of geometric properties of the euler equations of motion specifically we use lagrange poincare reduction to understand the coupling between a fluid and a set of lagrangian particles that are supposed to simulate it we substitute the use of principal connections in cendra et al 2001 13 with vector field valued interpolations from particle velocity data the consequence of writing evolution equations in terms of interpolation is two fold first it provides estimates on the error incurred when interpolation is used to derive the evolution of the system second this form of the equations of motion can inspire a family of particle and hybrid particle spectral methods where the error analysis is built in we also discuss the influence of other parameters attached to the particles such as shape orientation or higher order deformations and how they can help us achieve a particle centric version of kelvin s circulation theorem

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

geometrical analysis of free rotation of a rigid body rotational motion of a free rigid body without action of external torque is governed by the euler equation which is reformulated as a geodesic equation by introducing a riemannian metric and connection on the lie group so 3 the aim of this work is to investigate the stability of the motion from the viewpoint of geodesic variation and also to study the relationship between the instability and the riemannian sectional curvatures of variable signs the stability theorem of steady rotations known in the mechanics dynamical property is recovered by solving the jacobi equation for the variational field y a geometrical field existence of a conjugate point for the rigid body of any shape is confirmed and the condition for any jacobi field concerning the steady rotations such that to have a conjugate time is derived the sectional curvatures k s are calculated for the stable steady rotation the curvatures k s take either positive definite values or both positive and negative values in oscillatory manner depending on the inertia tensor however the time averages s are always positive for any y in the linearly stable case while there exist y s which make negative in the case of linear instability

wellposedness in energy space for the nonlinear klein gordon schrodinger system this paper is concerned with the wellposedness of the nonlinear klein gordon schrodinger nkgs equations under multi interactions in 3 dimensions by using the vanishing viscosity techniques and the compactness arguments we establish the existence of the global finite energy solutions for the nkgs equations in addition by introducing a time piecewise function with integral form we prove uniqueness and continuous dependence on the initial data

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

coarsening dynamics of the one dimensional cahn hilliard model the dynamics of one dimensional cahn hilliard model is studied the stationary and particle type solutions the bubbles are perused as a function of initial conditions boundary conditions and system size we characterize the bubble solutions which are involved in the coarsening dynamics and establish the bifurcation scenarios of the system a set of ordinary differential equation permits us to describe the coarsening dynamics in very good agreement with numerical simulations we also compare these dynamics with the bubble dynamics deduced from the classical kink interaction computation where our model seems to be more appropriated in the case of two bubbles we deduce analytical expressions for the bubble s position and the bubble s width besides a simple description of the ulterior dynamics is presented

a nonlinear hamiltonian structure for the euler equations abstract the euler equations for inviscid incompressible fluid flow have a hamiltonian structure in eulerian coordinates the hamiltonian operator though depending on the vorticity conservation laws arise from two sources one parameter symmetry groups which are completely classified yield the invariance of energy and linear and angular momenta degeneracies of the hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point wise conservation of vorticity it is conjectured that no further conservation laws exist indicating that the euler equations are not completely integrable in particular do not have soliton like solutions

finite mode analogs of 2d ideal hydrodynamics coadjoint orbits and local canonical structure abstract the algebraic finite mode hydrodynamic type systems that have o n integrals of motion for o n n modes and are intrinsically connected with two dimensional ideal fluid flows are studied special emphasis is given on the structure of their phase space and the problem of finding proper dynamical variables for its description in connection with appropriate problems for the original hydrodynamic equations various possibilities for embedding in the space of canonical variables are demonstrated on the simplest example of the su 3 model giving the finite dimensional analogs of clebsch variables

finite quotients of groups of i type abstract to every group of i type we associate a finite quotient group that plays the role that coxeter groups play for artin tits groups since groups of i type are examples of garside groups this answers a question of d bessis in the particular case of groups of i type groups of i type are related to finite set theoretical solutions of the yang baxter equation so our result provides a new tool to attack the problem of the classification of finite set theoretical solutions of the yang baxter equation

kato type estimates for nls equation in a scalar field and unique solvability of nkgs system in energy space abstract we consider schrodinger equation in r 2 1 with nonlinear scalar potential the potentials are time independent or determined as solutions to inhomogeneous wave equations by constructing a modified propagator we derive kato type smoothing estimates for the nonlinear schrodinger nls equation with the help of these results we prove the unique solvability of the nonlinear klein gordon schrodinger nkgs system for all time in the energy space

hyperkahler structures and infinite dimensional grassmannians abstract in this paper we describe an example of a hyperkahler quotient of a banach manifold by a banach lie group although the initial manifold is not diffeomorphic to a hilbert manifold not even to a manifold modelled on a reflexive banach space the quotient space obtained is a hilbert manifold which can be furthermore identified either with the cotangent space of a connected component gr res j j z of the restricted grassmannian or with a natural complexification of this connected component thus proving that these two manifolds are isomorphic hyperkahler manifolds moreover kahler potentials associated with the natural complex structure of the cotangent space of gr res j and with the natural complex structure of the complexification of gr res j are computed using kostant souriau s theory of prequantization

travelling waves and static structures in a two dimensional exactly solvable reaction diffusion system exact travelling wave solutions of a fisher type reaction diffusion equation in two spatial dimensions are obtained the waves have nontrivial geometries and possess velocities smaller and larger than that of a plane wave the dispersion relationship and velocity curvature dependence generated by these solutions are characterized static structures with and without circular symmetry are constructed

nonrelativistic approximation in the energy space for kgs system abstract in this paper we study the nonrelativistic limit of the cauchy problem for the damped and the conserved klein gordon schrodinger kgs system respectively we prove that any finite energy solution to the damped kgs system converges to the one of yukawa schrodinger ys system in the energy space h 1 h 1 and the solution to the conserved system goes to the corresponding one of a nonlinear schrodinger nls equation as well

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

on the global evolution of vortex filaments blobs and small loops in 3d ideal flows we consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions the proof exploits the existence of suitable hamiltonian functions the approximate models we analyze essentially discrete and continuous vortex filaments and vortex loops are related to some problem of classical physics concerning turbulence and also to the numerical approximation of flows with very high reynolds number finally we apply our strategy to discrete models for filaments used in numerical methods

some open problem abstract this conference proceedings covers a variety of topics related by a common theme solitons and coherent structures an objective of this paper is to exhibit some of the relations between topics by identifying some unsolved but well defined problems that either were or might have been discussed during the conference and by suggesting how they might be solved using methods that were discussed here the problems were chosen to identify unifying themes rather than to identify the most important problems in each area for example i will not discuss any of the important work of baxter despite its fundamental role in the development of solvable lattice models the problems discussed come from quantum field theory transonic flow water waves and the two dimensional motion of a perfect fluid

symmetries and conservation laws of two dimensional hydrodynamics abstract we shall briefly review the hamiltonian and lie algebraic structure of 2 d hydrodynamics and its known conservation laws energy enstrophy the moments of vorticity then we utilize the eigenmode expansion of the euler equation to represented it as an infinite ode system and the detailed analysis of the structure coefficients to show that there are no other nontrivial conserved integrals among all linear and quadratic functional of vorticity this result provides a partial justification of the kraichnan energy enstrophy theory of 2 d turbulence

statistical mechanics of an exactly integrable system the equilibrium thermodynamics of the nonlinear schrodinger model at finite temperature is calculated by means of the quantum inverse method working directly in an infinite volume we derive the equation of state and the integral equation which determines the excitation spectrum this integral equation is found to be closely related to the gelfand levitan expression for the charge density operator

uniform decay rate estimates for schrodinger and plate equations with nonlinear locally distributed damping on a compact n dimensional riemannian manifold m g we establish uniform decay rate estimates for the linear schrodinger and plate equations subject to an internal nonlinear damping locally distributed on the manifold our approach can be also employed for other equations provided that inverse inequality for the linear model occurs in the particular case of the wave equation where the well known geometric control condition gcc is equivalent to the observability inequality our method generalizes the results due to cavalcanti et al 2010 2009 9 10 regarding the optimal choice of dissipative regions

involutive yang baxter groups in 1992 drinfeld posed the question of finding the set theoretic solutions of the yang baxter equation recently gateva ivanova and van den bergh and etingof schedler and soloviev have shown a group theoretical interpretation of involutive non degenerate solutions namely there is a one to one correspondence between involutive non degenerate solutions on finite sets and groups of i type a group g of i type is a group isomorphic to a subgroup of fa n sym n so that the projection onto the first component is a bijective map where fa n is the free abelian group of rank n and sym n is the symmetric group of degree n the projection of g onto the second component sym n we call an involutive yang baxter group iyb group this suggests the following strategy to attack drinfeld s problem for involutive non degenerate set theoretic solutions first classify the iyb groups and second for a given iyb group g classify the groups of i type with g as associated iyb group it is known that every iyb group is solvable in this paper some results supporting the converse of this property are obtained more precisely we show that some classes of groups are iyb groups we also give a non obvious method to construct infinitely many groups of i type and hence infinitely many involutive non degenerate set theoretic solutions of the yang baxter equation with a prescribed associated iyb group

statistical behavior and coherent structures in two dimensional inviscid turbulence the ergodic and mixing questions are studied for the dynamical systems obtained by truncation of the fourier transformed two dimensional euler equations with periodic boundary conditions such statistical properties are important because they bear a close relationship to the existence of coherent structures in turbulent flow fields these theoretical relationships are derived in particular for the case of macroscopic vortex structures the main emphasis is on the numerical analysis of these statistical properties several different systems with various a priori statistical properties ergodic nonergodic unknown have been computed in this manner comparison standards are established for systems with unknown statistical properties a careful error analysis is presented for the results in particular it is found that final integration times for the modal truncations must be kept small relative to a relaxation time in order to insure numerical accuracy the consequences of this fact are important for th

set theoretic solutions of the yang baxter equation rc calculus and garside germs abstract building on a result by w rump we show how to exploit the right cyclic law x y x z y x y z in order to investigate the structure groups and monoids attached with involutive nondegenerate set theoretic solutions of the yang baxter equation we develop a sort of right cyclic calculus and use it to obtain short proofs for the existence both of the garside structure and of the i structure of such groups we describe finite quotients that play for the considered groups the role that coxeter groups play for artin tits groups

clebsch lagrange variational principle and geometric constraint analysis of relativistic field theories inspired by the clebsch optimal control problem we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry its special feature is that the lagrange multipliers couple to the configuration variables via the symmetry group action the resulting constraints are formulated as a condition on the momentum map of the gauge group action on the phase space of the system we discuss the hamiltonian picture and the reduction in the gauge symmetry by stages in this geometric setting we show that the yang mills higgs action and the einstein hilbert action fit into this new framework after a 1 3 splitting moreover we recover the gaus constraint of yang mills higgs theory and the diffeomorphism constraint of general relativity as momentum map constraints inspired by the clebsch optimal control problem we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry its special feature is that the lagrange multipliers couple to the configuration variables via the symmetry group action the resulting constraints are formulated as a condition on the momentum map of the gauge group action on the phase space of the system we discuss the hamiltonian picture and the reduction in the gauge symmetry by stages in this geometric setting we show that the yang mills higgs action and the einstein hilbert action fit into this new framework after a 1 3 splitting moreover we recover the gaus constraint of yang mills higgs theory and the diffeomorphism constraint of general relativity as momentum map constraints

invariants of the euler equations for ideal or barotropic hydrodynamics and superconductivity in d dimensions abstract the hamiltonian formalism for the euler equations of an ideal fluid superconductivity and a barotropic fluid on a d dimensional riemannian manifold is proposed we show that each of these equations has an infinite series of integrals if d is even generalized enstrophies and at least one integral if d is odd generalized helicity we prove that the magnetic hydrodynamics integral f v b is equal to the average linking number of vector fields rot v and b in terms of the ergodic theory all the invariants considered are casimir elements i e invariants of coadjoint action of the corresponding infinite dimensional lie algebras

global existence and uniqueness of solutions to the maxwell schrodinger equations the time local and global well posedness for the maxwell schrodinger equations is considered in sobolev spaces in three spatial dimensions the strichartz estimates of koch and tzvetkov type are used for obtaining the solutions in the sobolev spaces of low regularities one of the main results is that the solutions exist time globally for large data

poisson brackets and clebsch representations for magnetohydrodynamics multifluid plasmas and elasticity abstract poisson brackets are constructed by the same mathematical procedure for three physical theories ideal magnetohydrodynamics multifluid plasmas and elasticity each of these brackets is given a simple lie algebraic interpretation moreover each bracket is induced to physical space by use of a canonical poisson bracket in the space of clebsch potentials which are constructed for each physical theory by the standard procedure of constrained lagrangians

poisson brackets associated to the conformal geometry of curves in this paper we present an invariant moving frame in the group theoretical sense along curves in the mobius sphere this moving frame will describe the relationship between all conforrnal differential invariants for curves that appear in the literature using this frame we first show that the kac moody poisson bracket on lo n 1 1 can be poisson reduced to the space of conformal differential invariants of curves the resulting bracket will be the conformal analogue of the adler gel fand dikii bracket secondly a conformally invariant flow of curves induces naturally an evolution on the differential invariants of the flow we give the conditions on the invariant flow ensuring that the induced evolution is hamiltonian with respect to the reduced poisson bracket because of a certain parallelism with the euclidean case we study what we call frenet and natural cases we comment on the implications for completely integrable systems and describe conformal analogues of the hasimoto transformation

gaussian groups and garside groups two generalisations of artin groups it is known that a number of algebraic properties of the braid groups extend to arbitrary finite coxeter type artin groups here we show how to extend the results to more general groups that we call garside groups define a gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths and where left and right lowest common multiples exist a garside monoid is a gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition a gaussian group is the group of fractions of a gaussian monoid and a garside group is the group of fractions of a garside monoid braid groups and more generally finite coxeter type artin groups are garside groups we determine algorithmic criteria in terms of presentations for recognizing gaussian and garside monoids and groups and exhibit infinite families of such groups we describe simple algorithms that solve the word problem in a gaussian group show that these algorithms have a quadratic complexity if the group is a garside group and prove that garside groups have quadratic isoperimetric inequalities we construct normal forms for gaussian groups and prove that in the case of a garside group the language of normal forms is regular symmetric and geodesic has the 5 fellow traveller property and has the uniqueness property this shows in particular that garside groups are geodesically fully biautomatic finally we consider an automorphism of a finite coxeter type artin group derived from an automorphism of its defining coxeter graph and prove that the subgroup of elements fixed by this automorphism is also a finite coxeter type artin group that can be explicitly determined 1991 mathematics subject classification primary 20f05 20f36 secondary 20b40 20m05

semigroups ofi type abstract assume thatsis a semigroup generated by x1 xn and let u be the multiplicative free commutative semigroup generated by u1 un we say thatsis ofi typeif there is a bijectivev u ssuch that for alla u v u1a v una x1v a xnv a this condition appeared naturally in the work on sklyanin algebras by john tate and the second author in this paper we show that the condition for a semigroup to be ofi type is related to various other mathematical notions found in the literature in particular we show that semigroups ofi type appear in the study of the set theoretic solutions of the yang baxter equation in the theory of bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author

one sided orthogonality orthomodular spaces quantum sets and a class of garside groups abstract the classification of orthomodular lattices by means of their structure group is extended to non symmetric orthogonality relations spaces with a non hermitian sesquilinear form quantum sets and a new class of garside groups are given as examples

on singular hamiltonians the existence of quasi periodic solutions and nonlinear stability on considere l existence de solutions quasiperiodiques pour une classe de hamiltoniens singuliers et la stabilite non lineaire des singularites

global solutions of the klein gordon schrodinger system with rough data in r 2 1 abstract in this paper we consider the klein gordon schrodinger system with quadratic yakuwa coupling and cubic autointeractions in r 2 1 and prove the existence and uniqueness of global solution for rough data the techniques to be used are adapted from a general scheme originally introduced by j bourgain to split the data into the low and high frequency parts

painleve analysis of the two dimensional burgers equation a painleve analysis of the two dimensional burgers equation is carried out and used to obtain a restricted backlund transformation that maps a subclass of the solutions of the 1 2d burgers equation onto a linear heat like equation alternatively the backlund transformation can be expressed as a map onto the derivative of the one dimensional burgers equation in appropriate dependent and independent variables the singularity analysis also yields a further class of solutions obtained by solving a schwarzian differential equation

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system