nearly inviscid faraday waves in annular containers of moderately large aspect ratio nearly inviscid parametrically excited surface gravity capillary waves in two dimensional domains of finite depth and large aspect ratio are considered coupled equations describing the evolution of the amplitudes of resonant left and right traveling waves and their interaction with a mean flow in the bulk are derived and the conditions for their validity established under suitable conditions the mean flow consists of an inviscid part together with a viscous mean flow driven by a tangential stress due to an oscillatory viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer these forcing mechanisms are important even in the limit of vanishing viscosity and provide boundary conditions for the navier stokes equation satisfied by the mean flow in the bulk for moderately large aspect ratio domains the amplitude equations are nonlocal but decouple from the equations describing the interaction of the slow spatial phase and the viscous mean flow two cases are considered in detail gravity capillary waves and capillary waves in a microgravity environment

parametric instability of a liquid sheet when a liquid sheet is subject to vertical vibration parametric instabilities may occur and give rise to standing waves the present work develops the linear theory of this phenomenon for inviscid newtonian and viscoelastic liquids floquet theory is applied to the governing equations to obtain recursion relations for the temporal modes of the deformations of the free surfaces in contrast to the case where there is no vibration we find that the symmetric and antisymmetric deformations of the liquid sheet are coupled to each other for the inviscid case the recursion relations are shown to be equivalent to a pair of coupled mathieu equations for the newtonian and viscoelastic cases the recursion relations are converted into an eigenvalue problem which is solved numerically to obtain the critical vibration amplitude needed to excite the instability along with the corresponding critical wavenumber the results display behaviour which is similar in several ways to that of the classic faraday instability the parametric instabilities studied here may be an important mechanism in the initial stages of foam destruction by ultrasound as well as in several other practical applications

quantum logic state space geometry and operator algebras the problem of characterising those quantum logics which can be identified with the lattice of projections in a jbw algebra or a von neumann algebra is considered for quantum logics which satisfy the countable chain condition and which have no typei2 part a characterisation in terms of geometric properties of the quantum state space is given

electromagnetic scattering from general bi isotropic objects using time domain integral equations combined with pmchwt formulations electromagnetic scattering by general bi isotropic objects is calculated by using the time domain integral equations which are incorporated with the poggio miller chang harrington wu tsai pmchwt formulations by introducing a pair of equivalent electric and magnetic sources electromagnetic fields inside a homogeneous bi isotropic region can be represented by these sources over its boundary after applying fields splitting method a series of coupled surface integral equations are obtained after imposing boundary conditions these equations are then solved numerically using the method of moment mom which involves separate spatial and temporal testing procedures the rao wilton glisson rwg functions are used as the spatial expansion and testing functions and the weighted laguerre functions are derived as the temporal basis and testing functions numerical results such as transient currents far scattered fields and normalized radar cross sections are presented and compared with analytical results as well as mom based frequency domain analysis and good agreements are observed

time domain finite difference and finite element methods for maxwell equations in complex media extensions of finite difference time domain fdtd and finite element time domain fetd algorithms are reviewed for solving transient maxwell equations in complex media also provided are a few representative examples to illustrate the modeling capabilities of fdtd and fetd for complex media the term complex media refers here to media with dispersive bi anisotropic inhomogeneous and or nonlinear properties present in the constitutive tensors

stability of rolls and hexagonal patterns in non potential systems abstract the stability of rolls hexagons and non equilateral hexagonal patterns is considered in the framework of a system of amplitude equations containing non potential resonant quadratic terms the stability analysis of roll patterns reveals a new kind of instability which can generate steady non equilateral hexagons noise sensitive transitions between roll patterns or periodic oscillations the busse balloons for equilateral and non equilateral hexagons are calculated

bi fdtd a novel finite difference time domain formulation for modeling wave propagation in bi isotropic media this paper presents a newly developed finite difference time domain fdtd technique referred to as bi fdtd for modeling electromagnetic wave interactions with bi isotropic bi media the theoretical foundation for the bi fdtd method will be developed based on a wavefield decomposition the main advantage of this approach is that the two sets of wavefields are uncoupled and can be viewed as propagating in an equivalent isotropic medium which makes it possible to readily apply conventional fdtd analysis techniques the bi fdtd scheme will also be extended to include the dispersive nature of chiral media an important subclass of bi isotropic media this extension represents the first of its kind in the fdtd community validations of this new model are demonstrated for a chiral half space and a chiral slab

orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine gordon equation abstract in this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two degree of freedom hamiltonian systems our methods are fundamentally different than other global perturbation methods e g standard melnikov theory in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation our methods combine the higher dimensional melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds we apply our methods to a modified model of the forced and damped sine gordon equation developed by bishop et al we give explicit conditions in terms of the system parameters for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle focus type chaotic dynamics follow from a theorem of silnikov this provides a mechanism for chaotic dynamics geometrically similar to that observed by bishop et al namely a random jumping between two spatially dependent states with an intermediate passage through a spatially independent state however in order for this type of silnikov dynamics to exist we require a different and unphysical type of damping compared to that used by bishop et al

global bifurcations to strange attractors in a model for skew varicose instability in thermal convection abstract we present a global bifurcation study of a four dimensional system of differential equations proposed by f h busse and coworkers modeling instabilities of convection rolls in the rayleigh benard experiment the rayleigh and prandtl numbers are two natural parameters on which the system depends we focus on a detailed mathematical study combining numerical pathfollowing and bifurcation analysis of chaotic dynamics and transitions to chaotic dynamics numerical continuation makes clear how homoclinic and heteroclinic bifurcations organize the bifurcation diagram in the parameter plane combined with a theoretical bifurcation analysis this explains the development of patterns and the creation of chaotic spatio temporal dynamics in the model the organizing centers such as heteroclinic cycles with resonance conditions among eigenvalues and homoclinic loops with geometric degeneracies inclination flips are identified and their unfoldings are analyzed this ties the creation of strange attractors of various geometric structures to codimension two global bifurcations

modelling of wave propagation in wire media using spatially dispersive finite difference time domain method numerical aspects the finite difference time domain fdtd method is applied for modelling of wire media as artificial dielectrics both frequency dispersion and spatial dispersion effects in wire media are taken into account using the auxiliary differential equation method according to the authors knowledge this is the first time when the spatial dispersion effect is considered in the fdtd modelling the stability of developed spatially dispersive fdtd formulations is analyzed through the use of von neumann method combined with the routh hurwitz criterion the results show that the conventional stability courant limit is preserved using standard discretization scheme for wire media modelling flat sub wavelength lenses formed by wire media are chosen for validation of proposed spatially dispersive fdtd formulation results of the simulations demonstrate excellent sub wavelength imaging capability of the wire medium slabs the size of the simulation domain is significantly reduced using the modified perfectly matched layer mpml which can be placed in close vicinity of the wire medium it is demonstrated that the reflections from the mpml wire medium interface are less than 70 db that lead to dramatic improvement of convergence compared to conventional simulations

heteroclinic dynamics in a model of faraday waves in a square container we study periodic orbits associated with heteroclinic bifurcations in a model of the faraday system for containers with square cross section and single frequency forcing these periodic orbits correspond to quasiperiodic surface waves in the physical system the heteroclinic bifurcations are related to a continuum of heteroclinic connections in the integrable hamiltonian limit some of which persist in the presence of small damping the dynamics in the neighborhood of one of the heteroclinic bifurcations are examined in detail using approximate poincare maps with predictions that agree with numerical computations the results suggest a great richness of possible dynamics of faraday waves even in simple geometries and with single frequency forcing

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

scale disparities in the complex swift hohenberg equation for lasers the complex swift hohenberg csh equation is a generic order parameter equation that applies to many physical systems in the case of class c lasers it can be obtained from the maxwell bloch equations using the assumptions of slow envelope and small detuning we show that the resulting csh equation inevitably contains different asymptotic order terms associated with the dominance of the effect of dispersion over diffusion these scale disparities are usually overlooked or simply not mentioned in the literature assuming that a csh equation with all terms of the same order still provides qualitative information in this paper the asymptotically nonuniform csh equation is carefully deduced using a simpler scaling free procedure and a stability analysis of the simplest solutions together with some numerical simulations are presented in which the mentioned scale disparities are clearly seen

orbits homoclinic to resonances the hamiltonian case abstract in this paper we develop methods to show the existence of orbits homoclinic or heteroclinic to periodic orbits hyperbolic fixed points or combinations of hyperbolic fixed points and or periodic orbits in a class of two degree of freedom integrable hamiltonian systems subject to arbitrary hamiltonian perturbations our methods differ from previous methods in that the invariant sets periodic orbits fixed points are created and become hyperbolic as a result of the interaction of the perturbation with a resonance in the unperturbed system this results in a very degenerate situation that requires a combination of geometric singular perturbation theory higher dimensional melnikov type methods and transversality theory we establish a simple energy phase criterion which gives a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonance we apply our methods to a two mode truncation of the driven nonlinear schrodinger equation first studied by bishop et al in this example we show that as the energy is increased at resonance orbits homoclinic to hyperbolic periodic orbits are created in pairs in a global bifurcation that is best described as a saddle node bifurcation of homoclinic orbits

mean flow suppression by endwalls in oscillatory binary fluid convection abstract competition between traveling and standing waves in binary fluid mixtures with a negative separation ratio s is studied for realistic boundary conditions for typical experimental values of the prandtl and lewis numbers the suppression of net mean flow in traveling waves due to the presence of endwalls is found to have negligible effect 1 part in 10 4 on the coefficients of the nonlinear terms in the amplitude equations the dominant effect of the endwalls enters therefore at linear order the results are used to predict the range of s in which the initial instability is to stable chevron states

quantum logics and convex geometry the main result is a representation theorem which shows that for a large class of quantum logics a quantum logic q is isomorphic to the lattice of projective faces in a suitable convex setk as an application we extend our earlier results 4 which subject to countability conditions gave a geometric characterization of those quantum logics which are isomorphic to the projection lattice of a von neumann algebra or aj b w algebra

instabilities and spatio temporal chaos in hexagon patterns with rotation abstract the dynamics of hexagon patterns in rotating systems are investigated within the framework of modified swift hohenberg equations that can be considered as simple models for rotating convection with broken up down symmetry e g non boussinesq rayleigh benard or marangoni convection in the weakly nonlinear regime a linear stability analysis of the hexagons reveals long and short wave instabilities which can be steady or oscillatory the oscillatory short wave instabilities can lead to stable hexagon patterns that are periodically modulated in space and time or to a state of spatio temporal chaos with a fourier spectrum that precesses on average in time the chaotic state can exhibit bistability with the steady hexagon pattern there exist regimes in which the steady hexagon patterns are unstable at all wave numbers

measures on projections and physical states it is shown that a finitely additive measure on the projections of a von neumann algebra withouti2 andii1 summands is the restriction of a state a definition of a physical state is proposed and it is shown that such a physical state on a simplec algebra with unit is a state

dynamics of penta hepta defects in hexagonal patterns abstract the bound state of two dislocations known as a penta hepta defect phd is a generic object in hexagonal patterns its motion provides an important mechanism of wavevector s selection in this paper the structure and dynamics of penta hepta defects in hexagonal patterns is studied in the framework of envelope equations analytical solution for phase field of moving phd is found in the far field which generalizes the static solution due to pismen and nepomnyashchy 1993 the mobility tensor of phd is calculated using combined analytical and numerical approach the results for the velocity of phd climbing in slightly non optimal hexagonal patterns are in good agreement with numerical simulations of amplitude equations interaction of penta hepta defects in optimal hexagonal patterns is studied numerically

spatial period multiplying instabilities of hexagonal faraday waves abstract a recent faraday wave experiment with two frequency forcing reports two types of superlattice patterns that display periodic spatial structures having two separate scales physica d 123 1998 99 these patterns both arise as secondary states once the primary hexagonal pattern becomes unstable in one of these patterns so called superlattice two the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2 3 from the original scale of the hexagons in contrast the time averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale 3 larger than the original scale and apparently has 60 rotation symmetry we present a symmetry based approach to the analysis of this bifurcation taking as our starting point only the observed instantaneous symmetry of the superlattice two pattern presented in physica d 123 1998 99 and the subharmonic nature of the secondary instability we show a that a pattern with the same instantaneous symmetries as the superlattice two pattern can bifurcate stably from standing hexagons b that the pattern has a spatio temporal symmetry not reported in physica d 123 1998 99 and c that this spatio temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time averaged pattern but not for the apparent 60 rotation symmetry the approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns and does not rely on the primary pattern having small amplitude

numerical study of pattern formation in weakly damped parametric surface waves abstract we present a numerical study of parametrically driven surface waves in fluids of low viscosity the equations governing fluid motion in the bulk and the appropriate boundary conditions at the free surface are approximated by a nonlocal set of equations involving surface coordinates alone thus allowing the numerical analysis of the large aspect ratio limit the aspect ratio being the ratio between the size of the system and the wavelength of the waves the numerical calculation supports the predictions of a previously given multi scale asymptotic analysis yielding stable patterns of square symmetry above onset in the capillary dominated regime in the case of mixed capillary gravity waves patterns of eight fold symmetry are obtained again in agreement with the predictions of the asymptotic expansion these calculations provide indirect evidence about the stability of such patterns analysis that is too involved to be carried out analytically

quasipatterns in a parametrically forced horizontal fluid film abstract we shake harmonically a thin horizontal viscous fluid layer frequency forcing only one harmonic to reproduce the faraday experiment and using the system derived in rojas et al 2010 34 invariant under horizontal rotations when the physical parameters are suitably chosen there is a critical value of the amplitude of the forcing such that instability occurs with at the same time the mode oscillating at frequency 2 and the mode with frequency moreover at criticality the corresponding wave lengths k c and k c are such that if we define the family of 2 q equally spaced horizontal wave vectors k j on the circle of radius k c then k j k l k n with k j k l k c k n k c it results under the above conditions that 0 is an eigenvalue of the linearized operator in a space of time periodic functions frequency 2 having a spatially quasiperiodic pattern if q 4 restricting our study to solutions invariant under rotations of angle 2 q gives a kernel of dimension 4 in the spirit of rucklidge and silber 2009 29 we derive formally amplitude equations for perturbations possessing this symmetry then we give simple necessary conditions on coefficients for obtaining the bifurcation of formally stable time periodic frequency 2 quasipatterns in particular we obtain a solution such that a time shift by half the period is equivalent to a rotation of angle q of the pattern

phase and amplitude instabilities for benard marangoni convection in fluid layers with large aspect ratio pattern formation in fluids heated from below is examined in the presence of a free flat surface on that surface the marangoni effect is taken into account as a second instability mechanism it is shown that phase instabilities already known from rayleigh b enard convection confine the region of stable hexagons and shrink the band of stable wavelengths considerably the problem is attacked from two sides amplitude equations are derived explicitly from the basic hydrodynamic equations and are analyzed with regard to secondary instabilities this results in stability diagrams analogous to earlier calculations obtained for parallel rolls in simple rayleigh b enard convection on the other hand we developed a numerical scheme that allows for a direct integration of the fully three dimensional hydrodynamic equations this method is described in detail and time series for pattern evolution are presented showing phase and amplitude instabilities as expected from the formalism of amplitude equations finally we show the connection between amplitude equations and two dimensional generalized ginzburg landau equations these models may reproduce pattern formation near convective threshold in a quantitative way they have the advantage of being more general than the basic equations and they can be treated numerically in a much easier way this allows the computation of pattern evolution in very large aspect ratio systems

two frequency forced faraday waves weakly damped modes and pattern selection abstract recent experiments a kudrolli b pier j p gollub physica d 123 1998 99 111 on two frequency parametrically excited surface waves produced an intriguing superlattice wave pattern near a codimension two bifurcation point where both subharmonic and harmonic waves onset simultaneously but with different spatial wave numbers the superlattice pattern is synchronous with the forcing spatially periodic on a large hexagonal lattice and exhibits small scale triangular structure similar patterns have been shown to exist as primary solution branches of a generic 12 dimensional d 6 t 2 equivariant bifurcation problem and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities m silber m r e proctor phys rev lett 81 1998 2450 2453 here we use the spatial and temporal symmetries of the problem to argue that weakly damped harmonic waves may be critical to understanding the stabilization of this pattern in the faraday system we illustrate this mechanism by considering the equations developed by zhang and vinals j fluid mech 336 1997 301 330 for small amplitude weakly damped surface waves on a semi infinite fluid layer we compute the relevant nonlinear coefficients in the bifurcation equations describing the onset of patterns for excitation frequency ratios of 2 3 and 6 7 for the 2 3 case we show that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities near the codimension two point also we find that the 6 7 case is significantly different from the 2 3 case due to the presence of additional weakly damped harmonic modes these additional harmonic modes can result in a stabilization of the superpatterns

patterns and spatiotemporal chaos in parametrically forced surface waves a systematic survey at large aspect ratio abstract a systematic experimental survey of both the primary patterns and the secondary instabilities of parametrically forced surface waves faraday waves in the large system limit is presented the symmetry of the primary pattern stripes squares or hexagons depends on viscosity and driving frequency f o hexagons are observed at low f o over the whole viscosity range despite the subharmonic symmetry that tends to suppress them possible mechanisms for the occurrence of hexagons for single frequency forcing are discussed boundary induced distortion is absent for the hexagonal and square patterns but present for stripes phase defects occur between hexagonal domains differing in temporal phase by with respect to the forcing patterns of different symmetry coexist in certain parameter ranges the transition to spatiotemporal chaos stc depends on the symmetry of the primary patterns the hexagonal patterns undergo an order disorder transition in which the angular anisotropy in fourier space declines continuously to zero striped patterns at high viscosity become unstable via transverse amplitude modulations in regions of high curvature this instability results in a spatially nonuniform mixed state in which domains of stc coexist with stripes we propose that this phenomenon may be understood in terms of a critical curvature that depends on the acceleration a secondary oscillatory instability is also observed to deform the stripes within the mixed state at intermediate viscosities

mean flow in hexagonal convection stability and nonlinear dynamics weakly nonlinear hexagon convection patterns coupled to mean flow are investigated within the framework of coupled ginzburg landau equations the equations are in particular relevant for non boussinesq rayleigh benard convection at low prandtl numbers the mean flow is found to 1 affect only one of the two long wave phase modes of the hexagons and 2 suppress the mixing between the two phase modes as a consequence for small prandtl numbers the transverse and the longitudinal phase instability are expected to occur in sufficiently distinct parameter regimes that they can be studied separately through the formation of penta hepta defects they lead to different types of transient disordered states the results for the dynamics of the penta hepta defects shed light on the persistence of grain boundaries in such disordered states

multi pulse jumping orbits and homoclinic trees in a modal truncation of the damped forced nonlinear schro dinger equation abstract in this paper we prove the existence of multi pulse orbits homoclinic to a slow manifold in a two mode truncation of the damped forced nonlinear schrodinger equation first suggested by bishop et al these orbits are jumping i e the corresponding solutions keep switching in time between neighborhoods of the two charateristic breathers of the integrable limit in the case of no damping we find multi pulse smale horseshoes in the two mode model while in the dissipative case we establish the existence of structurally stable multi pulse heterolinic connections between two unstable equilibria the orbits we construct are not amenable to melnikov type perturbation methods in both the hamiltonian and the dissipative case we find homoclinic trees which describe the repeated bifurcations of multi pulse solutions to illustrate the theoretical predictions we also present visualizations of these complicated structures

fractak tracer distributions in turbulent field theories abstract we study the motion of passive tracers in a two dimensional turbulent velocity field generated by the kuramoto sivashinsky equation by varying the direction of the velocity vector with respect to the field gradient we can continuously vary the two lyapunov exponents for the particle motion and thereby find a regime in which the particle distribution is a strange attractor we compare the lyapunov dimension to the information dimension of actual particle distributions and show that there is good agreement with the kaplan yorke conjecture similar phenomena have been observed experimentally

heteroclinic dynamics in the nonlocal parametrically driven nonlinear schrodinger equation faraday waves are described under appropriate conditions by a damped nonlocal parametrically driven nonlinear schrodinger equation as the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics the origin of these transitions is explained using a careful study of a two mode galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves these connections are associated with cascades of gluing and symmetry switching bifurcations such bifurcations are located in the partial differential equations as well

two mechanisms of the transition to chaos in finite dimensional models of convection abstract the onset of chaos in systems of ordinary differential equations possessing a stationary solution with a one dimensional unstable manifold is studied both numerically and qualitatively with the help of an auxiliary piecewise monotomic discontinuous recursion relation a connection is established between the route to chaos and the ratio of two leading eigenvalues of the vector field linearized near the fixed point this connection is confirmed by numerical data obtained from the investigation of differential equations originating from a hydrodynamical problem two routes are considered a well known mechanism suggested by lorenz and another one which is due to the accumulation of bifurcations corresponding to the emergence of homoclinic orbits of a saddle point the asymptotical properties of the latter route prove to be entirely determined by the above ratio

the eckhaus instability in hexagonal patterns abstract the eckhaus instability of hexagonal patterns is studied within the model of three coupled envelope equations for the underlying roll systems the regions of instability in the parameter space are found analytically from both the phase approximation and a full system of amplitude equations beyond the stability limits of hexagons two different modes go unstable both provide symmetry breaking of an initially regular pattern via splitting of a triplet of rolls into two triplets of growing disturbances the parameters of fastest growing disturbances wavelength orientation growth rate are determined from the full set of linearized amplitude equations the nonlinear stage of the eckhaus instability is investigated numerically symmetry breaking due to the eckhaus instability indeed occurs within a certain range of parameters which for small supercriticality parameter leads to a metastable disordered hexagonal state with numerous line and point defects for larger the eckhaus instability triggers the transition of regular hexagonal pattern to disordered roll state the roll phase originates in the cores of defects and then spreads all over the pattern

disclinations in square and hexagonal patterns we report the observation of defects with fractional topological charges disclinations in square and hexagonal patterns as numerical solutions of several generic equations describing many pattern forming systems swift hohenberg equation damped kuramoto sivashinsky equation as well as nonlinear evolution equations describing large scale rayleigh benard and marangoni convection in systems with thermally nearly insulated boundaries it is found that disclinations in square and hexagonal patterns can be stable when nucleated from special initial conditions the structure of the disclinations is analyzed by means of generalized cross newell equations

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

standing wave description of nearly conservative parametrically driven waves in extended systems we consider the standing wavetrains that appear near threshold in a nearly conservative parametrically excited extended system that is invariant under space translations and reflection sufficiently close to threshold the relevant equation is a ginzburg landau equation whose cubic coefficient is extremely sensitive to wavenumber shifts which can only be understood in the context of a more general quintic equation that also includes two cubic terms involving the spatial derivative this latter equation is derived from the standard system of amplitude equations for counterpropagating waves whose validity is well established today the coefficients of the amplitude equation for standing waves are obtained for 1d faraday waves in a deep container to correct several gaps in former analyses in the literature this application requires to also consider the effect of the viscous mean flow produced by the surface waves which couples the dynamics of the surface waves themselves with the free surface deformation induced by the mean flow

modulational instability of two pairs of counter propagating waves and energy exchange in a two component system abstract the modulational instability of two pairs of plane counter propagating waves in a two component system is considered within the framework of two coupled sine gordon equations the emphasis is on the generic case when the system is not integrable and the group velocities of each pair of waves are arbitrary and usually different from each other we consider the dynamics of weakly nonlinear wave packets and using an asymptotic multiple scales expansion we obtain several sets of evolution equations describing the two components of the system depending on the wave packet length scale vis a vis the wave amplitude scale these evolution equations are either four non dispersive and nonlinearly coupled envelope equations or four non locally coupled nonlinear schrodinger equations we also consider a set of fully coupled nonlinear schrodinger equations even though in general this system contains small dispersive terms which are strictly beyond the leading order of the asymptotic multiple scales expansion procedure although each set of amplitude equations can be used to describe a range of dynamical phenomena we focus here on stability of plane wave solutions and show that they may be modulationally unstable we study these instabilities in the context of solutions exhibiting an energy exchange between the two physical components of the system we also perform numerical simulations of the original unapproximated coupled sine gordon equations to assess the accuracy of the instability predictions derived from the various asymptotic models and consider the range of validity of each asymptotic model we show that the instabilities can lead to the formation of localized structures and to a modification of the linear energy exchange which then continues for some time into the nonlinear regime as an energy exchange between these localized structures when the system is close to being integrable the time evolution is distinguished by a remarkable almost periodic sequence of energy exchange scenarios with spatial patterns alternating between approximately uniform wavetrains and localized structures which invites further theoretical study of this special case

the mackey gleason problem let a be a von neumann algebra with no direct summand of type i 2 and let p a be its lattice of projections let x be a banach space let m p a x be a bounded function such that m p q m p m q whenever p and q are orthogonal projections the main theorem states that m has a unique extension to a bounded linear operator from a to x in particular each bounded complex valued finitely additive quantum measure on p a has a unique extension to a bounded linear functional on a

temporal forcing of small amplitude waves in anisotropic systems we investigate the effect of resonant temporal forcing on an anisotropic system that exhibits a hopf bifurcation to obliquely traveling waves in the absence of this forcing we find that the forcing can excite various phase locked standing wave structures rolls rectangles and cross rolls at onset at most one of the two char22 rolls or rectangles char22 is stable the cross rolls can arise in a secondary bifurcation and can be stable experimentally they would appear as a periodic switching between a structure in which the zig component dominates and one with a dominating zag structure since there are two symmetry related states of this kind one may expect disordered structures to arise due to the breakup of the pattern into domains the results are consistent with recent experiments on electroconvection in nematic liquid crystals by de la torre ju arez and rehberg phys rev a 42 2096 1990 we also apply the general analysis to a model of the behavior near a lifshitz point where the angle of obliqueness vanishes this analysis indicates that phase locked standing rectangles are always unstable in this parameter regime

an energy equation for the weakly damped driven nonlinear schro dinger equations and its application to their attractors abstract in this note we present an energy equation concerning the h 2 norm of a forced one dimensional nonlinear schrodinger equation as an application we are able to corroborate recent results on attractors of weakly damped forced nonlinear schrodinger equation by proving that the attractor is an attractor in the strong topology of h 2 and its hausdorff and fractal dimensions are finite in h 2 the energy equation for the h 1 norm can be derived in the same fashion and it implies corresponding results for the attractor in h 1

n pulse homoclinic orbits in perturbations of resonant hamiltonian systems in this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable two degree of freedom resonant hamiltonian systems our energy phase method applies to both hamiltonian and dissipative perturbations and reveals families of multi pulse solutions which are not amenable to melnikov type methods as an example we study a two mode approximation of the nonlinear nonplanar oscillations of a parametrically forced inextensional beam in this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically

normal state spaces of jordan and von neumann algebras abstract normal state spaces of jordan and von neumann algebras are characterized among convex sets the normal state spaces of jbw algebras are precisely those which are spectral and elliptic along these the normal state spaces of von neumann algebras are distinguished by the global 3 ball property

parametrically driven surface waves in surfactant covered liquids vertical vibration will excite standing waves on a liquid free surface we perform a linear stability analysis for viscous and viscoelastic liquids of arbitrary depth to determine the role that insoluble surfactants play in the formation of these parametrically driven surface waves we find that in order to obtain time periodic solutions which involve marangoni forces in a non trivial way it is necessary to consider the high peclet number limit of the surfactant transport equation floquet theory is applied to the linearized governing equations to obtain a recursion relation for the temporal modes of the free surface deflection the recursion relation is then solved numerically to obtain the critical vibration amplitude needed to excite the surface waves and the corresponding wavenumber the results show that the presence of surfactants raises or lowers the critical amplitude and wavenumber depending on the spatial phase shift between the surfactant concentration variations and surface deflections

quantum measures and states on jordan algebras a problem of mackey for von neumann algebras has been settled by the conjunction of the early work of gleason and the recent advances of christensen and yeadon we show that mackey s conjecture holds in much greater generality leta be a jbw algebra and letl be the lattice of all projections ina a quantum measure onl is a countably additive map m froml into the real numbers our results imply thatm always has a unique extension to a bounded linear functional ona provided thata has no typei2 direct summand

hexagonal patterns in finite domains in many mathematical models for pattern formation a regular hexagonal pattern is stable in an infinite region however laboratory and numerical experiments are carried out in finite domains and this imposes certain constraints on the possible patterns in finite rectangular domains it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational in practice it is found experimentally that in a rectangular region patterns of irregular hexagons are often observed this work analyses the geometry and dynamics of irregular hexagonal patterns these patterns occur in two different symmetry types either with a reflection symmetry involving two wavenumbers or without symmetry involving three different wavenumbers the relevant amplitude equations are studied to investigate the detailed bifurcation structure in each case it is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls numerical simulations of a model partial differential equation are also presented to illustrate the behaviour