two extended toda fields in three dimensions two extended toda fields associated with saveliev vershik s continuum lie algebras are studied such fields satisfy three dimensional integro differential equations structures such as the fundamental poisson relation classical yang baxter equation chiral exchange algebra and dressing transformations are recovered which are in complete analogy to the two dimensional case the formal solution is also considered using leznov saveliev analysis but this does not lead to explicit solutions because we have insufficient knowledge about the highest weight representations of the continuum lie algebras

a new mixed finite element for the stokes and elasticity problems the stokes problem is approximated by a mixed finite element method using a new finite el ement which has properties analogous to the finite volume methods namely the local conservation of the momentum and the mass estimates of optimal order are derived for the errors in the velocity the pressure and the gradient of the velocity this new finite element also works for the elasticity problem and all estimates are valid uniformly with respect to the compressibility finally some numerical results for the incompressible navier stokes equations are presented

waves and bumps in neuronal networks with axo dendritic synaptic interactions we consider a firing rate model of a neuronal network continuum that incorporates axo dendritic synaptic processing and the finite conduction velocities of action potentials the model equation is an integral one defined on a spatially extended domain apart from a spatial integral mixing the network connectivity function with space dependent delays arising from non instantaneous axonal communication the integral model also includes a temporal integration over some appropriately identified distributed delay kernel these distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs they are interpreted as green s functions of some linear dierential operator exploiting this green s function description we discuss formal reductions of this non local system to equivalent partial dierential equation pde models we distinguish between those spatial connectivity functions that give rise to local pde models and those that give rise to pde models with delayed non local terms for cases in which local pdes are derived we investigate traveling wave solutions in a comoving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions we also calculate exact solutions parameterized by axonal conduction velocity for the heaviside firing rate function the sigmoidal firing rate function in the limit of infinte gain the inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses which we study analytically and numerically in terms of a global homoclinic orbit finally we consider the impact of dendritic interactions on waves and on static spatially localized solutions exact analysis for

localized turing patterns in nonlinear optical cavities abstract the subcritical turing instability is studied in two classes of models for laser driven nonlinear optical cavities in the first class of models the nonlinearity is purely absorptive with arbitrary intensity dependent losses in the second class the refractive index is real and is an arbitrary function of the intracavity intensity through a weakly nonlinear analysis a ginzburg landau equation with quintic nonlinearity is derived thus the maxwell curve which marks the existence of localized patterns in parameter space is determined in the particular case of the lugiato lefever model the analysis is continued to seventh order yielding a refined formula for the maxwell curve and the theoretical curve is compared with recent numerical simulation by gomila et al d gomila a scroggie w firth bifurcation structure of dissipative solitons physica d 227 2007 70 77

a continuous interior penalty method for viscoelastic flows in this paper we consider a finite element discretization of the oldroyd b model of viscoelastic flows the method uses standard continuous polynomial finite element spaces for velocities pressures and stresses inf sup stability and stability for convection dominated flows are obtained by adding a term penalizing the jump of the solution gradient over element faces to increase robustness when the deborah number is high we add a nonlinear artificial viscosity of shock capturing type the method is analyzed on a linear model problem and optimal a priori error estimates are proven that are independent of the solvent viscosity eta s finally we demonstrate the performance of the method on some known benchmark cases

kernel of star product for spin tomograms the spin state tomogram is presented as the modulus squared of the matrix element of the su 2 group irreducible representation for the spin system we established explicitly a realization of the set of operators defining for spin tomograms the star product formalism in terms of irreducible tensors on the set of spin tomograms the delta function and standard and moyal like kernels of the tomogram star product are calculated explicitly in terms of clebsch gordan and racah coefficients and matrix elements of the su 2 irreducible representation in the limit of infinite spin the spin tomogram is shown to become the tomogram of the harmonic oscillator state

homoclinic snaking in bounded domains homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time independent spatially localized states in a bistable spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side on the real line this process continues forever in finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions with periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states however with non neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state this behavior shown here in detail for the swift hohenberg equation explains the phenomenon of snaking without bistability recently observed in simulations of binary fluid convection by mercader et al phys rev e 80 025201 2009

super toda theories and w algebras from superspace wess zumino witten models using a superspace approach it is proved that an 1 super toda theory can be seen as a constrained wzw model based on a supergroup the gauge transformations which survive the constraints are then useda la drinfeld sokolov to determine explicity the super w algebra underlying this theory the conformal spin content of any such super walgebra is provided in the general case

non conservative perturbations of homoclinic snaking scenarios abstract homoclinic snaking refers to the continuation of homoclinic orbits to an equilibrium e near a heteroclinic cycle connecting e and a periodic orbit p typically homoclinic snaking appears in one parameter families of reversible conservative systems here we discuss perturbations of this scenario which are both non reversible and non conservative we treat this problem analytically in the spirit of the work 3 the continuation of homoclinic orbits happens with respect to both the original continuation parameter and the perturbation parameter the continuation curves are parametrised by the dwelling time l of the homoclinic orbit near p it turns out that l tends to zero while the vs l diagram displays isolas or criss cross snaking curves in a neighbourhood of the original snakes and ladder structure in the course of our studies we adapt both fenichel coordinates near p and the analysis of shilnikov problems near p to the present situation

conditions for superconvergence of hdg methods for stokes flow we provide an a priori error analysis of a wide class of finite element methods for the stokes equations the methods are based on the velocity gradient velocity pressure formulation of the equations and include new and old mixed and hybridizable discontinuous galerkin methods we show how to reduce the error analysis to the verification of some properties of an elementwise defined projection and of the local spaces defining the methods we also show that the projection of the errors only depends on the approximation properties of the projection we then provide sufficient conditions for the superconvergence of the projection of the error in the approximate velocity we give many examples of these methods and show how to systematically construct them from similar methods for the diffusion equation

on the transition from the ginzburg landau equation to the extended fisher kolmogorov equation abstract the ginzburg landau gl equation generically describes the behaviour of small perturbations of a marginally unstable basic state in systems on unbounded domains in this paper we consider the transition from this generic situation to a degenerate co dimension 2 case in which the gl approach is no longer valid instead of studying a general underlying model problem we consider a two dimensional system of coupled reaction diffusion equations in one spatial dimension we show that near the degeneration the behaviour of small perturbations is governed by the extended fisher kolmogorov efk equation at leading order the relation between the gl equation and the efk equation is quite subtle but can be analysed in detail the main goal of this paper is to study this relation which we do asymptotically the asymptotic analysis is compared to numerical simulations of the full reaction diffusion system as one approaches the co dimension 2 point we observe that the stable stationary periodic patterns predicted by the gl equation evolve towards various different families of stable stationary but not necessarily periodic so called multi bump solutions in the literature these multi bump patterns are shown to exist as solutions of the efk equation but there is no proof of the asymptotic stability of these solutions our results suggest that these multi bump patterns can also be asymptotically stable in large classes of model problems

stable localized solutions of arbitrary length for the quintic swift hohenberg equation abstract we show that localized solutions of arbitrary length are stable over a finite parameter interval of subcritical values for the quintic swift hohenberg equation with a destabilizing cubic term this equation is thought to model a weakly hysteretic transition to stationary patterns we argue that the stabilization of the localized states of arbitrary length can be traced back to the interaction between long wavelength modulations and spatial variations on the length scale of one unit cell these results are critically compared with other known mechanisms to stabilize localized states in various situations we also discuss for which experimental systems the states predicted here could be detected including e g the stationary onset of binary fluid convection

mixed hybrid finite elements for the simulation of thermal oxidation in semiconductors in this paper we present and analyze a dual mixed hybrid formulation capable of treating in a unified framework both compressible and incompressible problems in continuum mechanics the potentialities of this approach are exploited in the computation of the stress field required in the simulation of the thermal oxidation process in semiconductor device technology

w gauge structures and their anomalies an algebraic approach starting from flat two dimensional gauge potentials we propose the notion of w gauge structure in terms of a nilpotent becchi rouet stora brs differential algebra the decomposition of the underlying lie algebra with respect to an sl 2 subalgebra is crucial for the discussion of conformal covariance in particular the appearance of a projective connection different sl 2 embeddings lead to different w gauge structures we present a general soldering procedure which allows one to express zero curvature conditions for the w currents in terms of conformally covariant differential operators acting on the w gauge fields and to obtain at the same time the complete nilpotent brs differential algebra generated by w currents gauge fields and the ghost fields corresponding to w diffeomorphisms as illustrations we treat the cases of sl 2 itself and the two different sl 2 embeddings in sl 3 viz the w 1 3 and w 2 3 gauge structures in some detail in these cases we determine algebraically w anomalie

review and complements on mixed hybrid finite element methods for fluid flows mixed and hybrid finite element methods for the resolution of a wide range of linear and nonlinear boundary value problems linear elasticity stokes problem navier stokes equations boussinesq equations etc have known a great development in the last few years these methods allow simultaneous computation of the original variable and its gradient both of them being equally accurate moreover they have local conservation properties conservation of the mass and the momentum as in the finite volume methods the purpose of this paper is to give a review on some mixed finite elements developed recently for the resolution of stokes and navier stokes equations and the linear elasticity problem further developments for a quasi newtonian flow obeying the power law are presented

analysis of non singular solutions of a mixed navier stokes formulation abstract a mixed finite element method is used for the numerical solution of the stationary navier stokes equations in this formulation the pressure the velocity and its gradient are the most relevant unknowns an existence and uniqueness results for the finite element solution together with the convergence results are proved near a non singular solution optimal error estimates and numerical results are presented

regular conjugacy classes in the weyl group and integrable hierarchies generalized kdv hierarchies associated by drinfeld sokolov reduction with grade 1 regular semisimple elements from non equivalent heisenberg subalgebras of a loop algebra g x c lambda lambda 1 are studied the graded heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the weyl group w g of the simple lie algebra g a representative w epsilon w g of a regular conjugacy class can be lifted to an inner automorphism of g given by w exp 2i pi ad i0 m where i0 is the defining vector of an sl2 subalgebra of g the grading is then defined by the operator d m i0 m lambda d d lambda ad i0 and any grade 1 regular element lambda from the heisenberg subalgebra associated with w takes the form lambda c lambda c where i0 c m 1 c and c is included in an sl2 subalgebra containing i0 the largest eigenvalue of adi0 is m 1 except for some cases in f4 e6 7 8 we explain how these lie algebraic results follow from known results and apply them to construct integrable systems if the largest ad i0 eigenvalue is m 1 then using any grade 1 regular element from the heisenberg subalgebra associated with w we can construct a kdv system possessing the standard w algebra defined by i0 as its second poisson bracket algebra for g a classical lie algebra we derive pseudo differential lax operators for those non principal kdv systems that can be obtained as discrete reductions of kdv systems related to gln non abelian toda systems are also considered

theory of group representations and integration of nonlinear systems xa zz exp kx a abstract a constructive proof of complete integrability of the system xa zf z exp kx a and its explicit solution are given for k coinciding with the cartan matrix of simple lie algebras the technique for dealing with this system of essentially nonlinear two dimensional partial differential equations involves lax type representation and representation theory of the corresponding group

inf sup conditions for twofold saddle point problems necessary and sufficient conditions for existence and uniqueness of solutions are developed for twofold saddle point problems which arise in mixed formulations of problems in continuum mechanics this work extends the classical saddle point theory to accommodate nonlinear constitutive relations and the twofold saddle structure application to problems in incompressible fluid mechanics employing symmetric tensor finite elements for the stress approximation is presented

snaking of radial solutions of the multi dimensional swift hohenberg equation a numerical study abstract the bifurcation structure of localized stationary radial patterns of the swift hohenberg equation is explored when a continuous parameter n is varied that corresponds to the underlying space dimension whenever n is an integer in particular we investigate how 1d pulses and 2 pulses are connected to planar spots and rings when n is increased from 1 to 2 we also elucidate changes in the snaking diagrams of spots when the dimension is switched from 2 to 3

integrable graded manifolds and nonlinear equations a method is proposed for the classification of integrable embeddings of 2 2 dimensional supermanifoldsv2 2 into an enveloping superspace supplied with the structure of a lie superalgebra the approach is first applied to the even part of the scheme i e for the embeddings of 2 dimensional manifoldsv2 into riemannian or non riemannian enveloping space the general consideration is also illustrated by the example of superspaces supplied with the structure of the series sl n n 1 whose integrable supermanifolds are described by supersymmetrical 2 dimensional toda lattice type equations in particular forn 1 they are described by the supersymmetrical liouville and sine gordon equations

a discontinuous petrov galerkin method with lagrangian multipliers for second order elliptic problems we present a discontinuous petrov galerkin dpg method for the finite element discretization of second order elliptic boundary value problems the novel approach emanates from a one element weak formulation of the differential problem this procedure which is typical of discontinuous galerkin dg methods is based on introducing variables defined in the interior and on the boundary of the element the interface variables are suitable lagrangian multipliers that enforce interelement continuity of the solution and of its normal derivative thus providing the proper connection between neighboring elements the internal variables can be eliminated in favor of the interface variables using static condensation to end up with a system of reduced size in the sole lagrangian multipliers a stability and convergence analysis of the novel formulation is carried out and its connection with mixed hybrid and dg methods is explored numerical tests on several benchmark problems are included to validate the convergence performance and the flux conservation properties of the dpg method

the swift hohenberg equation with a nonlocal nonlinearity it is well known that aspects of the formation of localised states in a one dimensional swift hohenberg equation can be described by ginzburg landau type envelope equations this paper extends these multiple scales analyses to cases where an additional nonlinear integral term in the form of a convolution is present the presence of a kernel function introduces a new lengthscale into the problem and this results in additional complexity in both the derivation of envelope equations and in the bifurcation structure when the kernel is short range weakly nonlinear analysis results in envelope equations of standard type but whose coefficients are modified in complicated ways by the nonlinear nonlocal term nevertheless these computations can be formulated quite generally in terms of properties of the fourier transform of the kernel function when the lengthscale associated with the kernel is longer our method leads naturally to the derivation of two different novel envelope equations that describe aspects of the dynamics in these new regimes the first of these contains additional bifurcations and unexpected loops in the bifurcation diagram the second of these captures the stretched out nature of the homoclinic snaking curves that arises due to the nonlocal term

homoclinic and heteroclinic orbits underlying the post buckling of axially compressed cylindrical shells abstract a structural system with an unstable post buckling response that subsequently restabilizes has the potential to exhibit homoclinic connections from the fundamental equilibrium state to itself over a range of loads and heteroclinic connections between fundamental and periodic equilibrium states over a different smaller range of loads it is argued that such equilibrium configurations are important in the interpretation of observed behaviour and govern the minimum possible post buckling loads to illustrate this the classical problem of a long thin axially compressed cylindrical shell is revisited from three different perspectives asymptotic conjecture analogy with nonlinear dynamics and numerical continuation analysis of a partial spectral decomposition of the underlying equilibrium equations the nonlinear dynamics analogy demonstrates that the structure of the heteroclinic connections is more complicated than that indicated by the asymptotics this is confirmed by the numerics however when the asymptotic portrayal is compared to the numerics it turns out to be surprisingly accurate in its maxwell load prediction of the practically significant first minimum to appear in the post buckling regime

pohlmeyer reduction for superstrings in ads space the pohlmeyer reduced equations for strings moving only in the ads subspace of ads5 s5 have been used recently in the study of classical euclidean minimal surfaces for wilson loops and some semiclassical three point correlation functions we find an action that leads to these reduced superstring equations for example for a bosonic string in adsn such an action contains a liouville scalar part plus a k k gauged wzw model for the group k so n 2 coupled to another term depending on two additional fields transforming as vectors under k solving for the latter fields gives a non abelian toda model coupled to the liouville theory for n 5 we generalize this bosonic action to include the s5 contribution and fermionic terms the corresponding reduced model for the ads2 s2 truncation of the full ads5 s5 superstring turns out to be equivalent to super liouville theory our construction is based on taking a limit of the previously found reduced theory actions for bosonic strings in adsn s1 and superstrings in ads5 s5 this new action may be useful as a starting point for possible quantum generalizations or deformations of the classical pohlmeyer reduced theory we give examples of simple extrema of this reduced superstring action which represent strings moving in the ads5 part of the space expanding near these backgrounds we compute the corresponding fluctuation spectra and show that they match the spectra found in the original superstring theory

localization in a spanwise extended model of plane couette flow we consider a nine partial differential equation 1 space and 1 time model of plane couette flow in which the degrees of freedom are severely restricted in the streamwise and cross stream directions to study spanwise localization in detail of the many steady eckhaus spanwise modulational instabilities identified of global steady states none lead to a localized state spatially localized time periodic solutions were found instead which arise in saddle node bifurcations in the reynolds number these solutions appear global domain filling in narrow small spanwise domains yet can be smoothly continued out to fully spanwise localized states in very wide domains this smooth localization behavior which has also been seen in fully resolved duct flow s okino ph d thesis kyoto university kyoto 2011 indicates that an apparently global flow structure does not have to suffer a modulational instability to localize in wide domains

oscillon type structures and their interaction in a swift hohenberg model abstract motivated by the observation of localized circular excitations oscillons in vertically vibrated granular layers p umbanhowar f melo h swinney nature 382 1996 793 we numerically investigate an extension of a swift hohenberg model that exhibits a subcritical transition to square patterns for sufficiently strong damping of the basic state stable oscillon structures are found the localization mechanism is quite general and is due to non adiabatic effects bound structures of oscillons of equal and opposite polarity are found with bound states of like polarity being less robust much of the phenomena are consistent with the experimental observations and suggest that oscillons are not specific to patterns in granular media or to parametrically driven systems experimental tests are suggested that would determine whether this minimal framework is sufficient to describe the phenomena

effects of exponentially small terms in the perturbation approach to localized buckling localized buckling solutions are known to exist in the heuristic model of an elastic strut resting on an elastic winkler foundation the primary localized solution emerging from the critical buckling state consists of an amplitude envelope of approximately the form of a hyperbolic secant function which modulates a fast varying sinusoidal oscillation in previous works such solutions have been tracked for the entire post buckling regime both numerically and through a rayleigh ritz approach a very rich structure is known to exist in the subcritical load range but has been proved to exist for only a certain family of reversible systems studies have concentrated on symmetric homoclinic solutions and the asymmetric solutions which bifurcate from these solution paths the primary solution is known to exist for the entire subcritical parameter range and all other symmetric and associated asymmetric solutions exist strictly for values less than critical here we uncover a new family of antisymmetric solutions and some asymptotic analysis suggests that the primary antisymmetric solution exists over the same range as does the primary symmetric solution a perturbation approach can be used so as to describe the bifurcation hierarchy for the novel antisymmetric forms we illustrate a unified approach which is able to predict the circumstances under which non divergent localized solutions are possible and the results of the analysis are compared with some numerical solutions

weakly subcritical stationary patterns eckhaus instability and homoclinic snaking the transition from subcritical to supercritical stationary periodic patterns is described by the one dimensional cubic quintic ginzburg landau equation at a axx i a1 a 2 ax a2a 2 a b a 2 a a 4 a w here a x t represents the pattern amplitude and the coefficients a1 a2 a ndb are real the conditions for eckhaus instability of periodic solutions are determined and the resulting spatially modulated states are computed some of these evolve into spatially localized structures in the vicinity of a maxwell point while others resemble defect states the results are used to shed light on the behavior of localized structures in systems exhibiting homoclinic snaking during the transition from subcriticality to supercriticality

analysis of nonlinear poro elastic and poro visco elastic models we consider the initial and boundary value problem for a system of partial differential equations describing the motion of a fluid solid mixture under the assumption of full saturation the ability of the fluid phase to flow within the solid skeleton is described by the permeability tensor which is assumed here to be a multiple of the identity and to depend nonlinearly on the volumetric solid strain in particular we study the problem of the existence of weak solutions in bounded domains accounting for non zero volumetric and boundary forcing terms we investigate the influence of viscoelasticity on the solution functional setting and on the regularity requirements for the forcing terms the theoretical analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not viscoelasticity is present the theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite element discretization when the data are sufficiently regular the simulations show that the solutions satisfy the energy estimates predicted by the theoretical analysis interestingly the simulations also show that in the purely elastic case the darcy velocity and the related fluid energy might become unbounded if indeed the data do not enjoy the time regularity required by the theory

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

about the self dual chern simons system and toda field theories on the noncommutative plane the relation of the noncommutative self dual chern simons ncsdcs system to the noncommutative generalizations of toda and of affine toda field theories is investigated more deeply this paper continues the programme initiated in cabrera carnero i 2005 j high energy phys jhep10 2005 071 where it was presented how it is possible to define toda field theories through second order differential equation systems starting from the ncsdcs system here we show that using the connection of the ncsdcs to the noncommutative chiral model exact solutions of the toda field theories can also be constructed by means of the noncommutative extension of the uniton method proposed in lee k m 2004 j high energy phys jhep08 2004 054 by lee particularly some specific solutions of the nc liouville model are explicitly constructed

a new mixed finite element method for the stokes problem we propose a new mixed formulation of the stokes problem where the extra stress tensor is considered based on such a formulation a mixed finite element is constructed and analyzed this new finite element has properties analogous to the finite volume methods namely the local conservation of the momentum and the mass optimal error estimates are derived for the numerical implementation of this finite element a hybrid form is presented this work is a first step towards the treatment of viscoelastic fluid flows by mixed finite element methods

finite volume methods for two dimensional incompressible flows with complex boundaries abstract the paper discusses possible ways of developing accurate and efficient finite volume methods for complex incompressible flows the issues of boundary fitted grid generation choice of coupled or uncoupled procedures cartesian or grid aligned velocity components staggered or non staggered variable arrangement discretization methods solution algorithms and convergence acceleration by multigrid methods are addressed an example of a finite volume method for general two dimensional flows is described in some detail and applications of this method to a variety of flows are presented

on traveling wave solutions of the kuramoto sivashinsky equation abstract previous numerical results indicate that the kuramoto sivashinsky equation admits three classes of non periodic traveling wave solutions namely regular shocks oscillatory shocks and solitary waves however it has been shown that regular monotonic shocks cease to exist in the weak shock limit due to the radiation of oscillatory waves of exponentially small with respect to the shock strength but growing in space amplitude here oscillatory shocks and solitary waves are constructed by asymptotic analysis it thus transpires that in the weak shock limit oscillatory shocks can only be antisymmetric otherwise oscillatory and monotonic waves of exponentially small but growing in space amplitude would inevitably be excited under certain conditions however the growing waves can link a nearly antisymmetric oscillatory shock with a radiating regular shock to form a solitary wave the predictions of the asymptotic theory are supported by numerical results

zero curvature conditions and conformal covariance two dimensional zero curvature conditions with special emphasis on conformal properties are investigated in detail and the appearance of covariant higher order differential operators constructed in terms of a projective connection is elucidated the analysis is based on the kostant decomposition of simple lie algebras in terms of representations with respect to their principal sl 2 subalgebra

exponential asymptotics and stokes lines in nonlinear ordinary differential equations a technique for calculating exponentially small terms beyond all orders in singularly perturbed ordinary differential equations is presented the approach is based on the application of a wkbj type ansatz to the late terms in the naive asymptotic expansion and the identification of stokes lines and is closely related to the well known stokes line smoothing phenomenon in linear ordinary differential equations the method is illustrated by application to examples

general solutions of nonlinear equations in the geometric theory of the relativistic string general solutions for the system of nonlinear equations in the second order partial derivatives with two independent variables are obtained they determine the basic differential forms of the two dimensional minimal surface embedded inton dimensional pseudo euclidean space

front dynamics in a delayed feedback system with external forcing a scalar nonlinear delay differential equation of the kind arising in optically bistable systems is reduced in some physically relevant limit to a partial differential equation first and then to a system of ordinary differential equations these simplified equations are studied analytically we describe the continuous deformation of periodic solutions from sine waves to square waves away from the oscillation threshold and determine their stability we also study the effect of a weak periodic modulation it is found that a resonant forcing can stabilize a very large number of oscillation modes taking the form of square waves with missing front pairs

a mixed finite element method for a ladyzhenskaya model we study a mixed finite element approximation of a model proposed by ladyzhenskaya for stationary incompressible viscous flow we give existence and uniqueness results for the continuous problem and its approximation and we prove error bounds which improve the existing ones finally some numerical results are presented

localized states in the generalized swift hohenberg equation the swift hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states the presence of these states is related to a phenomenon called homoclinic snaking numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states the evolution of the localized states once they lose stability is illustrated using direct simulations in time i introduction ever since the observation that the subcritical complex ginzburg landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states the presence of these states has important consequences for other systems described by partial differential equations on the line since the ginzburg landau equation describes the behavior of such systems near bifurcation from the trivial state of the system specifically the complex ginzburg landau equation describes the evolution of a long wavelength oscillatory instability as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry in contrast near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real ginzburg landau equation and this equation possesses only unstable spatially localized states it is of interest therefore to examine what happens to these unstable states at larger amplitude where the real ginzburg landau equation no longer provides an adequate description of the system in this paper we show that the localized states can become stable at such amplitudes and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions we are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1 1 resonance and use this theory to construct a large number of such states the stability properties of these states are also determined and the evolution of nonstationary localized states is studied by numerical integration in time it is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation the theory was originally developed in the context of water waves where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ode the resulting localized states are called solitary waves and in some cases turn out to be solitons kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area specifically the ode is viewed as a dynamical system in space and localized states are sought as homoclinic orbits connecting the trivial state to itself whether such orbits are possible depends in part on the stability properties of the trivial state eigenvalues with positive real part indicate that a nontrivial state can grow from x while eigenvalues with negative real part indicate that such a state may return under appropriate conditions back to the trivial state as x the spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system in many cases and in particular in the case considered here the ode is reversible as a result the bifurcations that are encountered as a parameter is increased are nongeneric in the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1 1 resonance the unfolding of this resonance has been worked out in detail by iooss and peroueme 3 and can be used to understand the appearance of a variety of homoclinic orbits in this system and hence of localized states with different spatial structure

bicomplexes and conservation laws in non abelian toda models a bicomplex structure is associated with the leznov saveliev equation of integrable models the linear problem associated with the zero curvature condition is derived in terms of the bicomplex linear equation the explicit example of a non abelian conformal affine toda model is discussed in detail and its conservation laws are derived from the zero curvature representation of its equation of motion

coprimeness preserving non integrable extension to the two dimensional discrete toda lattice equation we introduce a so called coprimeness preserving non integrable extension to the two dimensional toda lattice equation we believe that this equation is the first example of such discrete equations defined over a three dimensional lattice we prove that all the iterates of the equation are irreducible laurent polynomials of the initial data and that every pair of two iterates is co prime which indicate confined singularities of the equation by reducing the equation to two or one dimensional lattices we obtain coprimeness preserving non integrable extensions to the one dimensional toda lattice equation and the somos 4 recurrence

vertex operators and soliton solutions of affine toda model with u 2 symmetry the symmetry structure of the non abelian affine toda model based on the coset sl 3 sl 2 u 1 is studied it is shown that the model possess non abelian noether symmetry closing into a q deformed sl 2 u 1 algebra specific two vertex soliton solutions are constructed

mixed methods for stationary navier stokes equations based on pseudostress pressure velocity formulation in this paper we develop and analyze mixed finite element methods for the stokes and navier stokes equations our mixed method is based on the pseudostress pressure velocity formulation the pseudostress is approximated by the raviart thomas brezzi douglas marini or brezzi douglasfortin marini elements the pressure and the velocity by piecewise discontinuous polynomials of appropriate degree it is shown that these sets of finite elements are stable and yield optimal accuracy for the stokes problem for the pseudostress pressure velocity formulation of the stationary navier stokes equations the well posedness and error estimation results are established by eliminating the pseudostress variables in the resulting algebraic system we obtain cell centered finite volume schemes for the velocity and pressure variables that preserve local balance of momentum

superposition formulas for exterior differential systems abstract in this article we solve an inverse problem in the theory of quotients for differential equations we characterize a family of exterior differential systems that can be written as a quotient of a direct sum of two associated systems that are constructed from the original the fact that a system can be written as a quotient can be used to find the general solution to these equations some examples are given to demonstrate the theory

a dual mixed hybrid formulation for fluid mechanics problems mathematical analysis and application to semiconductor process technology in this paper we propose a dual mixed hybrid formulation capable of treating under a unified framework both compressible and incompressible problems in continuum mechanics a theoretical analysis of the method is carried out and optimal error estimates are derived for both mixed and hybrid variables the potentialities of the novel approach are exploited in the computation of the stress field required in the simulation of the thermal oxidation process in semiconductor technology the numerical formulation is validated both on model problems in continuum mechanics and on a realistic example of the thermal oxidation process in a local oxidation structure

a mixed discontinuous galerkin method with symmetric stress for brinkman problem based on the velocity pseudostress formulation abstract the brinkman equations can be regarded as a combination of the stokes and darcy equations which model transitions between the fast flow in channels governed by stokes equations and the slow flow in porous media governed by darcy s law the numerical challenge for this model is the designing of a numerical scheme which is stable for both the stokes dominated high permeability and the darcy dominated low permeability equations in this paper we solve the brinkman model in n dimensions n 2 3 by using the mixed discontinuous galerkin mdg method which meets this challenge this mdg method is based on the pseudostress velocity formulation and uses a discontinuous piecewise polynomial pair p k 1 s p k k 0 where the stress field is symmetric the main unknowns are the pseudostress and the velocity whereas the pressure is easily recovered through a simple postprocessing a key step in the analysis is to establish the parameter robust inf sup stability through specific parameter dependent norms at both continuous and discrete levels therefore the stability results presented here are uniform with respect to the permeability thanks to the parameter robust stability analysis we obtain optimal error estimates for the stress in broken h div norm and velocity in l 2 norm furthermore the l 2 error estimate for pseudostress is derived under certain conditions finally numerical experiments are provided to support the theoretical results and to show the robustness accuracy and flexibility of the mdg method

analytical and numerical solutions of the generalized dispersive swift hohenberg equation the generalization of the swift hohenberg equation is studied it is shown that the equation does not pass the kovalevskaya test and does not possess the painleve property exact solutions of the generalized swift hohenberg equation which are very useful to test numerical algorithms for various boundary value problems are obtained the numerical algorithm which is based on the crank nicolson adams bashforth scheme is developed this algorithm is tested using the exact solutions the selforganization processes described by the generalization of the swift hohenberg equation are studied

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

point symmetries of generalized toda field theories ii symmetry reduction the lie symmetries of a large class of generalized toda field theories are studied and used to perform symmetry reduction reductions lead to generalized toda lattices on one hand and to periodic systems on the other boundary conditions are introduced to reduce theories on an infinite lattice to those on semi infinite or finite ones

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

homoclinic snaking near a heteroclinic cycle in reversible systems snaking curves of homoclinic orbits have been found numerically in a number of ode models from water wave theory and structural mechanics along such a curve infinitely many fold bifurcation of homoclinic orbits occur thereby the corresponding solutions spread out and develop more and more bumps oscillations about their own centre a common feature of the examples is that the systems under consideration are reversible in this paper it is shown that such a homoclinic snaking can be caused by a heteroclinic cycle between two equilibria one of which is a bi focus using lin s method a snaking of 1 homoclinic orbits is proved to occur in an unfolding of such a cycle further dynamical consequences are discussed as an application a system of boussinesq equations is considered where numerically a homoclinic snaking curve is detected and it is shown that the homoclinic orbits accumulate along a heteroclinic cycle between a real saddle and a bi focus equilibrium

stokes phenomenon and matched asymptotic expansions this paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting stokes phenomenon in particular the approach highlights the way in which the local structure and the possibility of finding stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions

a new mixed formulation for elasticity we propose a new mixed variational formulation for the equations of linear elasticity it does not require symmetric tensors and consequently is easy to discretize by adapting mixed finite elements developed for scalar second order elliptic equations

analysis of the t point hopf bifurcation abstract in a parameterized three dimensional system of autonomous differential equations a t point is a point of the parameter space where a special kind of codimension 2 heteroclinic cycle occurs if the parameter space is three dimensional such a bifurcation is located generically on a curve a more degenerate scenario appears when this curve reaches a surface of hopf bifurcations of one of the equilibria involved in the heteroclinic cycle we are interested in the analysis of this codimension 3 bifurcation which we call t point hopf in this work we propose a model based on the construction of a poincare map that describes the global behavior close to a t point hopf bifurcation the existence of certain kinds of homoclinic and heteroclinic connections between equilibria and or periodic orbits is proved the predictions deduced from this model strongly agree with the numerical results obtained in a modified van der pol duffing electronic oscillator

solitary wave interaction phenomena in a strut buckling model incorporating restabilisation the archetypal model of the buckling of a compressed long elastic strut resting on a nonlinear elastic foundation is studied localised buckling is investigated when the foundation has both quadratic and cubic terms which initially destabilise but subsequently restabilise the structure the primary solution can be detected by a double scale perturbation procedure and is reminiscent of a solitary wave essentially it consists of a fast periodic oscillation which is slowly modulated and decays exponentially in both directions particular interest is paid to the process of adapting the procedure to account for the post buckling behaviour of two packet or double humped solitary waves in this model we employ the methods of beyond all orders asymptotics to reveal terms formally exponentially small in the perturbation parameter which have macroscopic effects on the post buckling behaviour of the system including the interaction phenomenon of interest the analysis is reinforced by direct numerical computations which reveal the so called snaking behaviour in the subsidiary homoclinic orbits as is observed in the case of primary solutions however additional phenomena arise for these subsidiary forms including the formation of bridges linking solution paths and the appearance of a multitude of closed isolated loops disconnected from other features of the bifurcation diagram

point symmetries of generalized toda field theories a class of two dimensional field theories with exponential interactions is introduced the interaction depends on two coupling matrices and is sufficiently general to include all toda field theories existing in the literature lie point symmetries of these theories are found for an infinite semi infinite and finite number of fields special attention is accorded to conformal invariance and its breaking

classical mathcal w algebras in types a b c d and g we produce explicit generators of the classical mathcal w algebras associated with the principal nilpotents in the simple lie algebras of all classical types and in the exceptional lie algebra of type g 2 the generators are given by determinant formulas in the context of the poisson vertex algebras we also show that the images of the mathcal w algebra generators under the chevalley type isomorphism coincide with the elements defined via the corresponding miura transformations

on localised hotspots of an urban crime model abstract we investigate stationary spatially localised crime hotspots on the real line and the plane of an urban crime model of short et al m short m dorsogna a statistical model of criminal behavior mathematical models and methods in applied sciences 18 2008 1249 1267 extending the weakly nonlinear analysis of short et al we show in one dimension that localised hotspots should bifurcate off the background spatially homogeneous state at a turing instability provided the bifurcation is subcritical using path following techniques we continue these hotspots and show that the bifurcating pulses can undergo the process of homoclinic snaking near the singular limit we analyse the singular limit to explain the existence of spike solutions and compare the analytical results with the numerical computations in two dimensions we show that localised radial spots should also bifurcate off the spatially homogeneous background state localised planar hexagon fronts and hexagon patches are found and depending on the proximity to the singular limit these solutions either undergo homoclinic snaking or act like multi spot solutions finally we discuss applications of these localised patterns in the urban crime context and the full agent based model

front motion metastability and subcritical bifurcations in hydrodynamics metastability occurs whenever under given external conditions as defined by the reynolds number in hydrodynamics more than one linearly stable solution may exist in variational problems the preferred state is usually claimed to be the one with the lowest energy but in hydrodynamics and except for very special conditions no energy functional exists from an analogy between amplitude and reaction diffusion equations it is argued that the border separating in space two possible solutions of the flow equations as turbulent and laminar in pipe flows or boundary layers moves with a constant mean velocity depending on the control parameter the sign of this velocity allows to decide which state is stable and which is metastable possible physical implications of these ideas for various problems of hydrodynamics are discussed

two dimensional exactly and completely integrable dynamical systems monopoles instantons dual models relativistic strings lund regge model generalized toda lattice etc an investigation of two dimensional exactly and completely integrable dynamical systems associated with the local part of an arbitrary lie algebra mathfrak g whose grading is consistent with an arbitrary integral embedding of 3d subalgebra in mathfrak g has been carried out we have constructed in an explicit form the corresponding systems of nonlinear partial differential equations of the second order and obtained their general solutions in the sense of a goursat problem a method for the construction of a wide class of infinite dimensional lie algebras of finite growth has been proposed

localized periodic patterns for the non symmetric generalized swift hohenberg equation abstract a new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized swift hohenberg equation an analysis of this envelope equation results in parametric conditions for localized patterns furthermore it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case the analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing hamiltonian and lagrangian for periodic solutions one striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an o 1 distance from the bifurcation thus providing a novel approach for studying these localized patterns

a mixed formulation of boussinesq equations analysis of nonsingular solutions this paper is concerned with the mixed formulation of the boussinesq equations in two dimensional domains and its numerical approximation the paper deals first with existence and uniqueness results as well as the description of the regularity of any solution the problem is then approximated by a mixed finite element method where the gradient of the velocity and the gradient of the temperature quantities of practical importance are introduced as new unknowns an existence result for the finite element solution and convergence results are proved near a nonsingular solution quasi optimal error estimates are finally presented

w algebras and superalgebras from constrained wzw models a group theoretical classification we present a classification ofw algebras and superalgebras arising in abelian as well as non abelian toda theories each model obtained from a constrained wzw action is related with ansl 2 subalgebra resp osp 1 2 superalgebra of a simple lie algebra resp superalgebra g however the determination of anu 1 y factor commuting withsl 2 resp osp 1 2 appears when it exists particularly useful to characterize the correspondingw algebra the super conformal spin contents of eachw super algebra is performed the class of all the superconformal algebras i e with conformal spinss 2 is easily obtained as a byproduct of our general results

bifurcation structure of dissipative solitons in this paper we analyze in detail the structure of the phase space of a reversible dynamical system describing the stationary solutions of a model for a nonlinear optical cavity we compare our results with the general picture described in p d woods a r champneys physica d 129 1999 147 p coullet c riera c tresser phys rev lett 84 2000 3069 and find that the stable and unstable manifolds of homogeneous and patterned solutions present a much higher level of complexity than predicted including the existence of additional localized solutions and fronts this extra complexity arises due to homoclinic and heteroclinic intersections of the invariant manifolds of low amplitude periodic solutions and to the fact that these periodic solutions together with the high amplitude ones constitute a one parameter family generating a closed line on the symmetry plane

exponential asymptotics of localised patterns and snaking bifurcation diagrams abstract localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple scales analysis beyond all orders the model studied is the swift hohenberg equation of nonlinear optics which is equivalent to the classical swift hohenberg equation with a quadratic and a cubic nonlinearity applying the asymptotic technique away from the maxwell point first it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sech type amplitude in the vicinity of the maxwell point the beyond all orders calculation yields the pinning range of parameters where stable stationary fronts connect the homogeneous and periodic states the full bifurcation diagram for localised patterns is then computed analytically including snake and ladder bifurcation curves this last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a down front to this end a third super slow spatial scale needs to be introduced in which fronts appear as boundary layers in addition the location of the maxwell point and the oscillation wave number of localised patterns are required to fourth order accuracy in the oscillation amplitude