bounds for the solutions of the complex ginzburg landau equation in terms of the dispersion parameters abstract the diameter in the l norm of the global attractor of the complex ginzburg landau equation ut 1 i u ru 1 i u2 u is estimated by using weighted energy estimates for the solutions on the whole space r d for all parameters d and for which global existence is known we obtain the bound c d r 1 2 upper estimates for c are given which are polynomial in for d

spatial structure of time periodic solutions of the ginzburg landau equation abstract we study solutions of the form a x t a x ei t of the complex ginzburg landau equation i a t a xx a a 2 a near those of the integrable limit the nonlinear schrodinger equation i a t a xx a 2 a real we show that spatially periodic and quasiperiodic solutions as well as heteroclinic orbits to them exist in addition to the spatially decaying homoclinic solitions found by hocking and stewartson 1972 we speculate briefly on spatially chaotic solutions

numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues in this paper we derive explicit formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues these include all strong resonances as well as the degenerate flip and neimark sacker bifurcations applying our results to n dimensional maps one avoids the computation of the center manifold but one can directly evaluate the critical normal form coefficients in the original basis the formulas remain valid also for n 2 and allow one to avoid the transformation of the linear part of the map into jordan form the developed techniques are tested on two examples 1 a three dimensional map appearing in adaptive control 2 a periodically forced epidemic model we compute numerically the critical normal form coefficients for several codim 2 bifurcations occurring in these models to compute the required derivatives of the poincare map for the epidemic model the automatic differentiation package adol c is used

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

periodic solutions of the ginzburg landau equation abstract spatially periodic solutions to the ginzburg landau equation are considered in particular we obtain criteria for primary and secondary bifurcation limit cycle solutions nonlinear dispersion relations relating spatial and temporal frequencies only relatively simple tools appear in the treatment and as a result a wide range of parameter cases are considered finally we briefly treat the case of spatial bifurcations

approximation of the korteweg de vries equation by the nonlinear schrodinger equation abstract in 1968 zakharov derived the nonlinear schrodinger equation as an approximation equation for the water wave problem it is still an open problem if there are really solutions of the water wave problem which can be approximated by the solutions of the nonlinear schrodinger equation in answering this question we made some progress for a model problem the approximation of the korteweg de vries equation by the nonlinear schrodinger equation

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

non linear stability of modulated fronts for the swift hohenberg equation we consider front solutions of the swift hohenberg equation t u 1 x 2 2 u 2 u u 3 these are traveling waves which leave in their wake a periodic pattern in the laboratory frame using renormalization techniques and a decomposition into bloch waves we show the non linear stability of these solutions it turns out that this problem is closely related to the question of stability of the trivial solution for the model problem t u x t x 2 u x t 1 tanh x ct u x t u x t p with p 3 in particular we show that the instability of the perturbation ahead of the front is entirely compensated by a diffusive stabilization which sets in once the perturbation has hit the bulk behind the front

modulations of perturbed kdv wavetrains the modulations of n phase korteweg devries kdv wavetrains in the presence of external perturbations is investigated an invariant representation of these modulation equations in terms of differentials on a riemann surface is derived from averaged perturbed conservation laws in particular the explicit dependence of the representation on the external perturbation is obtained this invariant representation is used to place the equation in a riemann diagonal form whose dependence on the external perturbation is explicitly computed

symmetry classes of variable coefficient nonlinear schrodinger equations a variable coefficient nonlinear schrodinger vcnls equation involving three arbitrary complex functions of space time in 1 1 dimensions is analysed from the point of view of its symmetries all equations of the type studied having non trivial lie point symmetry groups g are identified the symmetry group is shown to be at most five dimensional and only when the equation is equivalent to the nls equation itself or to a rather special complex ginzburg landau equation lie point transformations are used to obtain solutions of specific vcnls equations that should be of interest in nonlinear optics or other branches of physics

on the turing patterns in one dimensional gradient skew gradient dissipative systems in this article fundamental properties concerning the turing patterns are considered in one dimensional dissipative systems with gradient skew gradient structure introduced in m kuwamura and e yanagida phys d 175 2003 pp 185 195 it is a natural extension of free energy which covers reaction diffusion systems of activator inhibitor type the theory based on this concept provides a new perspective on a fundamental problem of what unique turing pattern is to be selected among many

asymptotic symmetries and asymptotically symmetric solutions of partial differential equations symmetry methods for differential equations are a powerful tool to attack nonlinear problems in particular for determining solutions with given symmetries to nonlinear pdes since in real applications one is often interested in solutions which are asymptotically symmetric we propose here an approach to asymptotic symmetry based on the methods of lie theory we adopt translate in geometric language and develop the renormalization group approach recently proposed by bricmont and kupiainen 1992 for the ginzburg landau equation

convection patterns in large aspect ratio systems abstract a theory which should have widespread application is developed to treat the statics and slow dynamics of patterns of convective rolls encountered in large aspect ratio rayleigh benard boxes for case of presentation the theory is developed using model equations wavenumber selection the shape of patterns stability and the time dependence resulting from long wavelength instabilities are discussed the effects of adding the local mean drift important in low prandtl number situations are investigated our theory includes the notion of the busse stability balloon reduces near critical values of the stress parameter to the newell whitehead segel equations and contains the pomeau manneville phase equation it also gives a detailed description of the way in which the field amplitude which is slaved to the phase gradient away from threshold becomes an independent order parameter near the critical point

degenerate hyperbolic conservation laws with dissipation reduction to and validity of a class of burgers type equations a conservation law is said to be degenerate or critical if the jacobian of the flux vector evaluated on a constant state has a zero eigenvalue in this paper it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles burger s dynamics the case of k fold degeneracy is also treated and it is shown that it leads to a reduction to a quadratically coupled k fold system of burgers type equations validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces thereby capturing both finite and infinite energy solutions the theory is applied to some examples from stratified shallow water hydrodynamics that model the birth of hydraulic jumps

phase dynamics in the complex ginzburg landau equation abstract for 1 stable time periodic solutions a x t aqeiqx i qt are the locally preferred planform for the complex ginzburg landau equation t a a 1 i x 2 a 1 i a a 2 in order to describe the spatial global behavior an evolution equation for the local wave number q can be derived formally the local wave number q satisfies approximately a conservation law q h q it is the purpose of this paper to explain the extent to which the conservation law is valid by proving estimates for this formal approximation

modulation equation and spdes on unbounded domains we consider the approximation via modulation equations for nonlinear spdes on unbounded domains with additive space time white noise close to a bifurcation an infinite band of eigenvalues changes stability and we study the impact of small space time white noise on the dynamics close to this bifurcation as a first example we study the stochastic swift hohenberg equation on the whole real line here due to the weak regularity of solutions the standard methods for modulation equations fail and we need to develop new tools to treat the approximation as an additional result we sketch the proof for local existence and uniqueness of solutions for the stochastic swift hohenberg and the complex ginzburg landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions which is natural for translation invariant noise like space time white noise we use energy estimates to show that solutions of the ginzburg landau equation are holder continuous and have moments in those functions spaces this gives just enough regularity to proceed with the error estimates of the approximation result

numerical periodic normalization for codim 1 bifurcations of limit cycles explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous odes are derived they include second order coefficients for the fold limit point bifurcation as well as third order coefficients for the flip period doubling and neimark sacker torus bifurcations the formulas are independent of the dimension of the phase space and involve solutions of certain boundary value problems on the interval 0 t where t is the period of the critical cycle as well as multilinear functions from the taylor expansion of the right hand sides near the cycle the formulas allow us to distinguish between sub and supercritical bifurcations in agreement with earlier asymptotic expansions of the bifurcating solutions our formulation makes it possible to use robust numerical boundary value algorithms based on orthogonal collocation rather than shooting techniques which greatly expands its applicability the actual implementation is described in detail we include three numerical examples in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients

new planforms in systems of partial differential equations with euclidean symmetry dionne golubitsky 10 consider the classification of planforms bifurcating simultaneously in scalar partial differential equations that are equivariant with respect to the euclidean group in the plane in particular those planforms corresponding to isotropy subgroups with one dimensional fixed point space are classified many important euclidean equivariant systems of partial differential equations essentially reduce to a scalar partial differential equation but this is not always true for general systems we extend the classification of 10 obtaining precisely three planforms that can arise for general systems and do not exist for scalar partial differential equations in particular there is a class of one dimensional pseudoscalar partial differential equations for which the new planforms bifurcate in place of three of the standard planforms from scalar partial differential equations for example the usual rolls solutions are replaced by a nonstandard planform called anti rolls scalar and pseudoscalar partial differential equations are distinguished by the representation of the euclidean group

saint venant s problem and semi inverse solutions in nonlinear elasticity saint venant s problem consists in finding elastic deformations of an infinite prismatic body taking given values for the cross sectional resultants of force and moment using the center manifold approach we show that all deformations having sufficiently small bounded strains lie on a finite dimensional manifold in particular the flow on this manifold is described by a set of equations having exactly the form of the classical rod equations moreover the set of semi inverse solutions can be analyzed locally

breaking the hidden symmetry in the ginzburg landau equation in this paper we study localised traveling solutions to a ginzburg landau equation to which we have added a small o e 0 e 1 quintic term we consider this term as a model for the higher order nonlinearities which appear in the derivation of the ginzburg landau equation by a combination of a geometrical approach and an explicit perturbation analysis we are able to relate the family of bekki nozaki solutions of the cubic equation n bekki and b nozaki 1985 phys lett a 110 133 135 to a curve of co dimension 2 homoclinic bifurcations in parameter space thus we are able to interpret the hidden symmetry which has been conjectured to explain the existence of the bekki nozaki solutions from the point of view of bifurcation theory we show that the quintic term breaks this hidden symmetry and that the one parameter family of bekki nozaki solutions is embedded in a two parameter family of homoclinic solutions which exist at a co dimension 1 homoclinic bifurcation furthermore we show mainly by geometrical arguments that the addition of the small quintic term can create large families of traveling localised structures that cannot exist in the cubic case these solutions exist in open subsets of the parameter space and correspond to structurally stable multi or n circuit heteroclinic orbits in an ode reduction and have a monotonically decreasing increasing amplitude except for a number n of relatively fast jumps

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

mean field effects for counterpropagating traveling wave solutions of reaction diffusion systems in many problems e g in combustion or solidification one observes traveling waves that propagate with constant velocity and shape in the x direction say are independent of y and z and describe transitions between two equilibrium states e g the burned and the unburned reactants as parameters of the system are varied these traveling waves can become unstable and give rise to waves having additional structure such as traveling waves in the y and z directions which can themselves be subject to instabilities as parameters are further varied to investigate this scenario we consider a system of reaction diffusion equations with a traveling wave solution as a basic state we determine solutions bifurcating from the basic state that describe counterpropagating traveling waves in directions orthogonal to the direction of propagation of the basic state and determine their stability specifically we derive long wave modulation equations for the amplitudes of the counterpropagating traveling waves that

analyticity of ginzburg landau modes ginzburg landau formalism applies for dissipative systems defined on cylindrical domains which are close to the threshold of instability and for which the unstable fourier modes belong to non zero wave numbers in these situations the real part of the curve of critical eigenvalues as function drawn over the wave numbers k is positive of height o 2 and of width o here it is shown that the set of solutions which can be described by the ginzburg landau formalism is attractive to do this we demonstrate that in fourier space peaks appear at integer multiples of the critical wave number kc these peaks called ginzburg landau modes concentrate in time like e k mkc t for 0 t o 1 2 and m z the inverse fourier transform of such a ginzburg landau mode is an analytic function in a strip of width t this result extends a former work of w eckhaus

split step methods for the solution of the nonlinear schro dinger equation a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation the space variable is discretized by means of a finite difference and a fourier method these methods are analyzed with respect to various physical properties represented in the nls in particular it is shown how a conservation law dispersion and instability are reflected in the numerical schemes analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived these results are demonstrated by numerical examples

approximation and attractivity properties of the degenerated ginzburg landau equation abstract we are interested in spatially extended pattern forming systems close to the threshold of the first instability in case when the so called degenerated ginzburg landau equation takes the role of the classical ginzburg landau equation as the amplitude equation of the system this is the case when the relevant nonlinear terms vanish at the bifurcation point here we prove that in this situation every small solution of the pattern forming system develops in such a way that after a certain time it can be approximated by the solutions of the degenerated ginzburg landau equation in this paper we restrict ourselves to a swift hohenberg kuramoto shivashinsky equation as a model for such a pattern forming system

nonlinear stability of taylor vortices in infinite cylinders we consider the taylor couette problem in an infinitely extended cylindrical domain in the case when couette flow is weakly unstable and a family of spatially periodic equilibria called the taylor vortices has bifurcated from this trivial ground state we show that those taylor vortices which are not linearly unstable in the sense of eckhaus are in fact nonlinearly stable with respect to small spatially localized perturbations the main difficulty in showing this result stems from the fact that on unbounded cylindrical domains the taylor vortices are only linearly marginally stable with continuous spectrum up to the imaginary axis bloch wave representations of the solutions and renormalization theory allow us to show that the nonlinear problem behaves asymptotically like the linearized one which is under a diffusive regime

propagating fronts and the center manifold theorem we prove the existence of propagating front solutions for the swift hohenberg equation sh using the center manifold theorem we reduce the problem to a two dimensional system of ordinary differential equations they describe stationary solutions and front solutions of the partial differential equation sh we identify the well known amplitude equation as the lowest order approximation to the equation of motion on the center manifold

switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in odes the paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n dimensional odes we discuss the implementation and the performance of the algorithm in several examples including an extended lorenz 84 model and a laser system

slow time periodic solutions of the ginzburg landau equation in this paper we study the behaviour of solutions of the form if z t q z e i wt e 1 of the rescaled ginzburg landau equation k 1 1 ib l kl2 k 1 ia kzz for a ca b eb w plays the role of free parameter this leads to a perturbation analysis on a complex duffing equation similar to the analysis of holmes physica d 23 1986 84 we show that the spatial quasiperiodic solutions of the unperturbed e 0 case disappear due to the perturbation and prove the existence of degenerated periodic solutions which oscillate through the origin we also establish the existence of several types of heteroclinic orbits connecting counterrotating periodic patterns

instability and stability of rolls in the swift hohenberg equation we develop a method for the stability analysis of bifurcating spatially periodic patterns under general nonperiodic perturbations in particular it enables us to detect sideband instabilities we treat in all detail the stability question of roll solutions in the two dimensional swift hohenberg equation and derive a condition on the amplitude and the wave number of the rolls which is necessary and sufficent for stability moreover we characterize the set of those wave vectors which give rise to unstable perturbations

the structure of spiral domain patterns and shocks in the 2d complex ginzburg landau equation abstract cellular patterns appear spontaneously in a number of nonequilibrium systems governed by the dynamics of a complex field in the case of the complex ginzburg landau equation disordered cells of effectively frozen spirals appear separated by thin walls shocks on a scale much larger than the basic wavelength of the spirals we show that these structures can be understood in very simple terms in particular we show that the walls are to a good approximation segments of hyperbolae and this allows us to construct the wall pattern given the vortex centers and a phase constant for each vortex the fact that the phase is only defined up to an integer multiple of 2 introduces a quantization condition on the sizes of the smallest spiral domains the transverse structure of the walls is analyzed by treating them as heteroclinic connections of a system of ordinary differential equations the structure depends on the angle the wall makes with the local phase contours and the behavior can be either monotonic or oscillatory depending on the parameters

spiral wave solutions for reaction diffusion equations in a fast reaction slow diffusion limit abstract a two species reaction diffusion equation is studied in a distinguished limit of fast reaction and slow diffusion asymptotic matching yields a spiral solution whose rotation frequency is a unique function of the physical parameters and the constant pitch of the spiral using an idea of fife the problem is first reduced to a free boundary problem the solution is obtained by matching the core where diffusion balances the curvature of the free boundary to the far field where the effects of diffusion are small the equations of motion are averaged over the rotation frequency to yield steady equations in the two regions helically symmetric solutions to the averaged equation reduce to solving a fourth order ode in the core region matched to a second order ode in the far field speculations on the dynamics of the system and coupling between the core and far field are presented

bifurcations of poiseuille flow between parallel plates three dimensional solutions with large spanwise wavelength the spatial dynamics approach is applied to the analysis of bifurcations of the three dimensional poiseuille flow between parallel plates in contrast to the classical studies we impose time periodicity as well as spatial periodicity with period 2 in the streamwise direction however we make no assumptions on the behavior in the spanwise direction except the uniform closeness of the bifurcating solution to the basic flow in an abstract setting it is shown how the dimension of the critical eigenspace of the spatial dynamics analysis can be uniquely determined from the classical linear stability problem for the three dimensional poiseuille problem we are able to find all relevant coefficients from the analysis of the purely two dimensional problem moreover we are able to analyze precisely the influence of a spanwise pressure gradient and the associated spanwise mass flux the study of the reduced problem shows that there are two different kinds of solutions spirals and ribbons which are 2 p periodic in the spanwise direction as in the couette taylor problem and both of them bifurcate in the same direction

convergence properties of the 8 10 and 12 mode representations of quasipatterns spatial fourier transforms of quasipatterns observed in faraday wave experiments suggest that the patterns are well represented by the sum of 8 10 or 12 fourier modes with wavevectors equally spaced around a circle this representation has been used many times as the starting point for standard perturbative methods of computing the weakly nonlinear dependence of the pattern amplitude on parameters we show that nonlinear interactions of n such fourier modes generate new modes with wavevectors that approach the original circle no faster than a constant times n 2 and that there are combinations of modes that do achieve this limit as in kam theory small divisors cause difficulties in the perturbation theory and the convergence of the standard method is questionable in spite of the bound on the small divisors we compute steady quasipattern solutions of the cubic swift hohenberg equation up to 33rd order to illustrate the issues in some detail and argue that the standard method does not converge sufficiently rapidly to be regarded as a reliable way of calculating properties of quasipatterns

analysis of bifurcations of limit cycles with lyapunov exponents and numerical normal forms in this paper we focus on the combination of normal form and lyapunov exponent computations in the numerical study of the three codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous odes the normal form formulas are independent of the dimension of the phase space and involve solutions of certain linear boundary value problems the formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations where 3 tori and 4 tori can be present we apply our techniques to the study of a known laser model a novel model from population biology and a model of mechanical vibrations these models exhibit limit point neimark sacker period doubling neimark sacker and double neimark sacker bifurcations lyapunov exponents are computed to numerically confirm the results of the normal form analysis in particular with respect to the existence of stable invariant tori of various dimensions conversely the normal forms are essential to understand the significance of the lyapunov exponents

solutions without phase slip for the ginsburg landau equation we consider the ginsburg landau equation for a complex scalar field in one dimension and consider initial data which have two different stationary solutions as their limits in space asx if these solutions are not very different then we show that the initial data will evolve to a stationary solution by a phase melting process which avoids phase slips i e which does not go through zero amplitude

instabilities induced by a weak breaking of a strong spatial resonance abstract through multiple scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady state pattern forming instabilities in the case that the wavelengths of the instabilities are nearly in the ratio 1 2 in the case of exact 1 2 resonance the amplitude equations are odes here they are pdes we discuss the stability of spatially periodic solutions to long wavelength disturbances by including these modulational effects we are able to explore the relevance of the exact 1 2 results to spatially extended physical systems for parameter values near to this codimension two bifurcation point these new instabilities can be described in terms of reduced normal form pdes near various secondary codimension two points the robust heteroclinic cycle in the odes is destabilised by long wavelength perturbations and a stable periodic orbit is generated that lies close to the cycle an analytic expression giving the approximate period of this orbit is derived

kawahara dynamics in dispersive media abstract in a previous paper it was shown that in the long wave limit the water wave problem without surface tension can be described approximately by two decoupled kdv equations here we study a model problem and prove that in a degenerate case which occurs for the water wave problem with surface tension and near other codimension two points at which the coefficient in front of the leading order dispersive term in the equation of motion vanishes the long wave limit can be rigorously approximated by two decoupled kawahara equations

the non linear stability of front solutions for parabolic partial differential equations for the ginzburg landau equation and similar reaction diffusion equations on the line we show convergence of complex perturbations of front solutions towards the front solutions by exhibiting a coercive functional

on exact solutions of modified complex ginzburg landau equation abstract the one dimensional modified complex ginzburg landau equation has been studied by the use of the conte and musette method this method permits us to derive all the known exact solutions in a unified natural scheme these solutions are expressed in terms of solitary wave periodic unbounded wave and shock type wave we also find previously unknown exact propagating hole the degeneracies of modified complex ginzburg landau equation have also been examined as well as several of their solutions

modulation equations stochastic bifurcation in large domains we consider the stochastic swift hohenberg equation on a large domain near its change of stability we show that under the appropriate scaling its solutions can be approximated by a periodic wave which is modulated by the solutions to a stochastic ginzburg landau equation we then proceed to show that this approximation also extends to the invariant measures of these equations

almost global existence and transient self similar decay for poiseuille flow at criticality for exponentially long times abstract we consider nonlinear diffusion equations with critical exponent as tu x2u u3 with x r for small initial data in l1 l it is well known that almost all solutions of this system explode in finite time however we make the observation that in terms of the norm of the initial conditions it takes an exponentially long time moreover before explosion the l norm of such solutions becomes exponentially small which makes it almost impossible to observe the instability in experiments as an application we consider the long time transient self similar decay to unstable poiseuille flow at criticality for exponentially long times this together with a subcritical bifurcation and short time transient amplification is a principal obstruction in all attempts to measure the critical reynolds number for this experiment more and more precisely

hidden symmetries on partially unbounded domains abstract systems of reaction diffusion equations posed on bounded rectangular domains with neumann boundary conditions often exhibit behavior that seems degenerate given the physical symmetries of the problem it is now well understood that neumann boundary conditions lead to hidden symmetries that are responsible for subtle changes in the generic bifurcations of such systems in this paper we consider the analogous situation for partially unbounded domains such as the strip r 0 we show that hidden symmetries due to the assumption of neumann boundary conditions have remarkable consequences for the validity of ginzburg landau equations which govern the local bifurcations a single ginzburg landau equation which is universal for general boundary conditions on r 0 no longer suffices in general instead it is necessary to consider p coupled ginzburg landau equations where p is an arbitrary positive integer

toric geometry and equivariant bifurcations abstract many problems in equivariant bifurcation theory involve the computation of invariant functions and equivariant mappings for the action of a torus group we discuss general methods for finding these based on some elementary considerations related to toric geometry a powerful technique in algebraic geometry this approach leads to interesting combinatorial questions about cones in lattices which lead to explicit calculations of minimal generating sets of invariants from which the equivariants are easily deduced we also describe the computation of hilbert series for torus invariants and equivariants within the same combinatorial framework as an example we apply these methods to the interaction of two linear modes of a euclidean invariant pde on a rectangular domain with periodic boundary conditions

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

validity of the kdv equation for the modulation of periodic traveling waves in the nls equation abstract bethuel et al 1 2 and chiron and rousset 3 gave very nice proofs of the fact that slow modulations in time and space of periodic wave trains of the nls equation can approximately be described via solutions of the kdv equation associated with the wave train here we give a much shorter proof of a slightly weaker result avoiding the very detailed and fine analysis of 1 2 3 our error estimates are based on a suitable choice of polar coordinates a cauchy kowalevskaya like method and energy estimates

diffusive mixing of stable states in the ginzburg landau equation the ginzburg landau equation on the real line has spatially periodic steady states of the form with and for we construct solutions which converge for all t 0 to the limiting pattern as these solutions are stable with respect to sufficiently small perturbations and behave asymptotically in time like where is uniquely determined by the boundary conditions this extends a previous result of brk92 by removing the assumption that should be close to zero the existence of the limiting profile is obtained as an application of the theory of monotone operators and the long time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term

the existence of dendritic fronts in this paper we study a fourth order semilinear parabolic equation on the infinite real line we show that in a certain parameter range this equation has propagating front solutions solutions tending to 0 at and advancing to the right with a speedc which leave behind them aperiodic pattern in the laboratory frame this is thus an example of spontaneous pattern formation

global existence via ginzburg landau formalism and pseudo orbits of ginzburg landau approximations the so called ginzburg landau formalism applies for parabolic systems which are defined on cylindrical domains which are close to the threshold of instability and for which the unstable fourier modes belong to non zero wave numbers this formalism allows to describe an attracting set of solutions by a modulation equation here the ginzburg landau equation if the coefficient in front of the cubic term of the formally derived ginzburg landau equation has negative real part the method allows to show global existence in time in the original system of all solutions belonging to small initial conditions inl another aim of this paper is to construct a pseudo orbit of ginzburg landau approximations which is close to a solution of the original system up tot we consider here as an example the socalled kuramoto shivashinsky equation to explain the methods but it applies also to a wide class of other problems like e g hydrodynamical problems or reaction diffusion equations too

amplitude equation for lattice maps a renormalization group approach we consider the development of instabilities of homogeneous stationary solutions of discrete time lattice maps under some generic hypotheses we derive an amplitude equation which is the space time continuous ginzburg landau equation using dynamical renormalization group methods we control the accuracy of this approximation in a large ball of its basin of attraction

numerical methods for two parameter local bifurcation analysis of maps we discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits cycles of maps and their implementation in matcont a matlab toolbox for continuation and bifurcation analysis of dynamical systems this includes the numerical continuation of fixed points of iterates of the map with one control parameter detecting and locating their bifurcation points i e limit point period doubling and neimark sacker and their continuation in two control parameters as well as detection and location of all codimension 2 bifurcation points on the corresponding curves for all bifurcations of codim 1 and 2 the critical normal form coefficients are computed both numerically with finite directional differences and using symbolic derivatives of the original map using a parameter dependent center manifold reduction explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period these asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations we provide two examples illustrating the developed techniques a generalized henon map and a juvenile adult competition model from mathematical biology

fully nonlinear global modes in spatially developing media abstract global modes on a doubly infinite one dimensional domain wkbj approximations in the linear and nonlinear regions through appropriate transition layers the real global frequency is determined by requiring that spatial branches issuing from x and x be continuously connected at a saddle point of the local nonlinear dispersion relation n l k r 2 x between the frequency the wave number k and amplitude r at a given station x the results constitute a fully nonlinear generalization of the linear frequency selection criteria previously obtained by chomaz et al 1991 monkewitz et al 1993 and le dizes et al 1996

the time dependent amplitude equation for the swift hohenberg problem precise estimates for the validity of the amplitude approximation for the swift hohenberg equation are given in a fully time dependent framework it is shown that small solutions of ordero which are modulated like stationary solutions have an evolution which is well described in the amplitude approximation for a time of ordero 2 for the proofs we use techniques for nonlinear semigroups and oscillatory integrals

the validity of phase diffusion equations and of cahn hilliard equations for the modulation of pattern in reaction diffusion systems abstract so called phase diffusion equations and cahn hilliard equations can be derived via multiple scaling analysis in order to describe slow modulations in time and space of stable and slightly unstable spatially periodic pattern it is the purpose of this paper to explain the extent to which these formal approximations are valid by proving estimates between the formal approximations and true solutions of the original system the proof is given for an abstract reaction diffusion system as original system

a singular boundary value problem for uniformly elliptic operators abstract strong solvability in the sobolev space w 2 p is proved for the oblique derivative problem j i 1 n a ij x d ij u f x almost every where in u l x u x in the trace sense on in the case when the vector field l x has a contact of infinite order with at the points of some non empty subset e

n th order operator splitting schemes and nonreversible systems this paper is concerned with partitioned nth order accurate split operator schemes built using m distinct solution operators for semidiscrete equations of the form frac d dt u j f j 1 u k cdots f j m u k which arise among others from constant coefficient parabolic partial differential equations we prove that for every n geq 3 and m geqslant 2 each solution operator must be applied for at least one backward fractional time step during each complete time step this result has important consequences for applications to the complex ginzburg landau equation with periodic boundary conditions and other partial differential equations with both reversible and nonreversible components furthermore for the special case of n 3 and m 2 we analytically determine all possible schemes

slowly varying fully nonlinear wavetrains in the ginzburg landau equation abstract following the ideas of howard and kopell 9 a perturbation theory is developed for slowly varying fully nonlinear wavetrains i e solutions which appear locally as travelling waves but with frequencies and wavelengths which may vary widely on long length and time scales this perturbation theory is applied to the ginzburg landau equation the motion and stability of slowly varying wavetrains is shown to be governed by a simple wave equation which can develop shocks corresponding to rapid changes in wavenumber numerical results supporting this theory are presented a shock structure is proposed and numerically verified these results together with a winding invariant valid in the limit of slow variation suggest that over a large range of parameters many initial conditions relax to uniform wavetrains the evolution of a marginally diffusively stable wavetrain is also examined it is argued that the evolution is governed by a perturbed korteweg de vries equation

numerical normalization techniques for all codim 2 bifurcations of equilibria in ode s explicit computational formulas for the coefficients of the normal forms for all codim 2 equilibrium bifurcations of equilibria in autonomous odes are derived they include second order coefficients for the bogdanov takens bifurcation third order coefficients for the cusp and fold hopf bifurcations and coefficients of the fifth order terms for the generalized hopf bautin and double hopf bifurcations the formulas are independent on the dimension of the phase space and involve only critical eigenvectors of the jacobian matrix of the right hand sides and its transpose as well as multilinear functions from the taylor expansion of the right hand sides at the critical equilibrium the normal form coefficients for the fold hopf bifurcation in the new lorenz model are computed using the derived formulas proving the existence of a nontrivial invariant set in the system

diffusive stability of spatial periodic solutions of the swift hohenberg equation we are interested in the nonlinear stability of the eckhaus stable equilibria of the swift hohenberg equation on the infinite line with respect to small integrable perturbations the difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero the nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant i e that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime

computing periodic orbits and their bifurcations with automatic differentiation this paper formulates several algorithms for the direct computation of periodic orbits as solutions of boundary value problems the algorithms emphasize the use of coarse meshes and high orders of accuracy convergence theorems are given in the limit of increasing order with a fixed mesh the algorithms are implemented with the use of matlab and adol c a software package for automatic differentiation automatic differentiation enables accurate computation of high order derivatives of functions without the truncation errors inherent in finite difference calculations we embed the algorithms in a continuation framework and extend them to compute saddle node bifurcations of periodic orbits directly we present data from numerical studies of four test problems making some comparisons with other methods for computing periodic orbits these results demonstrate that high order methods based upon automatic differentiation are capable of high precision with small meshes

characterization of dispersive chaos and related states of binary fluid convection abstract we describe an extensive series of experiments on traveling wave convection in an ethanol water mixture with separation ratio 0 020 in a long narrow annular container as the stress parameter is quasi statically increased through the range 0 2 above onset this system evolves through a series of complex spatiotemporal states before settling into an extended pattern of steady spatially uniform convective rolls much of the dynamical behavior is influenced by strong nonlinear dispersion which causes the repeated and often irregular appearance and sudden collapse of spatially localized bursts of traveling waves small amplitude states seen very close to onset are analyzed quantitatively using a complex ginzburg landau equation complex dynamical states encountered further above onset are analyzed using a number of classical statistical techniques focusing mainly on correlation functions and fourier spectra in addition we use the technique of biorthogonal decomposition in an effort to extract the basic structures of the patterns and to characterize the complexty of their dynamics

computation of periodic solution bifurcations in odes using bordered systems we consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous odes namely the fold the period doubling or flip bifurcation and the torus or neimark sacker bifurcation in the fold and flip cases we append one scalar equation to the standard periodic bvp that defines the periodic solution in the torus case four scalar equations are appended evaluation of these scalar equations and their derivatives requires the solution of linear bvps whose sparsity structure after discretization is identical to that of the linearization of the periodic bvp therefore the calculations can be done using existing numerical linear algebra techniques such as those implemented in the software auto and colsys

attractivity of the ginzburg landau mode distribution for a pattern forming system with marginally stable long modes abstract the mathematical theory of the description of pattern forming systems close to the first instability via the ginzburg landau equation is based on approximation and attractivity results this theory allowed to prove global existence results and upper semicontinuity of attractors for classical hydrodynamical stability problems such as the couette taylor problem recently approximation results for the ginzburg landau approximation for pattern forming systems with marginally stable long modes such as the benard marangoni system have been shown it is the purpose of this paper to prove the second fundamental property namely the attractivity of the ginzburg landau mode distribution for such systems

weak and strong type estimates for the multilinear pseudo differential operators abstract in this paper we investigate the boundedness of the multilinear pseudo differential operator t first we establish the local exponential decay estimates for t in terms of the corresponding commutators t b we obtain the local subexponential decay estimates secondly we derive the weighted mixed weak type inequality for t which parallels sawyer s conjecture for calderon zygmund operators and covers the endpoint weighted inequalities last but not least we present the sharp weighted estimates for t and t b it is worth mentioning that our results are totally new even in the linear case

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

a rigorous upper bound on the propagation speed for the swift hohenberg and related equations we prove that if the initial condition of the swift hohenberg equation partial t u x t varepsilon 2 1 partial x 2 2 text u x t u 3 x t is bounded in modulus by ce x as x the solution cannot propagate to the right with a speed greater than mathop sup limits 0 gamma leqslant beta gamma 1 varepsilon 2 4 gamma 2 8 gamma 4 this settles a long standing conjecture about the possible asymptotic propagation speed of the swift hohenberg equation the proof does not use the maximum principle and is simple enough to generalize easily to other equations we illustrate this with an example of a modified ginzburg landau equation where the critical speed is not determined by the linearization alone

boundary effects and the onset of taylor vortices it is well established that the onset of spatially periodic vortex states in the taylor couette flow between rotating cylinders occurs at the value of reynolds number predicted by local bifurcation theory however the symmetry breaking induced by the top and bottom plates means that the true situation should be a disconnected pitchfork indeed experiments have shown that the fold on the disconnected branch can occur at more than double the reynolds number of onset this leads to an apparent contradiction why should taylor vortices set in so sharply at the reynolds number predicted by the symmetric theory given such large symmetry breaking effects caused by the boundary conditions this paper offers a generic explanation the details are worked out using a swift hohenberg pattern formation model that shares the same qualitative features as the taylor couette flow onset occurs via a wall mode whose exponential tail penetrates further into the bulk of the domain as the driving parameter increases in a large domain of length l we show that the wall mode creates significant amplitude in the centre at parameter values that are o l 2 away from the value of onset in the problem with ideal boundary conditions we explain this as being due to a hamiltonian hopf bifurcation in space which occurs at the same parameter value as the pitchfork bifurcation of the temporal dynamics the disconnected anomalous branch remains o 1 away from the onset parameter since it does not arise as a bifurcation from the wall mode c 2003 elsevier b v all rights reserved

global characterization of the normal form for a vector field near a closed orbit on deduit une generalisation naturelle de la caracterisation globale de elphick pour des champs de vecteurs pres d une orbite fermee

steady state bifurcation with euclidean symmetry we consider systems of partial differential equations equivariant under the euclidean group e n and undergoing steady state bifurcation with nonzero critical wavenumber from a fully symmetric equilibrium a rigorous reduction procedure is presented that leads locally to an optimally small system of equations in particular when n 1 and n 2 and for reaction diffusion equations with general n reduction leads to a single equation our results are valid generically with perturbations consisting of relatively bounded partial differential operators in analogy with equivariant bifurcation theory for compact groups we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of e n the representation theory of e n is driven by the irreducible representations of o n 1 for n 1 this constitutes a mathematical statement of the universality of the ginzburg landau equation on the line in recent work we addressed the validity of this equation using related techniques when n 2 there are precisely two significantly different types of reduced equation scalar and pseudoscalar corresponding to the trivial and nontrivial one dimensional representations of o 1 there are infinitely many possibilities for each n equal to or greater than 3

extensive properties of the complex ginzburg landau equation we study the set of solutions of the complex ginzburg landau equation in r d d 3 we consider the global attracting set i e the forward map of the set of bounded initial data and restrict it to a cube q l of side l we cover this set by a minimal number n q l of balls of radius in l infin q l we show that the kolmogorov entropy per unit length exists in particular we bound by which shows that the attracting set is smaller than the set of bounded analytic functions in a strip we finally give a positive lower bound

the validity of modulation equations for extended systems with cubic nonlinearities modulation equations play an essential role in the understanding of complicated systems near the threshold of instability here we show that the modulation equation dominates the dynamics of the full problem locally at least over a long time scale for systems with no quadratic interaction term we develop a method which is much simpler than previous ones it involves a careful bookkeeping of errors and an estimate of gronwall type as an example for the dissipative case we find that the ginzburg landau equation is the modulation equation for the swift hohenberg problem moreover the method also enables us to handle hyperbolic problems the nonlinear schrodinger equation is shown to describe the modulation of wave packets in the sine gordon equation

coarsening by ginzburg landau dynamics we study slowly moving solutions of the real ginzburg landau equation on the infinite line by a method inspired by the work of j carr and r l pego these solutions are functions taking alternately positive or negative values on large intervals we give a formula for the speed of the motion of the zeros of these functions and present a description of the collapse of two nearby zeros since we work on an infinite domain we can find initial conditions which lead to infinite sequences of collapses of kinks these results justify in part the approach to the coarsening problem by a simplified model proposed by a j bray b derrida and c godreche their model starts from a random distribution of domain lengths and evolves by successive elimination of the smallest domain our model is on one hand more realistic since it controls the original ginzburg landau equation in continuous time on the other hand we fail to gain sufficient control over the vanishing of pairs of kinks embedded within pairs of kinks at a greater distance