feasible and noninterior path following in constrained minimization with low multiplier regularity primal dual path following methods for constrained minimization problems in function space with low multiplier regularity are introduced and analyzed regularity properties of the path are proved the path structure allows us to define approximating models which are used for controlling the path parameter in an iterative process for computing a solution of the original problem the moreau yosida regularized subproblems of the new path following technique are solved efficiently by semismooth newton methods the overall algorithmic concept is provided and numerical tests including a comparison with primal dual path following interior point methods for state constrained optimal control problems show the efficiency of the new concept

random multi overlap structures for optimization problems we extend to the k sat and p xor sat optimization problems the results recently achieved by introducing the concept of random multi overlap structure for the viana bray model of diluted mean field spin glass more precisely we can prove a generalized bound and an extended variational principle for the free energy per site in the thermodynamic limit moreover a trial function implementing ultrametric breaking of replica symmetry is exhibited the ultrametric structure exhibits the same factorization property as the optimal structures for the viana bray model and the sherrington kirkpatrick nondiluted model

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

studies of phase turbulence in the one dimensional complex ginzburg landau equation the phase turbulent pt regime for the one dimensional complex ginzburg landau equation cgle is carefully studied in the limit of large systems and long integration times using an efficient integration scheme particular attention is paid to solutions with a nonzero phase gradient for fixed control parameters solutions with conserved average phase gradient ensuremath nu exist only for ensuremath nu less than some upper limit the transition from phase to defect turbulence happens when this limit becomes zero a lyapunov analysis shows that the system becomes less and less chaotic for increasing values of the phase gradient for high values of the phase gradient a family of nonchaotic solutions of the cgle is found these solutions consist of spatially periodic or aperiodic waves traveling with constant velocity they typically have incommensurate velocities for phase and amplitude propagation showing thereby a type of quasiperiodic behavior the main features of these traveling wave solutions can be explained through a modified kuramoto sivashinsky equation that rules the phase dynamics of the cgle in the pt phase the latter explains also the behavior of the maximal lyapunov exponents of chaotic solutions

stability of oscillating hexagons in rotating convection abstract breaking the chiral symmetry rotation induces a secondary hopf bifurcation in weakly nonlinear hexagon patterns which gives rise to oscillating hexagons we study the stability of the oscillating hexagons using three coupled ginzburg landau equations close to the bifurcation point we derive reduced equations for the amplitude of the oscillation coupled to the phase of the underlying hexagons within these equations we identify two types of long wave instabilities and study the ensuing dynamics using numerical simulations of the three coupled ginzburg landau equations

secondary instability mode of a roll pattern and transition to spatiotemporal chaos in the taylor dean system the secondary instability mode and transition to a weak chaotic regime in a one dimensional roll pattern have been investigated in the taylor dean system the spatiotemporal modulation of a roll pattern called a triplet state in phys rev lett 64 1729 1990 is quantitatively characterized using the demodulation technique by the hilbert transform a triplet pattern arises from the primary structure by a periodic modulation of the phase both in space and time the amplitude of the modulation chosen as an order parameter has been measured and it grows as a square root of the reduced control parameter the correlation length and time of the triplet pattern are maximal near the threshold and decrease to values comparable to the size and period of a triplet in the chaotic regime in the chaotic regime regular oscillations of triplets are observed with a finite frequency suggesting a phase modulation of the triplet pattern itself s1063 651x9802209 0

replica bounds for optimization problems and diluted spin systems in this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by guerra cond mat 0205123 to prove that the replica method generates variational bounds for disordered systems we analyze a family of models that includes the viana bray model the diluted p spin model or random xor sat problem and the random k sat problem showing that the replica cavity method at the various levels of approximation provides systematic schemes to obtain lower bounds of the free energy at all temperatures and of the ground state energy in the case of k sat and xor sat it thus gives upper bounds of the satisfiability threshold our analysis underlines deep connections with the cavity method which are not evident in the long range case

barrier methods for optimal control problems with state constraints we study barrier methods for state constrained optimal control problems with pdes the focus of our analysis is the path of minimizers of the barrier subproblems with the aim to provide a solid theoretical basis for function space oriented path following algorithms we establish results on existence continuity and convergence of this path moreover we consider the structure of barrier subdifferentials which play the role of dual variables

sources sinks and wavenumber selection in coupled cgl equations and experimental implications for counter propagating wave systems we study the coupled complex ginzburg landau cgl equations for traveling wave systems and show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems we focus on the regime in which sources and sinks separate patches of left and right traveling waves i e the case that these modes suppress each other we present in detail the framework to analyze these coherent structures and show that the theory predicts a number of general properties which can be tested directly in experiments our counting arguments for the multiplicities of these structures show that independently of the precise values of the coefficients in the equations there generally exists a symmetric stationary source solution which sends out waves with a unique frequency and wave number sinks on the other hand occur in two parameter families and play an essentially passive role being sandwiched between the sources these simple but general results imply that sources are important in organizing the dynamics of the coupled cgl equations simulations show that the consequences of the wavenumber selection by the sources is reminiscent of a similar selection by spirals in the 2d complex ginzburg landau equations sources can send out stable waves convectively unstable waves or absolutely unstable waves we show that there exists an additional dynamical regime where both single and bimodal states are unstable the ensuing chaotic states have no counterpart in single amplitude equations a third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate e this is related to the fact that the standard coupled cgl equations are not uniform in e in particular when the group velocity term dominates over the linear growth term no stationary source can exist however sources displaying nontrivial dynamics can often survive here our results for the existence multiplicity wavelength selection dynamics and scaling of sources and sinks and the patterns they generate are easily accessible by experiments we therefore advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment in addition they bring up a large number of new research issues and open problems which are listed explicitly in the concluding section

linearity inside nonlinearity exact solutions to the complex ginzburg landau equation abstract systems of two linear partial differential equations pdes with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear pdes for the one dimensional complex ginzburg landau equation the usual representations of the complex field re a or a are multivalued which makes difficult the search for exact solutions in this manner fortunately the painleve analysis naturally introduces an elementary single valued like representation of a by two fields z grad respectively complex abd real uniquely defined by an explicit expression for and a ze i another important feature is the invariance by parity on a which increases the class of expected solutions this allows to retrieve quite easily the four famous solutions of nozaki and bekki represented by the constant coefficients of two linear partial differential equations and a finite set of constants

an interior point algorithm with inexact step computation in function space for state constrained optimal control we consider an interior point method in function space for pde constrained optimal control problems with state constraints our emphasis is on the construction and analysis of an algorithm that integrates a newton path following method with adaptive grid refinement this is done in the framework of inexact newton methods in function space where the discretization error of each newton step is controlled by adaptive grid refinement in the innermost loop this allows to perform most of the required newton steps on coarse grids such that the overall computational time is dominated by the last few steps for this purpose we propose an a posteriori error estimator for a problem suited norm

several path following methods for a class of gradient constrained variational inequalities path following splitting and semismooth newton methods for solving a class of problems related to elasto plastic material deformations are proposed analyzed and tested numerically while the splitting techniques result in alternating minimization schemes which are typically linearly convergent the proposed moreau yosida regularization based semismooth newton technique and an associated lifting step yield local superlinear convergence in function space the lifting step accounts for the fact that the operator associated with the linear system in the newton iteration need not be boundedly invertible uniformly along the iterates for devising an efficient update strategy for the path following parameter regularity properties of the path are studied and utilized within an inexact path following scheme for all approaches the paper ends by a report on numerical tests of the different approaches

bounds for diluted mean fields spin glass models in an important recent paper 2 s franz and m leone prove rigorous lower bounds for the free energy of the diluted p spin model and the k sat model at any temperature we show that the results for these two models are consequences of a single general principle our calculations are significantly simpler than those of 2 even in the replica symmetric case

on the time splitting spectral method for the complex ginzburg landau equation in the large time and space scale limit we are interested in the numerical approximation of the complex ginzburg landau equation in the large time and space limit there are two interesting regimes in this problem one being the large space time limit and one being the nonlinear schrodinger limit these limits have been studied analytically in for example t colin and a soyeur asymptot anal 13 1996 pp 361 372 f h lin comm pure appl math 51 1998 pp 385 441 f h lin and j x xin comm math phys 200 1999 pp 249 274 we study a time splitting spectral method for this problem in particular we are interested in whether such a scheme is asymptotic preserving ap with respect to these two limits our results show that the scheme is ap for the first limit but not the second one for the large space time limit our numerical experiments show that the scheme can capture the correct physical behavior without resolving the small scale dynamics even for transitional problems where small and large scales coexist

modulated amplitude waves and defect formation in the one dimensional complex ginzburg landau equation the transition from phase chaos to defect chaos in the complex ginzburg landau equation cgle is related to saddle node bifurcations of modulated amplitude waves maws first the spatial period p of maws is shown to be limited by a maximum psn which depends on the cgle coefficients maw like structures with period larger than psn evolve to defects second slowly evolving near maws with average phase gradients 0 and various periods occur naturally in phase chaotic states of the cgle as a measure for these periods we study the distributions of spacings p between neighbouring peaks of the phase gradient a systematic comparison of p and psn as a function of coefficients of the cgle shows that defects are generated at locations where p becomes larger than psn in other words maws with period psn represent critical nuclei for the formation of defects in phase chaos and may trigger the transition to defect chaos since rare events where p becomes sufficiently large to lead to defect formation may only occur after a long transient the coefficients where the transition to defect chaos seems to occur depend on system size and integration time we conjecture that in the regime where the maximum period psn has diverged phase chaos persists in the thermodynamic limit 2001 published by elsevier science b v

microscopic simulation of limit cycle behavior in spatially extended systems the onset of homogeneous oscillations in spatially extended system is considered the master equation formulation shows that in a one dimensional system there exists a finite length beyond which the homogeneous oscillations are destroyed microscopic simulations are used to investigate the status of this prediction and quantitative agreement is obtained the origin of the desynchronization mechanism is clarified

exact periodic solutions of the complex ginzburg landau equation three new exact periodic solutions of the complex ginzburg landau equation are obtained in terms of the weierstrass elliptic function furthermore the new periodic solutions and other shock solutions appear as their bounded limits along the real axis for particular relationships between the coefficients in the equation in the general case bounded limits are nothing but the already known pulse hole and shock solutions it is also shown that the shapes of the solutions are quite different from the shape of the usual envelope wave solution in particular the spatial structure of the new bounded periodic solutions varies in time while the pulse solution may exhibit breather like behavior

differential flow instability in the ginzburg landau and swift hohenberg approximations abstract the effects of a differential flow of the components of a reaction diffusion system which is close to its stability boundary are described within the long wavelength approximation in the vicinity of the hopf bifurcation the system s evolution is governed by a complex ginzburg landau equation modified by a purely imaginary convective term if the system is near the zero real eigenvalue bifurcation the governing equation is a modified swift hohenberg equation in both cases the homogeneous stable reference steady state may be destabilized by the differential flow in the ginzburg landau equation the destabilization occurs as long as the flow velocity exceeds some critical value v cr which tends to zero as the system approaches the hopf bifurcation in the modified swift hohenberg equation the flow has either a destabilizing or stabilizing effect depending on the sign of one of the system parameters destabilization occurs when the flow velocity exceeds some threshold however in this case the threshold remains finite even at the bifurcation point in both ginzburg landau and swift hohenberg equations the differential flow instability produces traveling plane waves the stability analysis shows that once a periodic plane wave is established its spatial period remains unchanged over a finite range of the flow velocity and changes in discrete steps the phenomenon of wavenumber locking wavenumber locking is verified in numerical experiments with the ginzburg landau equation near hopf bifurcation the benjamin feir instability may occur in this case irregular traveling waves are found but a regular component of the wave pattern survives depending on a parameter the differential flow either promotes or deters the benjamin feir instability as a result the increasing flow may swithc the periodic wave pattern into a irregular state or conversely may stabilize the previously induced irregular pattern and produce periodic waves

secondary structures in a one dimensional complex ginzburg landau equation with homogeneous boundary conditions experiments in extended systems such as the counter rotating couette taylor flow or the taylor dean flow system have shown that patterns with vanishing amplitude may exhibit periodic spatio temporal defects for some range of control parameters these observations could not be interpreted by the complex ginzburg landau equation cgle with periodic boundary conditions we have investigated the one dimensional cgle with homogeneous boundary conditions we found that in the benjamin feir stable region the basic wave train bifurcates to state with periodic spatio temporal defects the numerical results match the observations quite well we have built a new state diagram in the parameter plane spanned by the criticality or equivalently the linear group velocity and the nonlinear frequency detuning

stability limits of traveling waves and the transition to spatiotemporal chaos in the complex ginzburg landau equation abstract due to the convective nature of the eckhaus instability in oscillatory media there exists a conventively unstable wavenumber band in the benjamin feir unstable region this appears to explain at least qualitatively some features of spatiotemporal chaos found recently in the ginzburg landau theory

control of an elliptic problem with pointwise state constraints this paper deals with a quadratic control problem for elliptic equations with pointwise state constraints existence and uniqueness of the solution is proved optimality conditions are given and regularity of the optimal solution is investigated

dynamics of defects and traveling waves in an interfacial finger pattern we present the results of an experimental study of one dimensional traveling finger patterns dominated by source and sink defects the system studied is a driven fluid air interface and the traveling wave state arises via a subcritical bifurcation we analyze space time images of the patterns to obtain the amplitude and local wave number of the left and right going traveling waves near the defects we determine the width of the defects and find that the width of the sinks does not vary significantly with distance above the bifurcation over the range of our experiments while the source width increases as the bifurcation is approached from above we also observe both localized depressions or enhancements in wave amplitude and periodic modulations of the pattern amplitude and wave number

primary and secondary noninear global instability we study oscillating solutions of the complex ginzburg landau equation in a semi infinite domain asymptotic to a saturated traveling wave at c1 and subject to a homogeneous upstream boundary condition at x d 0 this inlet condition breaks the galilean invariance giving the advection velocity the role of a control parameter we give a criterion for the existence of a nontrivial solution called a nonlinear global mode or a self sustained resonance and we obtain the selected frequency the threshold and the frequency are shown to be determined by the linear absolute instability transition we undertake a singular perturbation analysis which first proves the validity of the criterion and secondly yields scaling laws for the frequency the growth length and the slope at the origin of the nonlinear global modes as functions of the criticality parameter comparisons with direct numerical simulation of the ginzburg landau model validate these predictions furthermore the results are in excellent agreement with numerical simulation of the taylor couette problem with throughflow by buchel et al p buchel m lucke d roth r schmitz pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem taylor vortices in axial flow phys rev e 53 1996 4764 4777 and of rayleigh benard flow with an added poiseuille flow by muller et al h w muller m lucke m kamps convective patterns in horizontal flow europhys lett 10 1989 451 456 h w muller m lucke m kamps transversal convection patterns in horizontal shear flow phys rev a 45 1992 3714 3726 the numerical simulations indicate that the nonlinear global modes are stable if the saturated traveling wave to which the nonlinear global mode is asymptotic i e the asymptotic downstream part of the global modes is not absolutely unstable to perturbations a complete analysis of the dispersion relation of the secondary instability of saturated traveling waves is given 1999 elsevier science b v all rights reserved

an interior point method in function space for the efficient solution of state constrained optimal control problems we propose and analyze an interior point path following method in function space for state constrained optimal control our emphasis is on proving convergence in function space and on constructing a practical path following algorithm in particular the introduction of a pointwise damping step leads to a very efficient method as verified by numerical experiments

synchronized disorder in a 2d complex ginzburg landau equation the phenomenon of spiral pair synchronization by oscillating strips nozaki bekki solution in a 2d cgle is investigated analytically the equations describing the interaction of the strips with one another and with spirals are derived analysis of the equations shows that under certain conditions the strips lead to frequency and phase locking of the spirals in this case the spiral pair dipole is aligned parallel to the strips with the position along the strips being arbitrary thus the interaction with strips may transform the spatio temporal chaos of spirals to the regime of periodically oscillating spatial disorder the dynamics of circular strips is investigated and their lifetime is estimated the behavior of the spirals bounded by circular stript is analysed

an admm numerical approach to linear parabolic state constrained optimal control problems optimal control problems arising from systems modeled by linear parabolic equations may be difficult for both theoretical analysis and algorithmic design for the case where there are additional constraints on the state variables restrictive regularity assumptions are usually required to guarantee the existence of the associated lagrange multiplier and thus some regularization type methods such as the moreau yosida and lavrentiev methods have been discussed in the literature in this article we study the application of the alternating direction method of multipliers admm to linear parabolic state constrained optimal control problems and propose an admm numerical approach we prove the convergence of the admm algorithm without any existence or regularity assumption on the lagrange multiplier and estimate its worst case convergence rate in both the ergodic and nonergodic senses an important feature of the admm approach is that it decouples the state constraints and the parabolic optimal control problems inside each iteration we show the efficiency of the admm approach by testing some control problems in two space dimensions

spatiotemporal chaos in the one dimensional complex ginzburg landau equation the dynamical behavior of a large one dimensional system obeying the cubic complex ginzburg landau equation is studied numerically as a function of parameters near a supercritical bifurcation two types of chaotic behavior can be distinguished beyond the benjamin feir instability a phase turbulence regime with a conserved phase winding number and no phase dislocations space time defects and a defect regime with a nonzero density of defects the transition between the two can either be continuous or discontinuous hysteretic depending on parameters the spatial decay of the phase correlation function is inferred to be exponential in both regimes with a sharp decrease of the correlation length upon entering the defect phase the temporal decay of correlations is exponential in the defect regime

nonlinear analysis of the eckhaus instability modulated amplitude waves and phase chaos with nonzero average phase gradient abstract we analyze the eckhaus instability of plane waves in the one dimensional complex ginzburg landau equation cgle and describe the nonlinear effects arising in the eckhaus unstable regime modulated amplitude waves maws are quasi periodic solutions of the cgle that emerge near the eckhaus instability of plane waves and cease to exist due to saddle node sn bifurcations these maws can be characterized by their average phase gradient and by the spatial period p of the periodic amplitude modulation a numerical bifurcation analysis reveals the existence and stability properties of maws with arbitrary and p maws are found to be stable for large enough and intermediate values of p for different parameter values they are unstable to splitting and attractive interaction between subsequent extrema of the amplitude defects form from perturbed plane waves for parameter values above the sn of the corresponding maws the break down of phase chaos with average phase gradient 0 wound up phase chaos is thus related to these sns a lower bound for the break down of wound up phase chaos is given by the necessary presence of sns and an upper bound by the absence of the splitting instability of maws

wound up phase turbulence in the complex ginzburg landau equation we consider phase turbulent regimes with nonzero winding number in the one dimensional complex ginzburg landau equation we find that phase turbulent states with winding number larger than a critical one are only transients and decay to states within a range of allowed winding numbers the analogy with the eckhaus instability for nonturbulent waves is stressed the transition from phase to defect turbulence is interpreted as an ergodicity breaking transition that occurs when the range of allowed winding numbers vanishes we explain the states reached at long times in terms of three basic states namely quasiperiodic states frozen turbulence states and riding turbulence states justification and some insight into them are obtained from an analysis of a phase equation for nonzero winding number rigidly moving solutions of this equation which correspond to quasiperiodic and frozen turbulence states are understood in terms of periodic and chaotic solutions of an associated system of ordinary differential equations a short report of some of our results has already been published r montagne et al phys rev lett 77 267 1996

laminarizing effects of dispersion in an active dissipative nonlinear medium abstract the kuramoto sivashinsky equation has become a popular prototype for systems which exhibit spatial temporal chaos we show here that a linear dispersion term h xxx tends to arrest such irregular behavior in favor of spatially periodic cellular structures as is consistent with prior numerical and experimental observations the study includes a normal form analysis and a numerical investigation of periodic and solitary stationary wave solutions of the equation it is shown that by 1 1 the infinite families of stationary periodic waves of the kuramoto sivashinsky equation each ending in a solitary wave have been annihilated successively such that only a lone family of periodic waves consisting of one hump kdv pulses for 3 7 remains as the only periodic stationary wave attractors of the system these periodic waves have much larger domains of attraction than the strange attractors and hence tend to dominate spatial temporal chaos in an extended domain with significant dispersion