boundary effects on localized structures in spatially extended systems abstract we present a general method of analyzing the influence of finite size and boundary effects on the dynamics of localized solutions of non linear spatially extended systems the dynamics of localized structures in infinite systems involve solvability conditions that require projection onto a goldstone mode our method works by extending the solvability conditions to finite sized systems by incorporating the finite sized modifications of the goldstone mode and associated nonzero eigenvalue we apply this method to the special case of non equilibrium domain walls under the influence of dirichlet boundary conditions in a parametrically forced complex ginzburg landau equation where we examine exotic nonuniform domain wall motion due to the influence of boundary conditions

time varying radial basis functions we introduce radial basis functions rbfs whose time varying coefficients determine not only the amplitude and position of each rbf but also their shape the intended use of these time varying rbfs tv rbfs is in the local in time representation of low dimensional approximations of functions that arise in solving spatiotemporal evolution problems in particular for time varying spatially localized solutions with a temporal translation component such as traveling waves modulated pulses or soliton like solutions of evolutionary differential equations this paper is restricted to the one dimensional spatial case we also present an algorithm that places the time varying rbfs tv rbfs over spatiotemporal data that may come from experiments from finely discretized pde simulations or even from multiscale particle based simulations it first approximates the function at a single time instant a temporal snapshot as a sum of rbfs using a novel weighted minimization that causes each rbf to primarily approximate one of the localized parts of the function it then extends that approximation to tv rbfs over a sequence of snapshots of the function at different times we conclude by discussing the potential uses of these tv rbfs

bifurcations of relative periodic orbits in nls gp with a triple well potential abstract the nonlinear schrodinger gross pitaevskii nls gp equation is considered in the presence of three equally spaced potentials the problem is reduced to a finite dimensional hamiltonian system by a galerkin truncation families of oscillatory orbits are sought in the neighborhoods of the system s nine branches of standing wave solutions normal forms are computed in the neighborhood of these branches various hamiltonian hopf and saddle node bifurcations showing how the oscillatory orbits change as a parameter is increased numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and nls gp near the hamiltonian hopf bifurcations and some subtle disagreements near the saddle node bifurcations due to exponentially small terms in the asymptotics

rotating chemical waves in the gray scott model a set of reaction diffusion equations is considered known as the gray scott model defined on a circle and the stability of rotating wave solutions formed via hopf bifurcations that break the circular o 2 symmetry is investigated using a hybrid numerical analytical technique center manifold normal form reductions are performed to analyze symmetry breaking hopf bifurcations degenerate hopf bifurcations and hopf hopf mode interactions it is found that stable rotating waves exist over broad ranges of parameter values and that the bifurcation behavior of this relatively simple model can be quite complex e g two and three frequency motions exist

renormalization group reduction of pulse dynamics in thermally loaded optical parametric oscillators we derive a perturbed parametrically forced nonlinear schrodinger equation to model pulse evolution in an optical parametric oscillator with absorption induced heating we apply both a rigorous renormalization group rg technique and the formal wilsonian renormalization group to obtain a low dimensional system of equations which captures the mutual interaction of pulses as well as their response to the thermally induced potential we compare the methodologies of the two approaches and find that the two reduced systems agree to leading order and compare well to simulations of the full equation predicting the formation of stably bound pulse pairs

soliton train dynamics in a weakly nonlocal non kerr nonlinear medium we analyze chainlike n soliton dynamics in a weakly nonlocal essentially nonintegrable system described by the cubic quintic nonlinear schrodinger equation quintic nonlinearity is not assumed to be small this system is reduced to a generalized complex toda chain model numerical simulations demonstrate adverse action of both cubic and quintic nonlocal responses in their own right on the quasi equidistant train propagation with a development of a chaotic regime from the toda chain model we predict a possibility of mutually compensating both types of nonlocality induced distortion restoring thereby a deterministic mode of the train propagation in a weakly nonlocal medium analytical predictions corroborate well with numerical results

the immersed boundary lattice boltzmann method for solving fluid particles interaction problems a new computational method the immersed boundary lattice boltzmann method is presented this method is a combination and utilizes the most desirable features of the lattice boltzmann and the immersed boundary methods the method uses a regular eulerian grid for the flow domain and a lagrangian grid to follow particles that are contained in the flow field the rigid body conditions for the fluid and the particles are enforced by a penalty method which assumes that the particle boundary is deformable with a high stiffness constant the velocity field of the fluid and particles is solved by adding a force density term into the lattice boltzmann equation this novel method preserves the advantages of lbm in tracking a group of particles and at the same time provides an alternative and better approach to treating the solid fluid boundary conditions the method also solves the problems of fluctuation of the forces and velocities on the particles when the bounce back boundary conditions are applied this method enables one to simulate problems with particle deformation and fluid structure deformation its results are validated by comparison with results from other methods

force evaluation in the lattice boltzmann method involving curved geometry the present work investigates two approaches for force evaluation in the lattice boltzmann equation the momentum exchange method and the stress integration method on the surface of a body the boundary condition for the particle distribution functions on curved geometries is handled with second order accuracy based on our recent works mei et al j comput phys 155 307 1999 ibid 161 680 2000 the stress integration method is computationally laborious for two dimensional flows and in general difficult to implement for three dimensional flows while the momentum exchange method is reliable accurate and easy to implement for both two dimensional and three dimensional flows several test cases are selected to evaluate the present methods including i two dimensional pressure driven channel flow ii two dimensional uniform flow past a column of cylinders iii two dimensional flow past a cylinder asymmetrically placed in a channel with vortex shedding iv three dimensional pressure driven flow in a circular pipe and v three dimensional flow past a sphere the drag evaluated by using the momentum exchange method agrees well with the exact or other published results

weakly interacting pulses in synaptically coupled neural media we use singular perturbation theory to analyze the dynamics of n weakly interacting pulses in a one dimensional synaptically coupled neuronal network the network is modeled in terms of a nonlocal integro differential equation in which the integral kernel represents the spatial distribution of synaptic weights and the output activity of a neuron is taken to be a mean firing rate we derive a set of n coupled ordinary differential equations odes for the dynamics of individual pulses establishing a direct relationship between the explicit form of the pulse interactions and the structure of the long range synaptic coupling the system of odes is used to explore the existence and stability of stationary n pulses and traveling wave trains

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

domain structures in fourth order phase and ginzburg landau equations abstract in pattern forming systems competition between patterns with different wave numbers can lead to domain structures which consist of regions with differing wave numbers separated by domain walls for domain structures well above threshold we employ the appropriate phase equation and obtain detailed qualitative agreement with recent experiments close to threshold a fourth order ginzburg landau equation is used which describes a steady bifurcation in systems with two competing critical wave numbers the existence and stability regime of domain structures is found to be very intricate due to interactions with other modes in contrast to the phase equation the ginzburg landau equation allows a spatially oscillatory interaction of the domain walls thus close to threshold domain structures need not undergo the coarsening dynamics found in the phase equation far above threshold and can be stable even without phase conservation we study their regime of stability as a function of their quantized length domain structures are related to zig zags in two dimensional systems the latter are therefore expected to be stable only when quenched far enough beyond the zig zag instability

drift bifurcation of dissipative solitons due to a change of shape experiment and theory abstract the subject of this paper is self organized solitary current filaments being observed experimentally in a planar dc gas discharge system with high ohmic barrier we report on the dynamical behavior of these objects which we refer to as dissipative solitons an application of stochastic data analysis methods enables the separation of intrinsic dynamics from stochastic contributions using the externally applied voltage as control parameter we observe a drift bifurcation that is accompanied by a change of shape of the bifurcating dissipative soliton in order to interpret the experimental finding a theoretical model in form of a three component reaction diffusion system is considered allowing for a qualitative understanding of the obtained results we discuss an analytical expression for the velocity of the dissipative soliton as a function of an appropriate control parameter and demonstrate that the change of the shape can induce the drift bifurcation this analytical finding is supported by numerical simulations of the model equations

existence and stability of spatially localized patterns abstract spatially localized patterns have been observed in numerous physical contexts and their bifurcation diagrams often exhibit similar snaking behavior symmetric solution branches connected by bifurcating asymmetric solution branches wind back and forth in an appropriate parameter previous papers have addressed existence of such solutions here we address their stability taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions we then show that under appropriate assumptions temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane in a companion paper we analyze the behavior of eigenvalues at 0 and inside the essential spectrum our results show that localized snaking solutions are stable if and only if the underlying fronts and backs are stable unlike localized non oscillatory solutions no interaction eigenvalues are present we use the planar swift hohenberg system to illustrate our results

patterns of propagating pulses the complex dynamics that arise in certain nonlinear partial differential equations in time and in one space dimension are studied in the general case considered the equation admits a solitary wave in the form of a pulse tailing off exponentially fore and aft with possibly oscillatory character complicated solutions are described by a superposition of many such solitary structures in interaction the description is asymptotic in terms of a parameter that becomes exponentially small as the ratio of typical pulse separation to pulse width becomes large the outcome is a set of dynamical equations for the motion of the individual pulses with nearest neighbor interactions this system of ordinary differential equations odes admits a wide range of patterns both regular and chaotic the stability theory of such patterns is sketched and the continuum limit of the lattice dynamical equations of the pulses is given

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

the accumulation of eigenvalues in a stability problem abstract when two waves propagating in a one dimensional medium are locked together as a composite wave a natural question arises as to whether the new wave is stable an interesting and novel instability mechanism is exposed here in which a cascade of eigenvalues accumulates at a distinguished point in the unstable half plane the underlying assumption is that the transition between the two waves occurs at an unstable homogeneous steady state of the partial differential equations this causes the individual waves to have an unstable continuous spectrum but the instability of the full wave cannot be predicted from the configuration of these spectra alone

the hodge conjecture for self products of certain k3 surfaces abstract we use a result of van geemen 2008 vg4 to determine the endomorphism algebra of the kuga satake variety of a k3 surface with real multiplication this is applied to prove the hodge conjecture for self products of double covers of p 2 which are ramified along six lines

a perturbation theory for soliton systems abstract a simplified perturbational approach appropriate for systems of solitons governed by the perturbed integrable equations is described some applications of this method are reviewed among them there are soliton structure of the shock waves in dispersive media double sine gordon equation etc

cascades of reversible homoclinic orbits to a saddle focus equilibrium abstract in a reversible system we consider a homoclinic orbit being bi asymptotic to a saddle focus equilibrium as was proved by devaney 1977 there exists a one parameter family of periodic orbits accumulating onto this homoclinic orbit in the present paper we show that for any n 2 there exist infinitely many n homoclinic orbits in a neighborhood of the primary homoclinic orbit each of them is accompanied by one or more families of periodic orbits moreover we indicate how these families of periodic orbits correspond to branches of subharmonic periodic orbits

effect of quintic nonlinearity on soliton collisions in optical fibers abstract we investigate the effect of quintic nonlinearity on collisions between solitons in different frequency channels in optical fibers we calculate the intensity and dynamics of the collision induced emitted radiation and changes in the soliton parameters due to the collision this is done by means of a perturbation theory with two small parameters the quintic nonlinearity coefficient q and the reciprocal of the inter channel frequency difference 1 we find that the amplitude of the collision induced emitted radiation is proportional to q this radiation can be described as originating from a variation in the kerr nonlinearity coefficient with respect to distance along the fiber we also find that the only other effect of quintic nonlinearity on the collision up to second order in the perturbation theory is an additional phase shift proportional to q the results obtained by our numerical simulations of the perturbed nonlinear schrodinger equation are in very good agreement with theory

integrability of an n coupled nonlinear schrodinger system for polarized optical waves in an isotropic medium via symbolic computation considering the simultaneous propagation of multicomponent fields in an isotropic medium an n coupled nonlinear schr odinger system with the self phase modulation cross phase modulation and energy exchange terms is investigated in this paper first via symbolic computation the painlev e singularity structure analysis shows that such a system admits the painlev e property then with the ablowitz kaup newell segur scheme the linear eigenvalue problem lax pair associated with this model is constructed in the frame of the block matrices with the hirota bilinear method the bright one and two soliton solutions of this system are presented in addition the bright multisoliton solutions of the system for n 2 are straightforwardly derived by the linear superposition of soliton solutions of two independent scalar nonlinear schr odinger equations furthermore through the analysis for the soliton solutions the corresponding propagation behavior and applications for soliton pulses in nonlinear optical fibers are considered finally three significant conserved quantities i e energy momentum and hamiltonian are also given

existence of traveling wave trains in nerve axon equations we establish the existence of two discrete classes of traveling wave solutions to nerve axon equations under conditions on the linearization of the system about the rest state each solution in one class consists of a finite train of pulses while each solution in the other class consists of an infinite train that arises from the rest state in the leading direction the solutions are distinguished by the spacing of the pulses in the train and indexed by n tuples and sequences of integers respectively in the two classes this result is obtained from study of the resulting parametrized dynamical system which has a a single homoclinic orbit for one parameter value these assumptions were previously used to show the existence of the double pulse solutions and are known to hold in for some nerve models

the role of curvature and wavefront interactions in spiral wave dynamics abstract a theory of rotating spiral waves in excitable media is presented that allows the study of dynamical aspects such as nonsteady rotation an approximate spiral wave solution is proposed in the form of a superposition of curved solitary wavefronts parallel to each other this form is used by means of singular perturbation theory to derive an evolution equation for the spiral arm numerical solutions of that equation are found that describe one and two frequency rotations the transition to two frequency dynamics tip meandering is attributed to a destabilizing curvature effect

numerical bifurcation and stability analysis of solitary pulses in an excitable reaction diffusion medium abstract we present a systematic computer assisted study of the bifurcations and instabilities of solitary pulses in an excitable medium capable of displaying both stable pulse propagation and spatiotemporally chaotic dynamics over intervals of parameter space the reaction diffusion model used is of the activator inhibitor type only the activator diffuses in this medium the control parameters are the ratio of time scales of the activator and inhibitor dynamics and the excitation threshold this study focuses on travelling pulses their domain of existence and the bifurcations that render them unstable these pulses are approximated as a homoclinic orbits in a travelling wave ode frame and b as solutions of the full partial differential equation pde with periodic boundary conditions in large domains a variety of bifurcations in the travelling wave ode frame are observed including heteroclinic loops so called t points a r champneys and y a kuznetsov numerical detection and continuation of codimension 2 homoclinic bifurcations int j bif chaos 4 1994 785 h kokobu homoclinic and heteroclinic bifurcations of vectorfields japan j appl math 5 1988 455 instabilities in the full pde frame include both hopf bifurcations to modulated travelling waves involving the discrete pulse spectrum as well as transitions involving the continuous spectrum such as the so called backfiring transition m bar m hildebrand m eiswirth m falcke h engel and m neufeld chemical turbulence and standing waves in a surface reaction model the influence of global coupling and wave instabilities chaos 4 1994 499 the stability of modulated pulses is computed through numerical floquet analysis and a cascade of period doubling bifurcations is observed as well as certain global bifurcations these results corroborated by observations from direct numerical integration provide a skeleton around which many features of the overall complex spatiotemporal dynamics of the pde are organized

discrete lattice effects on the forcing term in the lattice boltzmann method we show that discrete lattice effects must be considered in the introduction of a force into the lattice boltzmann equation a representation of the forcing term is then proposed with the representation the navier stokes equation is derived from the lattice boltzmann equation through the chapman enskog expansion several other existing force treatments are also examined

transient behavior of collapsing ring solutions in the critical nonlinear schrodinger equation abstract the critical nonlinear schrodinger equation nls possesses nearly self similar ring profile solutions we address the question of whether this profile is maintained all the way until the point of singularity a perturbative analysis of the rescaled pde and the resulting self similar profile uncover slow dynamics that eventually drive the ring structure to the classical peak shaped collapse instead a numerical scheme capable of resolving self similar behavior to high resolutions confirms our analysis we also consider ring type blowup arising either from azimuthally polarized solutions of a coupled nls system or as vortex solutions of the usual nls in this case the ring profile is maintained up to the time of singularity

oscillations of fronts and front pairs in two and three component reaction diffusion systems abstract we investigate multi co ponent reaction diffusion systems on a finite one dimensional domain the first component is assumed both to react on a short time scale and to have a small diffusion length leading to patterns with sharp fronts to analyze stability we apply the slep method by nishiura and fujii this method is superior to standard perturbation methods we illustrate this using a two component system in the three component system in a situation where both the influence of the first component on the other components and the interaction of the latter is only weak we show that there is a unique hopf destabilization of stationary fronts when changing time constants as an example we treat a system with global coupling used e g for semi conductor devices for filaments front pairs we find two types of hopf destabilizations breathing and swinging the global coupling controls which type occurs this corresponds to recent experimental results and is confirmed via numerical calculations

a universal separatrix map for weak interactions of solitary waves in generalized nonlinear schrodinger equations abstract it is known that weak interactions of two solitary waves in generalized nonlinear schrodinger nls equations exhibit fractal dependence on initial conditions and the dynamics of these interactions is governed by a universal two degree of freedom ode system y zhu j yang universal fractal structures in the weak interaction of solitary waves in generalized nonlinear schrodinger equations phys rev e 75 2007 036605 in this paper this ode system is analyzed comprehensively using asymptotic methods along separatrix orbits a simple second order map is derived this map does not have any free parameters after variable rescalings and thus is universal for all weak interactions of solitary waves in generalized nls equations comparison between this map s predictions and direct simulations of the ode system shows that the map can capture the fractal scattering phenomenon of the ode system very well both qualitatively and quantitatively

local stability theory of solitary pulses in an active medium abstract if the spatio temporal chaos of an extended active medium is driven by repulsive or attractive interaction among localized coherent structures such as kinks and pulses its dynamics can often be described by a low dimensional dynamical system this welcomed reduction in the degrees of freedom only occurs when the localized structures remain stable during their mutual interaction it is shown that the structures can be destroyed by two types of local disturbance an expanding radiation packet and a localized disturbance which travels with the structure these two instabilities and the bifurcations they induce select the proper coherent structure and specify the dynamics of the generalized kuramoto sivashinsky gks equation in an extended domain the possibility of describing pulse radiation interaction by resonance poles also suggests that a more general coherent structure theory can be developed

existence and stability of n pulses on optical fibers with phase sensitive amplifiers abstract the propagation of pulses in optical communication systems in which attenuation is compensated by phase sensitive amplifiers is investigated a central issue is whether optical fibers are capable of carrying several pieces of information at the same time in this paper multiple pulses are shown to exist for a fourth order nonlinear diffusion model due to kutz and co workers 1994 moreover criteria are derived for determining which of these pulses are stable the pulses arise in a reversible orbit flip a homoclinic bifurcation investigated here for the first time numerical simulations are used to study multiple pulses far away from the actual bifurcation point they confirm that properties of the multiple pulses including their stability are surprisingly well predicted by the analysis carried out near the bifurcation

breakup of a multisoliton state of the linearly damped nonlinear schrodinger equation we address the breakup splitting of multisoliton solutions of the nonlinear schrodinger equation nlse occurring due to linear loss two different approaches are used for the study of the splitting process the first one is based on the direct numerical solution of the linearly damped nlse and the subsequent analysis of the eigenvalue drift for the associated zakharov shabat spectral problem the second one involves the multisoliton adiabatic perturbation theory applied for studying the evolution of the solution parameters with the linear loss taken as a small perturbation we demonstrate that in the case of strong nonadiabatic loss the evolution of the zakharov shabat eigenvalues can be quite nontrivial we also demonstrate that the multisoliton breakup can be correctly described within the framework of the adiabatic perturbation theory and can take place even due to small linear loss eventually we elucidate the occurrence of the splitting and its dependence on the phase mismatch between the solitons forming a two soliton bound state

lattice boltzmann model for dense suspended particles based on improved bounce back method abstract we develop an efficient and accurate lattice boltzmann model for dense suspended particles based on improved bounce back it is difficult to model flow with dense suspended particles because there are not enough fluid nodes surrounding each particle to apply many existing algorithms and mass leakage due to interpolated bounce back is more significant when dealing with many particles we propose a new curved boundary treatment and fresh node initialization scheme both of which are still effective when two particles get close to contact we further propose a method to calculate the hydrodynamic interaction accurately when two solid particles get very close to each other without leaving any fluid node between their surfaces an easy redistributing method is developed to fix mass leakage resulting from curved boundaries to make the algorithm more stable we combine implicit particle velocity updating and galilean invariant momentum exchange we validate the second order accuracy of this curved boundary method by channel flow and other simulations are carried out to show that this model is more accurate and stable in dealing with particle particle interaction finally simulations of magnetic particle flows are conducted to show that this model is suitable for dense suspended particles

nonlinear waves and solitons in physical systems abstract advances in nonlinear science have been plentiful in recent years in particular interest in nonlinear wave propagation continues to grow stimulated by new applications such as fiber optic communication systems as well as the many classical unresolved issues of fluid dynamics what is arguably the turning point for the modern perspective of nonlinear systems took place at los alamos over 40 years ago with the pioneering numerical simulations of fermi pasta and ulam a decade later this research initiated the next major advance of zabusky and kruskal that motivated the revolution in completely integrable systems with this in mind the conference on nonlinear waves in solitions in physical systems nwsps was organized by the center for nonlinear studies cnls at los alamos national laboratory in may of 1997 to assess the current state of the art in this very active field papers from the conference attendees as well as from researchers unable to attend the conference were collected in this special volume of physica d in this paper the contributions to the conference and to this special issue are reviewed with an emphasis on the many unifying principles that all these works share

on the exponentially slow motion of a viscous shock using formal asymptotic methods we study the internal layer behavior associated with the following viscous shock problem in the limit e 0 the convex nonlinearity f u satisfies f f for the steady problem we show that the method of matched asymptotic expansions fails to uniquely determine the location of the equilibrium shock layer solution this indeterminacy resulting from neglecting certain exponentially small effects is eliminated by using the projection method which exploits certain properties of the spectrum associated with the linearized operator for the time dependent problem we show that the viscous shock which is formed from initial data drifts towards the equilibrium solution on an exponentially long time interval of the order o ec e for some c 0 this exponentially slow behavior is analyzed by deriving an equation of motion for the location of the viscous shock for burgers equation f u u2 2 the results give an analytical characterization of the slow shock layer motion observed numerically in kreiss and kreiss see 11 we also show that the shock layer behavior is very sensitive to small changes in the boundary operator in addition using a wkb type method the slow viscous shock motion is studied numerically for small e the results comparing favorably with corresponding analytical results finally we relate the slow viscous shock motion to similar slow internal layer motion for the allen cahn equation

on nonlinear boundary value problems ghosts parasites and discretizations ghosts are a new class of approximate solutions to nonlinear boundary value problems bvps they are bounded away from true solutions but solve an initial value problem ivp for the differential equations exactly and approximate boundary and internal matching conditions with exponentially decaying error as a control parameter approaches infinity ghosts are characteristic of singularly perturbed differential equations in which solutions are localized and the ivp depends sensitively on initial data they correspond to shifted localized solutions and they can be observed in imperfect physical experiments and be approximated by convergent discretization schemes the latter property suggests that the concept of the numerically irrelevant solution introduced for discretized second order bvps should be re examined

information transmission using the nonlinear fourier transform part iii spectrum modulation motivated by the looming capacity crunch in fiber optic networks information transmission over such systems is revisited among numerous distortions interchannel interference in multiuser wavelength division multiplexing wdm is identified as the seemingly intractable factor limiting the achievable rate at high launch power however this distortion and similar ones arising from nonlinearity are primarily due to the use of methods suited for linear systems namely wdm and linear pulse train transmission for the nonlinear optical channel exploiting the integrability of the nonlinear schrodinger nls equation a nonlinear frequency division multiplexing nfdm scheme is presented which directly modulates noninteracting signal degrees of freedom under nls propagation the main distinction between this and previous methods is that nfdm is able to cope with the nonlinearity and thus as the signal power or transmission distance is increased the new method does not suffer from the deterministic crosstalk between signal components which has degraded the performance of previous approaches in this paper emphasis is placed on modulation of the discrete component of the nonlinear fourier transform of the signal and some simple examples of achievable spectral efficiencies are provided

stability of discrete solitons in nonlinear schrodinger lattices we consider the discrete solitons bifurcating from the anti continuum limit of the discrete nonlinear schrodinger nls lattice the discrete soliton in the anti continuum limit represents an arbitrary finite superposition of in phase or anti phase excited nodes separated by an arbitrary sequence of empty nodes by using stability analysis we prove that the discrete solitons are all unstable near the anti continuum limit except for the solitons which consist of alternating anti phase excited nodes we classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons

collisions of optical ultra short vector pulses asymptotic two pulse solutions of a modified via attaching a nonlinear derivative term vector nonlinear schrodinger equation mvnse are analyzed using the hirota bilinearization method this equation is known to describe the propagation of ultra short optical pulses in nonlinear fibers in particular in the optical crystal fibers some pulse parameter regimes of applicability of the solutions to study the pulse collisions are estimated it is found for these regimes that the collision of ultra short vector pulses of almost equal velocities results in a transformation of their polarizations similar as that of the manakov solitons whenever the pulse interaction effects can be neglected

a direct approach to studying soliton perturbations starting with an integrable nonlinear evolution equation the author investigates perturbations about a one soliton solution through the inversion of a linear equation for the first order correction to the soliton solution this inversion differs from past methods as the proposed method takes place in coordinate space not spectral space while it employs some of the tools of inverse scattering theory the method is applied to the korteweg de vries nonlinear schrodinger and sine gordon equations the first order corrections are then obtained

turning points and traveling waves in fitzhugh nagumo type equations abstract consider the following fitzhugh nagumo type equation u t u x x f u w w t g u w where f u w u u a w 1 u for some smooth function a w and g u w u w by allowing a w to cross zero and one the corresponding traveling wave equation possesses special turning points which result in very rich dynamics in this work we examine the existence of fronts backs and pulses solutions in particular the co existence of different fronts will be discussed

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

multi hump pulses in systems with reflection and phase invariance abstract we investigate the existence and stability of standing and travelling multi hump waves in partial differential equations with reflection and phase symmetries we focus on 2 and 3 pulse solutions that arise near bi foci and apply our results to the complex cubic quintic ginzburg landau equation

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

universal fractal structures in the weak interaction of solitary waves in generalized nonlinear schrodinger equations weak interactions of solitary waves in the generalized nonlinear schr odinger equations are studied it is first shown that these interactions exhibit similar fractal dependence on initial conditions for different nonlinearities then by using the karpman solov ev method a universal system of dynamical equations is derived for the velocities amplitudes positions and phases of interacting solitary waves these dynamical equations contain a single parameter which accounts for the different forms of nonlinearity when this parameter is zero these dynamical equations are integrable and the exact analytical solutions are derived when this parameter is nonzero the dynamical equations exhibit fractal structures which match those in the original wave equations both qualitatively and quantitatively thus the universal nature of fractal structures in the weak interaction of solitary waves is analytically established the origin of these fractal structures is also explored it is shown that these structures bifurcate from the initial conditions where the solutions of the integrable dynamical equations develop finite time singularities based on this observation an analytical criterion for the existence and locations of fractal structures is obtained lastly these analytical results are applied to the generalized nonlinear schr odinger equations with various nonlinearities such as the saturable nonlinearity and predictions on their weak interactions of solitary waves are made

regular article an accurate curved boundary treatment in the lattice boltzmann method the lattice boltzmann equation lbe is an alternative kinetic method capable of solving hydrodynamics for various systems major advantages of the method are due to the fact that the solution for the particle distribution functions is explicit easy to implement and natural to parallelize because the method often uses uniform regular cartesian lattices in space curved boundaries are often approximated by a series of stairs that leads to reduction in computational accuracy in this work a second order accurate treatment of the boundary condition in the lbe method is developed for a curved boundary the proposed treatment of the curved boundaries is an improvement of a scheme due to o filippova and d hanel 1998 j comput phys 147 219 the proposed treatment for curved boundaries is tested against several flow problems 2 d channel flows with constant and oscillating pressure gradients for which analytic solutions are known flow due to an impulsively started wall lid driven square cavity flow and uniform flow over a column of circular cylinders the second order accuracy is observed with a solid boundary arbitrarily placed between lattice nodes the proposed boundary condition has well behaved stability characteristics when the relaxation time is close to 1 2 the zero limit of viscosity the improvement can make a substantial contribution toward simulating practical fluid flow problems using the lattice boltzmann method

nonlinearity saturation as a singular perturbation of the nonlinear schroźdinger equation saturation of a kerr type nonlinearity in the nonlinear schro dinger equation nlse can be regarded as a singular perturbation which regularizes the well known blow up phenomenon in the cubic nlse an asymptotic expansion is proposed which takes into account multiple scale behavior both in the longitudinal and transverse directions in one dimension this leads to a free boundary problem reduction where the solitary wave acts solely to reflect impinging waves and is accelerated by an elastic transfer of momentum in two dimensions we find that interaction of a solitary wave and an adjacent wave field is governed by behavior of certain eigenfunctions of the linearized fast scale operator this leads to an outer solution with a free logarithmic singularity whose position evolves by virtue of a large transfer of momentum between the ambient and solitary waves however for a certain value of wave power we find there is essentially no interaction and the solitary wave is asymptotically transparent to the ambient field we test our results by numerical simulation of both the full equation and free boundary reductions

interpolated boundary condition for lattice boltzmann simulations of flows in narrow gaps several different interpolation schemes have been proposed for improving the accuracy of lattice boltzmann simulations in the vicinity of a solid boundary however these methods require at least two or three fluid nodes between nearby solid surfaces a condition that may not be fulfilled in dense suspensions or porous media for example here we propose an interpolation of the equilibrium distribution which leads to a velocity field that is both second order accurate in space and independent of viscosity the equilibrium interpolation rule infers population densities on the boundary itself to reduce the span of nodes needed for interpolation it requires a minimum of one grid spacing between the nodes by contrast the linear interpolation rule requires two fluid nodes in the gap and leads to a viscosity dependent slip velocity while the multireflection rule is viscosity independent but requires a minimum of three fluid nodes

lattice approach to the dynamics of phase coded soliton trains we address the collective dynamics of a soliton train propagating in a medium described by the nonlinear schrodinger equation our approach uses the reduction of train dynamics to the discrete complex toda chain ctc model for the evolution of parameters for each train constituent such a simplification allows one to carry out an approximate analysis of the dynamics of positions and phases of individual interacting pulses here we employ the ctc model to the problem which has relevance to the field of fibre optics communications where each binary digit of transmitted information is encoded via the phase difference between the two adjacent solitons our goal is to elucidate different scenarios of the train distortions and the subsequent information garbling caused solely by the intersoliton interactions first we examine how the structure of a given phase pattern affects the initial stage of the train dynamics and explain the general mechanisms for the appearance of unstable collective soliton modes then we further discuss the nonlinear regime concentrating on the dependence of the lax scattering matrix on the input phase distribution this allows one to classify typical features of the train evolution and determine the distance where the soliton escapes from its slot in both cases we demonstrate deep mathematical analogies with the classical theory of crystal lattice dynamics

stable patterns for fourth order parabolic equations we consider fourth order parabolic equations of gradient type for the sake of simplicity the analysis is carried out for the specific equation u

multireflection boundary conditions for lattice boltzmann models we present a general framework for several previously introduced boundary conditions for lattice boltzmann models such as the bounce back rule and the linear and quadratic interpolations the objectives are twofold first to give theoretical tools to study the existing link type boundary conditions and their corresponding accuracy second to design boundary conditions for general flows which are third order kinetic accurate using these new boundary conditions couette and poiseuille flows are exact solutions of the lattice boltzmann models for a reynolds number mathrm re 0 stokes limit for arbitrary inclination with the lattice directions numerical comparisons are given for stokes flows in periodic arrays of spheres and cylinders linear periodic array of cylinders between moving plates and for navier stokes flows in periodic arrays of cylinders for mathrm re l200 these results show a significant improvement of the overall accuracy when using the linear interpolations instead of the bounce back reflection up to an order of magnitude on the hydrodynamics fields further improvement is achieved with the new multireflection boundary conditions reaching a level of accuracy close to the quasianalytical reference solutions even for rather modest grid resolutions and few points in the narrowest channels more important the pressure and velocity fields in the vicinity of the obstacles are much smoother with multireflection than with the other boundary conditions finally the good stability of these schemes is highlighted by some simulations of moving obstacles a cylinder between flat walls and a sphere in a cylinder

nonlinear stability of gardner breathers abstract we show that breather solutions of the gardner equation a natural generalization of the kdv and mkdv equations are h 2 r stable through a variational approach we characterize gardner breathers as minimizers of a new lyapunov functional and we study the associated spectral problem through i the analysis of the spectrum of explicit linear systems spectral stability and i i controlling degenerated directions by using low regularity conservation laws

the semistrong limit of multipulse interaction in a thermally driven optical system abstract we consider the semistrong limit of pulse interaction in a thermally driven parametrically forced nonlinear schrodinger tdnls system modeling pulse interaction in an optical cavity the tdnls couples a parabolic equation to a hyperbolic system and in the semistrong scaling we construct pulse solutions which experience both short range tail tail interactions and long range thermal coupling we extend the renormalization group rg methods used to derive semistrong interaction laws in reaction diffusion systems to the hyperbolic parabolic setting of the tdnls system a key step is to capture the singularly perturbed structure of the semigroup through the control of the commutator of the resolvent and a re scaling operator the rg approach reduces the pulse dynamics to a closed system of ordinary differential equations for the pulse locations

frozen spatial chaos induced by boundaries csic uib e 07071 palma de mallorca spain received 4 june 1999 we show that rather simple but nontrivial boundary conditions could induce the appearance of spatial chaos that is stationary stable but spatially disordered con gurations in extended dynamical systems with verysimple dynamics we exemplify the phenomenon with a nonlinear reaction diffusion equation in a two dimensional undulated domain concepts from the theory of dynamical systems and a transverse single modeapproximation are used to describe the spatially chaotic structures s1063 651x 99 01812 7 pacs number s 05 45 2ai introduction

a perturbational approach to the two soliton systems abstract the two soliton systems of the perturbed non linear schrodinger nls and sine gordon sg equations are considered by means of a simple quasi particle approach in which the soliton interaction and the action of an external perturbation are described by the one soliton perturbation theory the method is quite general it does not assume the knowledge of the exact unperturbed multi soliton solutions it is restricted however by the requirement that the distances between the solitons must be large enough the validity of the approach is checked by applying it to some cases where it is possible to make a comparison with the inverse scattering method different types of soliton collisions and bound soliton systems are investigated in particular detailed descriptions of the two soliton systems of the double sg equations are presented

soliton soliton interaction with dual power law nonlinearity abstract the intra channel collision of optical solitons with dual power law nonlinearity is studied in this paper by the aid of quasi particle theory the perturbation terms that are considered in this paper are of hamiltonian type the suppression of soliton soliton interaction in presence of these perturbation terms is acheived the numerical simulations support the quasi particle theory

solitary wave interactions in dispersive equations using manton s approach we generalize the approach first proposed by manton nucl phys b 150 397 1979 to compute solitary wave interactions in translationally invariant dispersive equations that support such localized solutions the approach is illustrated using as examples solitons in the korteweg de vries equation standing waves in the nonlinear schr odinger equation and kinks as well as breathers of the sine gordon equation

stability and evolution of solitary waves in perturbed generalized nonlinear schrodinger equations in this paper we study the stability and evolution of solitary waves in perturbed generalized nonlinear schrodinger nls equations our method is based on the completeness of the bounded eigenstates of the associated linear operator in l2 space and a standard multiple scale perturbation technique unlike the adiabatic perturbation method our method details all instability mechanisms caused by perturbations of such equations and explicitly specifies when such instabilities will occur in particular our method uncovers the instability caused by bifurcation of nonzero discrete eigenvalues of the linearization operator as an example we consider the perturbed cubic quintic nls equation in detail and determine the stability regions of its solitary waves in the instability region we also specify where the solitary waves decay collapse develop movingfronts or evolve into a stable spatially localized and temporally periodic state the generalization of this method to other perturbed nonlinear wave systems is also discussed

lyapunov schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies abstract this article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits in addition tangencies in the intersection of un stable manifolds are allowed the main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time and their parameter values a leading order expansion is given in terms of the time spent near vertices and if applicable the location on the non trivial tangent directions the only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit the essential assumptions are hyperbolicity of the vertices and transversality of parameters using the result conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven finally equilibrium to periodic orbit heteroclinic cycles of various types are considered

persistence and stability of discrete vortices in nonlinear schrodinger lattices abstract we study discrete vortices in the two dimensional nonlinear schrodinger lattice with small coupling between lattice nodes the discrete vortices in the anti continuum limit of zero coupling represent a finite set of excited nodes on a closed discrete contour with a non zero charge using the lyapunov schmidt reductions we analyze continuation and termination of the discrete vortices for small coupling between lattice nodes an example of a square discrete contour is considered that includes the vortex cell also known as the off site vortex we classify families of symmetric and asymmetric discrete vortices that bifurcate from the anti continuum limit we predict analytically and confirm numerically the number of unstable eigenvalues associated with each family of such discrete vortices