spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

homoclinic orbits in the near integrable double discrete sine gordon equation we establish the existence of homoclinic orbits for the near integrable double discrete sine gordon ddsg equation under periodic boundary conditions the hyperbolic structure and homoclinic orbits are constructed through the backlund transformation and lax pair a geometric perturbation method based on mel nikov analysis is used to establish necessary criteria for the persistent of temporally homoclinic orbits for the class of ddsg equations with dissipative perturbations

improved mean square error estimate for the lms transversal equalizer with narrowband interference when the least mean square lms algorithm is used to adapt an adaptive transversal equalizer that is subject to strong narrowband interference a so called non wiener or nonlinear effect takes place this results in the mean square error mse performance of the adaptive equalizer being better than that of the fixed wiener filter of equivalent structure reuter and zeidler proposed a transfer function based approach to provide an estimate of the mse performance of the equalizer in such an environment we have recently shown that the mean of the lms weights in this adaptive equalizer problem shifts away from the wiener filter solution as a result we propose an mse model for the lms equalizer that is an improvement over the existing reuter zeidler model the new model uses the same transfer function based approach but incorporates the shift in the mean of the weights numerical simulations are provided to illustrate the improvement

complex ginzburg landau equations as perturbations of nonlinear schrodinger equations a melnikov approach abstract we study the persistence of quasiperiodic and homoclinic solutions of generalized nonlinear schrodinger equations under ginzburg landau perturbations in this paper the first of a series melnikov criteria for the persistence of quasiperiodic and homoclinic solutions are derived directly from the governing partial differential equations via an averaging technique for families of tori of quasiperiodic solutions such as rotating waves and traveling waves that arise within critical sets of linear combinations of conserved functionals we find that usually only isolated tori will satisfy these selection criteria moreover in some simple cases these criteria are sufficient to conclude that a torus persists we also demonstrate the nonpersistence of solutions that are homoclinic to rotating waves under a broad class of ginzburg landau perturbations which satisfy a convexity condition

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

numerical algorithms for diffusion reaction problems with non classical conditions abstract parabolic equations with nonlocal boundary conditions have been given considerable attention in recent years in this paper new high order algorithms for the linear diffusion reaction problem are derived the convergence of the new schemes is studied and numerical examples are given to show the efficiency of the new methods to solve linear and nonlinear diffusion reaction equations with these non classical conditions

chaotic dynamics for perturbations of infinite dimensional hamiltonian systems where w t z r is the transverse de9ection of the axis of the beam w t 0 w t 1 wzz t 0 wzz t 1 0 is an external load 0 is a ratio indicating the extensional rigidity and is the damping the rst result on the existence of a chaotic dynamics for system h has been given by holmes and marsden in 10 for a speci c periodic forcing perturbation of the type p t w t z f z cos t they use the theory of invariant manifolds and of non linear semigroups in order to extend the classical melnikov approach for planar ordinary di erential equations to system h

homoclinic crossings and pattern selection the correlations between a homoclinic orbit hmo and coherent patterns in the nonlinear schr odinger model are discussed in a hamiltonian situation two independent patterns are revealed to exist one corresponds to a motion within an hmo and the other outside of an hmo the study further illustrates when the hamiltonian constraint is released the significance of an hmo on the patterns dynamics by presenting the irregular hmo crossings and the resulting chaotic selections between the patterns

coupled oscillator models with no scale separation we consider a class of spatially discrete wave equations that describe the motion of a system of linearly coupled oscillators perturbed by a nonlinear potential we show that the dynamical behavior of this system cannot be understood by considering the slowest modes only there is an inverse cascade in which the effects of changes in small scales are felt by the largest scales and the mean field closure does not work despite this a one and a half degree of freedom model is derived that includes the influence of the small scale dynamics and predicts global conformational changes accurately thus we provide a reduced model for a system in which there is no separation of scales we analyze a specific coupled oscillator system that models global conformation change in biomolecules introduced in i mezic on the dynamics of molecular conformation proc natl acad sci 103 20 2006 7542 7547 in this model the conformational states are stable to random perturbations yet global conformation change can be quickly and robustly induced by the action of a targeted control we study the efficiency of small scale perturbations on conformational change and show that zipper traveling wave perturbations provide an efficient means for inducing such change a visualization method for the transport barriers in the reduced model yields insight into the mechanism by which the conformation change occurs

numerical chaos symplectic integrators and exponentially small splitting distances two discrete versions of duffing s equation are derived as reductions of discretizations of the nonlinear schrodinger nls equation inheriting the properties of the nsl discretizations one is shown to be integrable and using a mel nikov type analysis the other discretization is shown to be nonintegrable arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation as h 0 this suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous hamiltonian systems

spectral and modulational stability of periodic wavetrains for the nonlinear klein gordon equation abstract this paper is a detailed and self contained study of the stability properties of periodic traveling wave solutions of the nonlinear klein gordon equation u t t u x x v u 0 where u is a scalar valued function of x and t and the potential v u is of class c 2 and periodic stability is considered both from the point of view of spectral analysis of the linearized problem spectral stability analysis and from the point of view of wave modulation theory the strongly nonlinear theory due to whitham as well as the weakly nonlinear theory of wave packets the aim is to develop and present new spectral stability results for periodic traveling waves and to make a solid connection between these results and predictions of the formal modulation theory which has been developed by others but which we review for completeness

rogue waves dissipation and downshifting abstract we investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one dimensional dysthe equation significantly rogue waves do not develop after the downshifting becomes permanent thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves using the inverse spectral theory of the nls equation simulations of the damped dysthe equation for sea states characterized by jonswap spectrum consistently show that rogue wave events are well predicted by proximity to homoclinic data as measured by the spectral splitting distance the cut off distance c u t o f f decreases as the strength of the damping increases indicating that for stronger damping the jonswap initial data must be closer to homoclinic data for rogue waves to occur

the semiclassical limit of the defocusing nls hierarchy we establish the semiclassical limit of the one dimensional defocusing cubic nonlinear schrodinger nls equation complete integrability is exploited to obtain a global characterization of the weak limits of the entire nls hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows consequently this also establishes the zero dispersion limit of the modified korteweg de vries equation that resides in that hierarchy we have adapted and clarified the strategy introduced by lax and levermore to study the zero dispersion limit of the korteweg de vries equation expanding it to treat entire integrable hierarchies and strengthening the limits obtained a crucial role is played by the convexity of the underlying log determinant with respect to the times associated with the commuting flows c 1999 john wiley sons inc

spatiotemporal chaos in nonintegrable three wave interactions abstract the nonliear three wave interaction is ubiquitous having many physical manifestations the conservative equations are integrable by inverse scattering transforms and can possess soliton solutions two nonintegrable three wave models were numerically simulated spatiotemporal chaos where coherent structures evolve and interact chaotically was observed in the long time large system limit correlation functions were measured the behaviour was analysed by using perturbation theory around the inverse scattering transform solutions of the nonperturbed integrable equations saturation amplitudes correlation times and lengths were estimated

discretizations integrable systems and computation discretizations and associated numerical computation of solutions of certain integrable systems such as the nonlinear schrodinger equation nls and the sine gordon sg equations with periodic boundary values can lead to instabilities chaotic and spurious results the chaos can be due to truncation errors or even roundoff errors and can be traced to the fact that these integrable systems are strongly unstable when the initial values are in the neighbourhood of homoclinic manifolds by using the associated nonlinear spectral transform of the nls equation and tracking the evolution of relevant eigenvalues one can observe and relate crossing of homoclinic manifolds to the temporal chaos in the waveforms when the initial data is even for general initial values even though there is no crossing of the unperturbed homoclinic manifolds the waveforms still exhibit chaotic phenomena which can be related to the evolution of the spectrum this paper reviews the current understanding of this intriguing phenomena and also compares the implementation of certain symplectic integrators and runge kutta algorithms for the nls and sg equations in regions of phase space proximate to the homoclinic manifolds

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

multi dimensional homoclinic jumping and the discretized nls equation we consider a class of dynamical systems that arise frequently in multi mode truncations and discretizations of partial differential equations including the perturbed nls we develop a general method to detect the existence of multi pulse solutions that are doubly asymptotic to an invariant manifold with two different time scales we use our method together with some recent results of li and mclaughlin to show the existence of several families of multi pulse orbits for the ablowitz ladik discretization of the perturbed nls these orbits include n pulse heteroclinic orbits and n pulse silnikov type orbits for arbitrarily large n

frequency and phase locking of spatially periodic perturbed sine gordon breather trains this paper considers the damped driven sine gordon equation with even periodic boundary conditions on a finite length nearly integrable perturbation methods are applied to infer approximate solution branches of space and time periodic breather trains of order one amplitude these approximate solutions consist of coherent spatial structures either homogeneous or with one breather per period which lock in frequency and phase to the small time periodic perturbation in the presence of uniform weak damping the predicted resonant breather trains are then verified by overman via direct simulation of the perturbed sine gordon pde and shown to be accurate within the order of the perturbation theory

benjamin feir type instability of sine gordon equation and spectrum of lame equation the benjamin feir type instability of the nonhomoclinic wavetrain solution of the sine gordon equation against the spatially periodic small perturbation is observed by relating the fourier coefficients of the first order approximate solution of fourier type to the band structure of the spectrum of the lame equation

numerical simulation of quasi periodic solutions of the sine gordon equation abstract analytic solutions of the sine gordon equation corresponding to periodic boundary conditions can be complicated especially if initial values are chosen in the vicinity of homoclinic orbits for such initial values it has been demonstrated that numerical solutions develop instabilities and may become chaotic because of the analytical complexities one cannot easily calculate the accuracy of the numerical solutions and therefore compare different numerical schemes in a straightforward manner in this note we evaluate numerical methods in terms of the nonlinear spectrum in particular it provides one with a means of comparing symplectic and nonsymplectic integrators for integrable infinite dimensional hamiltonian systems

on unipotent blocks and their ordinary characters there is a great deal of evidence suggesting that block theory for finite reductive groups should lead to an enrichment of the harish chandra theory for those groups see s 5 1 fs2 p2 3 4 h 6 1 10 an e generalized harish chandra theory for e a positive integer is introduced in bmm i a the context is as follows let g be a connected reductive group defined over lfq with associated frobenius endomorphism f let e 1 an integer when l is an f stable levi subgroup of g one denotes by r and re the lusztig functor and its adjoint a unipotent e pair l l is defined by an e split levi subgroup l of g and a unipotent character z d l 1 one writes li z1 l2 j 2 if and only if l1 c l 2 and zl l2 l 1 2 l 4 0 see 3 l the notion of unipotent e cuspidal pairs and the resulting partition of g g v 1 are defined as in the classical case corresponding to e 1 if f is a prime not dividing q and e is the order of q mod f it seems natural in connection with blocks to ask if the partition of the unipotent characters 8 g v l resulting from e theory coincides with f blocks the blocks of g v which are defined by unipotent characters are called unipoteut f blocks we prove

simplified analysis of the lms adaptive filter using a transfer function approximation a simple approximation to the behavior of the lms adaptive filter as a discrete transfer function is developed this representation is a valid description for both deterministic inputs and for the expected results with random inputs including correlated inputs the results are shown to be exact for some classes of input including periodic signals one result of this analysis is the demonstration that the lms filter can produce results which are biased from the least squares solution under the combined conditions of a nonzero mean primary and a correlated reference input

normally elliptic singular perturbations and persistence of homoclinic orbits abstract we consider a dynamical system possibly infinite dimensional or non autonomous with fast and slow time scales which is oscillatory with high frequencies in the fast directions we first derive and justify the limit system of the slow variables assuming a steady state persists we construct the stable unstable center stable center unstable and center manifolds of the steady state of a size of order o 1 and give their leading order approximations finally using these tools we study the persistence of homoclinic solutions in this type of normally elliptic singular perturbation problems

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

the geometry of real sine gordon wavetrains the characterization ofreal n phase quasiperiodic solutions of the sine gordon equation has been an open problem in this paper we achieve this result employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature a significant by product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds and an understanding of how the real isospectral manifolds are embedded by placing the problem in this general context these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions

conservation of phase space properties using exponential integrators on the cubic schrodinger equation the cubic nonlinear schrodinger nls equation with periodic boundary conditions is solvable using inverse spectral theory the nonlinear spectrum of the associated lax pair reveals topological properties of the nls phase space that are difficult to assess by other means in this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach the initial condition used is a perturbation of the unstable plane wave solution which is difficult to numerically resolve our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step while a lower order multisymplectic is not as accurate and too slow to compete

parametric instabilities in the discrete sine gordon equation abstract the one dimensional sine gordon equation is an example of an exactly solvable nonlinear partial differential equation when discretized on a periodic lattice in space and in time it corresponds to a lattice of pendula coupled by linear springs we show that the discretized system is unstable to a parametric mode when the frequency of time discretization is sufficiently small and we obtain the instability condition and growth rate by considering the effects of two finite amplitude modes we also obtain the nonlinear saturation of the instability we also examine how solutions of the exact sine gordon equation behave under this map both in and away from the parametrically unstable regime

mel nikov analysis of numerically induced chaos in the nonlinear schro dinger equation abstract certain hamiltonian discretizations of the periodic focusing nonlinear schrodinger equation nls have been shown to be responsible for the generation of numerical instabilities and chaos in this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the nls using heuristic mel nikov methods the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system the structure of the persistent homoclinic orbits that are predicted by the mel nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos the mel nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments in a neighborhood of the persistent homoclinic orbits the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study

non wiener mean weight behavior of lms transversal equalizers with sinusoidal interference the least mean square lms algorithm is generally expected to operate near the corresponding wiener filter solution an important exception occurs when the algorithm is used to adapt a transversal equalizer in the presence of additive narrowband interference for this application the steady state lms equalizer behavior does not correspond to that of the fixed wiener equalizer the mean of its weights is different from the wiener weights and its mean square error performance may be significantly better than the wiener performance assuming a large interference to signal ratio isr we present the solution for the mean of the lms weight vector in steady state based on the butterweck expansion of the weight update equation the analytical results are valid for all mean square stable step sizes simulation results are included to support the analytical results and show that the analytical results predict the simulation results accurately over a wide range of isr

existence of homoclinic tubes in the sine gordon equation with hamiltonian perturbation we consider the sine gordon equation under hamiltonian perturbation with even periodic boundary conditions we give an analytic expression for homoclinic orbits and produce a useful representation of the gradient of an important integral of motion we establish the existence of homoclinic tubes based on mel nikov analysis and an implicit function theorem argument

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

discrete singular convolution for the sine gordon equation abstract this paper explores the utility of a discrete singular convolution dsc algorithm for the integration of the sine gordon equation the initial values are chosen close to a homoclinic manifold for which previous methods have encountered significant numerical difficulties such as numerically induced spatial and temporal chaos a number of new initial values are considered including a case where the initial value is exactly on the homoclinic orbit the present algorithm performs extremely well in terms of accuracy efficiency simplicity stability and reliability

homoclinic orbits for a perturbed nonlinear schrodinger equation in recent years there have been extensive studies on the existence of homoclinic orbits for nearly integrable hamiltonian pdes which are closely related to chaos in this work we consider a perturbed nonlinear schrodinger equation for u even and periodic in x the diffusion ieuxx is an unbounded perturbation term when the diffusion is replaced by its bounded fourier truncation li mclaughlin shatah and wiggins 26 proved the existence of homoclinic orbits for the perturbed equation the method was based on invariant manifolds foliations and melnikov analysis the unboundedness of the diffusion prevents the equation from being solved for t 0 for general initial values and destroys some geometric structures however we overcome these difficulties and prove the existence of homoclinic orbits for the diffusively perturbed nls 2000 john wiley sons inc

on the backlund gauge transformation and homoclinic orbits of a coupled nonlinear schrodinger system abstract the backlund gauge transformation for a system of coupled nls nonlinear schrodinger equations with a degenerate associated spectral operator is derived from an algebraic perspective extending aspects of other results m boiti tu guizhang il nuovo cimento 71b 1982 253 264 d h sattinger v d zurkowski physica d 26 1 3 1987 225 250 that apply in the context of non degenerate spectral operators moreover we demonstrate how the backlund gauge transformation can be used to explicitly construct the entire unstable manifold via superpositions of homoclinic orbits of a plane wave solution with both self phase instabilities and coupling instabilities this work builds on the results of ercolani et al n ercolani m g forest d w mclaughlin physica d 18 1986 472 474 n ercolani m g forest d w mclaughlin physica d 43 2 3 1990 349 384 for the sine gordon equation and forest et al m g forest d w mclaughlin d j muraki o c wright j nonlinear sci in press m g forest s p sheu o c wright phys lett a in press for the integrable coupled nls system

persistent homoclinic orbits for a perturbed nonlinear schrodinger equation the persistence of homoclinic orbits for certain perturbations of the integrable nonlinear schrodinger equation under even periodic boundary conditions is established more specifically the existence of a symmetric pair of homoclinic orbits is established for the perturbed nls equation through an argument that combines melnikov analysis with a geometric singular perturbation theory for the pde 1996 john wiley sons inc

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

nonlinear effects in lms adaptive equalizers an adaptive transversal equalizer based on the least mean square lms algorithm operating in an environment with a temporally correlated interference can exhibit better steady state mean square error mse performance than the corresponding wiener filter this phenomenon is a result of the nonlinear nature of the lms algorithm and is obscured by traditional analysis approaches that utilize the independence assumption current filter weight vector assumed to be statistically independent of the current data vector to analyze this equalizer problem we use a transfer function approach to develop approximate analytical expressions of the lms mse for sinusoidal and autoregressive interference processes we demonstrate that the degree to which lms may outperform the corresponding wiener filter is dependent on system parameters such as signal to noise ratio snr signal to interference ratio sir equalizer length and the step size parameter

stability and bifurcation of spatially coherent solutions of the damped driven nls equation an analytical study is conducted of the structure stability and bifurcation of the spatially dependent time periodic solutions of the damped driven sine gordon equation in the nonlinear schrodinger approximation locked states are found for which the spatial structure consists of coherent excitations localized about x 0 or l 2 a bifurcation analysis reveals the relationship of these spatially localized solutions to the spatially independent ones and provides a cutoff wavenumber above which there are no spatially dependent solutions this establishes an upper bound on the number of local excitations comprising the spatial pattern a linear stability analysis shows that the spatially localized solutions undergo a hopf bifurcation to temporal quasi periodicity as the driver amplitude gamma is increased for sufficiently high driver frequencies the temporally periodic solution regains its stability via another hopf bifurcation in a gamma window of finite width before undergoing a third hop

chaotic transport and integrable instabilities in a near integrable many particle hamiltonian lattice abstract this paper is a report of detailed numerical experiments on a large dimensional hamiltonian lattice which arises as a nearly integrable discretization of the periodic sine gordon equation through four different one parameter families of initial data we focus on the interplay between numerically observable near integrable behavior quantified by modern statistical diagnostics and the integrable pde solutions and phase space geometry associated with the initial data transitions are observed within each family from quasiperiodic behavior to chaotic motion which is sometimes accompanied by transport toward equipartition these transitions in the near integrable system are correlated with integrable features of the initial data in particular transitions from neutrally stable to exponentially unstable sine gordon solutions and associated homoclinic orbits geometrically these correspond to transitions from neutrally stable tori to whiskered tori our results indicate that 1 positive lyapunov exponents our messure of chaos in near integrable orbits coincide one to one with integrable pde instabilities and homoclinic orbits 2 the statistical transition to equipartition of energy on a long fixed timescale occurs only if but not necessarily if there are integrable instabilities and associated whiskered tori the rate of transport appears to depend on the amplitude of the homoclinic degrees of freedom 3 isolated cases of discreteness induced instabilities and action transport are observed these return to quasiperiodic flow at finer discretizations 4 a simple measure such as energy of the initial data is not an accurate predictor of chaotic or recurrent behavior one family of initial data exhibits as the energy increases alternating windows of quasiperiodic and chaotic orbits

morse and melnikov functions for nls pde s the theory of the focusing nls equation under periodic boundary conditions together with the floquet spectral theory of its associated zakharov shabat linear operator hat l is developed in sufficient detail for later use in studies of perturbations of the nls equation counting lemmas for the non selfadjoint operator hat l are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discd in the complex eigenvalue plane the radius of the discd is controlled by theh 1 norm of the potential overrightarrow q for this integrable nls hamiltonian system unstable tori are identified and backlund transformations are then used to construct global representations of their stable and unstable manifolds whiskered tori for the nls pde the floquet discriminant delta lambda overrightarrow q is used to introduce a natural sequence of nls constants of motion f j overrightarrow q equiv delta lambda lambda j c overrightarrow q overrightarrow q where denotes thej th critical point of the floquet discriminant a taylor series expansion of the constants f j overrightarrow q with explicit representations of the first and second variations is then used to study neighborhoods of the whiskered tori in particular critical tori with hyperbolic structure are identified through the first and second variations of f j overrightarrow q which themselves are expressed in terms of quadratic products of eigenfunctions of hat l the second variation permits identification within the disc d of important bifurcations in the spectral configurations of the operator hat l the constant f j overrightarrow q as the height of the floquet discriminant over the critical point admits a natural interpretation as a morse function for nls isospectral level sets this morse interpretation is studied in some detail it is valid globally for the infinite tail f j overrightarrow q j n which is associated with critical points outside the discd within this disc the interpretation is only valid locally with the same obstruction to its global validity as to a global ordering of the spectrum nevertheless this local morse theory together with the backlund representations of the whiskered tori produces extremely clear pictures of the stratification of nls invariant sets near these whiskered tori pictures which are useful in the study of perturbations of nls finally a natural connection is noted between the constants f j overrightarrow q of the integrable theory and melnikov functions for the theory of perturbations of the nls equation this connection generates a simple but general representations of the melnikov functions

on the endomorphism algebra of modular gelfand graev representations abstract we study the endomorphism algebras of a modular gelfand graev representation of a finite reductive group by investigating modular properties of homomorphisms constructed by curtis and curtis shoji

spectral stability analysis for periodic traveling wave solutions of nls and cgl perturbations abstract we consider cnoidal traveling wave solutions to the focusing nonlinear schrodinger equation nls that have been shown to persist when the nls is perturbed to the complex ginzburg landau equation cgl we show that while these periodic traveling waves are spectrally stable solutions of nls with respect to periodic perturbations they are unstable with respect to bounded perturbations furthermore we use an argument based on the fredholm alternative to find an instability criterion for the persisting solutions to cgl

autoresonant excitation of multiphase waves in the sine gordon model abstract autoresonant excitation and control of multiphase waves of the sine gordon equation is discussed the autoresonance a persistent nonlinear phase locking is achieved by applying chirped frequency wave like perturbations and passage through resonances with inactive dormant degrees of freedom in the system successful phase locking requires the amplitude of the driving perturbation to exceed a threshold that scales as 3 4 power of the driving frequency chirp rate the spectral theory of the inverse scattering transform method is used for diagnostics and analysis of the excited solutions

a quasi periodic route to chaos in a near integrable pde abstract pattern formation and transitions to chaos are described for the damped ac driven one dimensional periodic sine gordon equation in a nonlinear schrodinger regime a generic quasi periodic route to intermittent chaos is exhibited in detail using a range of dynamical systems diagnostics in addition a nonlinear spectral transform is exploited i to identify and quantify coordinates of space time attractors in terms of a small number of soliton modes of the underlying integrable system ii to use these analytic coordinates to identify homoclinic orbits as possible sources of chaos and iii to demonstrate the significance of energy transfer between coherent and extended states in this chaotic system

numerical inverse spectral transform for the periodic sine gordon equation theta function solutions and their linearized stability abstract the inverse spectral transform for integrable nonlinear ordinary and partial differential equations such as the toda lattice korteweg de vries sine gordon and nonlinear schrodinger equations provides explicit algorithms to generate exact solutions under periodic or quasiperiodic boundary conditions these oscillatory wavetrains may be prescribed a priori to consist of a nonlinear superposition of n phases j x t j x j t 0 j j 1 n where the wave is 2 periodic independently in each phase this paper exhibits the numerical implementation of the inverse spectral solution of the sine gordon equation the general construction is outlined and then implemented for n 1 2 and 3 we compute 1 the exact theta function solutions 2 the floquet spectrum of x periodic solutions 3 the labelling of linearized instabilities of n phase solutions in terms of spectral data and 4 the linearized growth rate in each unstable mode the associated surfaces q n x t are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system