signal sampling and recovery under dependent errors the paper examines the impact of the additive correlated noise on the accuracy of the signal reconstruction algorithm originating from the whittaker shannon ws sampling interpolation formula a class of band limited signals as well as signals which are non band limited are taken into consideration the proposed reconstruction method is a smooth post filtering correction of the classical ws interpolation series we assess both the point wise and global accuracy of the proposed reconstruction algorithm for a broad class of dependent noise processes this includes short and long memory stationary errors being independent of the sampling rate we also examine a class of noise processes for which the correlation function depends on the sampling rate whereas the short memory errors have relatively small influence on the reconstruction accuracy the long memory errors can greatly slow down the convergence rate in the case of the noise model depending on the sampling rate further degradation of the algorithm accuracy is observed we give quantitative explanations of these phenomena by deriving rates at which the reconstruction error tends to zero we argue that the obtained rates are close to be optimal in fact in a number of special cases they agree with known optimal min max rates the problem of the limit distribution of the l2 distance of the proposed reconstruction algorithm is also addressed this result allows us to tackle an important problem of designing non parametric lack of fit tests the theory of the asymptotic behavior of quadratic forms of stationary sequences is utilized in this case

symbolic computation on the darboux transformation for a generalized variable coefficient higher order nonlinear schrodinger equation from fiber optics considering the propagation of ultrashort pulse in the realistic fiber optics a generalized variable coefficient higher order nonlinear schrodinger equation is investigated in this paper under certain constraints a new 3 3 lax pair for this equation is obtained through the ablowitz kaup newell segur procedure furthermore with symbolic computation the darboux transformation and nth iterated potential transformation formula for such a model are explicitly derived the corresponding features of ultrashort pulse in inhomogeneous optical fibers are graphically discussed by the one and two soliton like solutions

new solitons for the hirota equation and generalized higher order nonlinear schrodinger equation with variable coefficients in this paper multisoliton solutions of the hirota equation with variable coefficients are obtained by the darboux transformation based on the ablowitz kaup newell segur technology as an example we discuss the evolutional behaviour of a two soliton solution in a soliton control fibre system the results reveal that one may control the interaction between the pulses by choosing the third order dispersion parameters d4 and h appropriately meanwhile more generalized forms of bright soliton and dark soliton solutions of generalized higher order nonlinear schrodinger equations ghonlse with variable coefficients are obtained by the extended tanh function method moreover new bright and dark combined solitary wave kink solitary wave and m shaped solitary wave to ghonlse with variable coefficients are firstly reported in this paper especially the term proportional to 1 resulting from the group velocity decides the group velocity and the phase shift of these new solitary waves

sasa satsuma complex modified korteweg de vries ii and the complex sine gordon ii equation revisited recursion operators nonlocal symmetries and more we present a new symplectic structure and a hereditary recursion operator for the sasa satsuma equation which is widely used in nonlinear optics using an integrodifferential substitution relating this equation to a third order symmetry flow of the complex sine gordon ii equation enabled us to find a hereditary recursion operator and higher hamiltonian structures for the latter equation we also show that both the sasa satsuma equation and the third order symmetry flow for the complex sine gordon ii equation are bi hamiltonian systems and we construct several hierarchies of local and nonlocal symmetries for these systems

exponential weighting algorithms for reconstruction of bandlimited signals the problem of reconstruction of bandlimited signals from discrete noisy observations is considered whittaker shannon interpolation series based estimates involving the weighting factors are proposed it is shown that they converge as the sampling rate increases to infinity the rate of convergence of the double exponential weighting algorithm is established these results are corroborated in computer simulations

magnitude of the truncation error in sampling expansions of band limited signals bounds are established for the truncation error in the sampling theorem for signals with radian bandwidth w w 0 the assumptions are either differenfiability conditions upon the fourier transform of the signal or on the rate of decay of the first derivative of the signal itself finally our theorems are applied to two new examples worked out in detail

the visual cortex as a crystal a theory of pattern formation in primary visual cortex v1 is presented that takes into account its crystalline like structure the cortex is partitioned into fundamental domains or hypercolumnsof a lattice describing the distribution of singularities or pinwheels in the orientation preference map each hypercolumn is modelled as a network of orientation and spatial frequency selective cells organised around a pair of pinwheels which are associated with high and low spatial frequency domains respectively the network topology of the hypercolumn is taken to be a sphere with the pinwheels located at the poles of the sphere interactions between hypercolumns are mediated by anisotropic long range lateral connections that link cells with similar feature preferences using weakly nonlinear analysis we investigate the spontaneous formation of cortical activity patterns through the simultaneous breaking of an internal o 3 symmetry and a discrete lattice symmetry the resulting patterns are characterised by states in which each hypercolumn exhibits a tuned response to both orientation and spatial frequency and the distribution of optimal responses across hypercolumns is doubly periodic or quasi periodic with respect to the underlying lattice 2002 published by elsevier science b v

persistence of solutions to higher order nonlinear schrodinger equation abstract applying an abstract interpolation lemma we showed persistence of solutions of the initial value problem to higher order nonlinear schrodinger equation also called airy schrodinger equation in weighted sobolev spaces x 2 for 0 1

perturbation induced radiative losses in collision of nse solitons the total energy and its spectral density emitted during the collision of two solitons described by the conservatively perturbed non linear schrodinger equation nse is calculated by means of perturbation theory based on the inverse scattering transform it is shown that the total emitted energy at large relative velocity does not depend on the relative internal phase of the colliding solitons the same results are obtained for solitons in a system of two weakly coupled non linear schrodinger equations

symmetrically coupled higher order nonlinear schrodinger equations singularity analysis and integrability the integrability of a system of two symmetrically coupled higher order nonlinear schrodinger equations with parameter coefficients is tested by means of singularity analysis it is proven that the system passes the painleve test for integrability only in ten distinct cases of which two are new for one of the new cases a lax pair and a multi field generalization are obtained for the other one the equations of the system are uncoupled by a nonlinear transformation

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

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

sampling theory for not necessarily band limited functions a historical overview shannon s sampling theorem is one of the most powerful results in signal analysis the aim of this overview is to show that one of its roots is a basic paper of de la vallee poussin of 1908 the historical development of sampling theory from 1908 to the present especially the matter dealing with not necessarily band limited functions which includes the duration limited case actually studied in 1908 is sketched emphasis is put on the study of error estimates as well as on the delicate point wise behavior of sampling sums at discontinuity points of the signal to be reconstructed

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

algebro geometric solutions for the gerdjikov ivanov hierarchy this paper is dedicated to provide theta function representations of algebro geometric solutions and related crucial quantities for the gerdjikov ivanov gi hierarchy our main tools include the polynomial recursive formalism the hyperelliptic curve with finite number of genus the baker akhiezer functions the meromorphic function the dubrovin type equations for auxiliary divisors and the associated trace formulas with the help of these tools the explicit representations of the baker akhiezer functions the meromorphic function and the algebro geometric solutions are obtained for the entire gi hierarchy

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

on uniqueness and decay of solution for hirota equation abstract we address the question of the uniqueness of solution to the initial value problem associated to the equation t u i x 2 u x 3 u i u 2 u u 2 x u u 2 x u 0 x t r and prove that a certain decay property of the difference u1 u2 of two solutions u1 and u2 at two different instants of times t 0 and t 1 is sufficient to ensure that u1 u2 for all the time

the riemann problem method for solving a perturbed nonlinear schrodinger equation describing pulse propagation in optic fibres we used the riemann problem method with a 3 3 matrix system to find the femtosecond single soliton solution for a perturbed nonlinear schrodinger equation which describes bright ultrashort pulse propagation in properly tailored monomode optical fibres compared with the gel fand levitan marchenko approach the major advantage of the riemann problem method is that it provides the general single soliton solution in a simple and compact form unlike the standard nonlinear schrodinger equation here the single soliton solution exhibits periodic evolution patterns

estimates of low sobolev norms of solutions for some nonlinear evolution equations abstract in this work we obtain results on the estimates of low sobolev norms for solutions of some nonlinear evolution equations in particular we apply our method for the complex modified korteweg de vries type equation and benjamin ono equation

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

moving average restoration of bandlimited signals from noisy observations the purpose of this paper is to describe the extension of the whittaker shannon sampling theorem to reconstruction of bandlimited functions in the presence of zero mean uncorrelated noise it is shown that the classical whittaker shannon sampling scheme is not consistent in the case of noisy measurements and new reconstruction algorithms based on the moving average smoothing are proposed the weak and strong consistency of the algorithms is established and the rate of convergence is investigated the theory is verified in the computer simulations

types of solutions of the variable coefficient nonlinear schrodinger equation with symbolic computation by using hirota s bilinear method and symbolic computation solutions for a variable coefficient nonlinear schrodinger equation are obtained theoretically it is found that the type of the solutions changes with the different choices of the group velocity dispersion coefficient beta 2 z according to those solutions the relevant properties and features of physical and optical interest are illustrated in addition an effective technique for controlling the shape of the pulses is presented the results of this paper will be valuable to the study of the future development of ultrahigh rate and long distance optical communication systems

darboux transformation and soliton like solutions for the gerdjikov ivanov equation an explicit n fold darboux transformation with multiparameters for a kind of coupled derivative nonlinear schrodinger equation is constructed with the help of a gauge transformation of a spectral problem as a reduction a darboux transformation of the gerdjikov ivanov equation is obtained furthermore the explicit soliton like solutions of the gerdjikov ivanov equation are given by applying its darboux transformation

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

exact and explicit solutions of higher order nonlinear equations of schrodinger type abstract new exact solutions for the higher order nonlinear schrodinger equation and coupled higher order nonlinear schrodinger equations are obtained by using the generalized jacobi elliptic function method solutions in the limiting cases are also studied

error analysis of frame reconstruction from noisy samples this paper addresses the problem of reconstructing a continuous function defined on rd from a countable collection of samples corrupted by noise the additive noise is assumed to be i i d with mean zero and variance sigma2 we sample the continuous function on the uniform lattice 1 m zd and show for large enough m that the variance of the error between the frame reconstruction fepsiv m from noisy samples of f and the function f satisfy var fepsiv m x f x ap sigma2 md cx where cx is the best constant for every x isin rd we also prove a similar result in the case that our data are weighted average samples of corrupted by additive noise

identification of miso nonlinear regressions in the presence of a wide class of disturbances a nonparametric algorithm based on multiple fourier series for identification of miso multidimensional input single output systems is investigated it is shown that the algorithm converges for a wide class of noise

maxwell s equations with arbitrary self action nonlinearity in a waveguiding theory guided modes and asymptotic of eigenvalues abstract the paper treats a nonlinear eigenvalue problem that describes propagation of a transverse magnetic wave in a plane dielectric waveguide having perfectly conducted walls at both sides the dielectric s permittivity is characterised by an arbitrary monotonically increasing self action nonlinearity the full set of guided modes is described by eigenvalues of the corresponding nonlinear maxwell operator with appropriate boundary conditions we give a comprehensive analysis of this problem and develop an original approach to study its solvability and properties of solutions several results about existence of the eigenvalues are proved their distribution and asymptotic are found zeros of the eigenfunctions and their location are also determined criterion of periodicity for the eigenfunctions is found comparison theorem is derived etc it is shown that unbounded nonlinearities lead to appearance of nonlinearised solutions this results in the existence of a novel guided regime that cannot be described within the framework of perturbation of linear guided wave regimes

exact travelling wave solutions for a generalized nonlinear schrodinger equation a wide class of exact travelling wave solutions of a generalized nonlinear schrodinger equation gnls is obtained and analysed in detail this class of solutions incorporates bright and dark solitary waves periodic waves unbounded waves and other solitary waves as asymptotic limits of the periodic or unbounded modes the method of analysis adopted is based on reducing the gnls to an ordinary differential equation and studying the phase plane of the resulting dynamical system application of the obtained results to the problem of propagation of femtosecond duration pulses in nonlinear optical fibres is also discussed

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

finite gap integration of the derivative nonlinear schrodinger equation a riemann hilbert method abstract in this paper we retrieve finite gap solutions of the gerdjikov ivanov type derivative nonlinear schrodinger equation by using the algebro geometric method and the riemann hilbert method we show that the baker akhiezer function of the derivative nonlinear schrodinger equation can be described in terms of two solvable matrix riemann hilbert problems on with 2 and 3 symmetry conditions based on the technique developed in the study of long time asymptotics our main tools include matrix baker akhiezer function asymptotic analysis algebraic curve and riemann theta function matrix riemann hilbert problem and associated deformation procedures

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

soliton solutions liouville integrability and gauge equivalence of sasa satsuma equation exact integrability of the sasa satsuma equation sse in the liouville sense is established by showing the existence of an infinite set of conservation laws the explicit form of the conserved quantities in terms of the fields are obtained by solving the riccati equation for the associated 3 3 lax operator the soliton solutions in particular one and two soliton solutions are constructed by the hirota s bilinear method the one soliton solution is also compared with that found through the inverse scattering method the gauge equivalence of the sse with a generalized landau lifshitz equation is established with the explicit construction of the new equivalent lax pair

soliton like solutions of a generalized variable coefficient higher order nonlinear schrodinger equation from inhomogeneous optical fibers with symbolic computation for the long distance communication and manufacturing problems in optical fibers the propagation of subpicosecond or femtosecond optical pulses can be governed by the variable coefficient nonlinear schrodinger equation with higher order effects such as the third order dispersion self steepening and self frequency shift in this paper we firstly determine the general conditions for this equation to be integrable by employing the painleve analysis based on the obtained 3 3 lax pair we construct the darboux transformation for such a model under the corresponding constraints and then derive the nth iterated potential transformation formula by the iterative process of darboux transformation through the one and two soliton like solutions we graphically discuss the features of femtosecond solitons in inhomogeneous optical fibers

prolongation structures of a higher order nonlinear schrodinger equation a higher order schrodinger equation containing parameters which is used to describe pulse propagation in optical fibres is shown to admit an infinite dimensional prolongation structure for exactly four combinations of the parameters besides the classical nls equation for each of these cases the structure of the resulting prolongation algebra is determined explicitly for the first three cases the prolongation algebra is essentially a sub algebra of a1 1 the fourth case turns out to be sub algebra of the twisted kac moody algebra a2 2 using vector field representations related systems of differential equations for the pseudo potential functions are given for each of the cases the cases found here correspond exactly to the derived nls equations i and ii the hirota equation and the equation recently considered by sasa and satsuma 1991 the result of this paper strongly indicates that the considered higher order nls equation is completely integrable for precisely these four cases

integrable evolution systems based on gerdjikov ivanov equations bi hamiltonian structure finite dimensional integrable systems and n fold darboux transformation a spectral problem and the associated gerdjikov ivanov gi hierarchy of nonlinear evolution equations is presented as a reduction the well known gi equation of derivative nonlinear schrodinger equations is obtained it is shown that the gi hierarchy is integrable in a liouville sense and possesses bi hamiltonian structure moreover the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenfunctions in particular an explicit n fold darboux transformation for the gi equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique some explicit solitonlike solutions of the gi equation are given by applying its darboux transformation

the structure function method for asymmetric solitary waves we present a method for the reconstruction of the waveform of asymmetric solitary waves which are observed in weakly dispersive nonlinear media starting from the morphology of the waveform we first give local differential laws which govern the waveform about its extrema and at infinity then we analytically extend these laws to find a differential equation of the generalized korteweg de vries family for the waveform holding over the whole real axis a multi valued singular functional dependence of the nonlinear term of this equation on the amplitude of the solitary wave results which was not considered before and which is intimately connected with the asymmetric nature of the solitary wave if cast in terms of a suitable combination of the wave amplitude a combination which we call the structure function of the solitary wave the differential equation reduces to a modified korteweg de vries equation with a polynomial nonlinear term which we solve by quadrature including up to sixth order nonlinearities in particular the fourth order accurate waveforms are given in terms of elementary functions and they are in excellent agreement with observations for a given maximum wave amplitude several waveforms are found each having a different width this width decreases as the maximum wave amplitude increases

quasi random sampling for signal recovery abstract the problem of reconstruction of band limited signals from nonuniformly sampled and and noisy observations is studied it is proposed to sample a signal at quasi random points that form a deterministic sequence with properties resembling a random variable being uniformly distributed such quasi random points can be easily and efficiently generated the reconstruction method based on the modified whittaker shannon cardinal expansion is proposed and its asymptotical properties are examined in particular the sufficient conditions for the convergence of the mean integrated squared error are found the key difference between the proposed reconstruction algorithm and the classical whittaker shannon scheme is in treating the sampling rate and the reconstruction rate differently this distinction is necessary to ensure consistency of the reconstruction algorithm in the presence of noise

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

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

recovering band limited signals under noise we consider the problem of recovering a band limited signal f t from noisy data yk f k spl tau spl gt epsilon sub k where spl tau is the sampling rate starting from the truncated whittaker shannon cardinal expansion with or without sampling windows both cases yield inconsistent estimates of f t we propose estimators that are convergent to f t in the pointwise and uniform sense the basic idea is to cut down high frequencies in the data and to use suitable oversampling spl tau spl les spl pi spl omega spl omega being the bandwidth maximum frequency of f t the simplest estimator we propose is given by f spl circ sub n t spl tau spl sigma t k spl tau spl les n spl tau yksin spl omega t k spl tau spl pi t k spl tau t spl les n spl tau generalizations of f spl circ sub n including sampling windows are also examined the main aim is to examine the mean squared error mse properties of such estimators in order to determine the optimal choice of the sampling rate spl tau yielding the fastest possible rate of convergence the best rate for the mse we obtain is o in n n

averaging principle for the higher order nonlinear schrodinger equation with a random fast oscillation this work concerns the problem associated with averaging principle for a higher order nonlinear schrodinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction diffusion equation evolving with respect to the fast time this model can be translated into a multiscale stochastic partial differential equations stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time scales to be more precise under suitable conditions we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear schrodinger equation is an average with respect to the stationary measure of the fast varying process finally by using the khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation and as a consequence the system can be reduced to a single higher order nonlinear schrodinger equation with a modified coefficient

finite sampling approximations for non band limited signals finite sampling approximations along with bounds on the approximation error are derived for certain deterministic and random signals which are not band limited

long time asymptotics for a mixed nonlinear schrodinger equation with the schwartz initial data abstract in this paper we define a general analytical domain and two reflection coefficients to study the long time asymptotics for a defocusing mixed nonlinear schrodinger equation with the schwartz initial data it is shown that the solution of the initial value problem for the mixed schrodinger equation can be expressed in terms of the solution associated with a riemann hilbert problem the long time asymptotic of the mixed schrodinger equation is further obtained via the deift zhou nonlinear steepest descent method the long time asymptotic for the classical schrodinger equation derivative schrodinger equation and modified schrodinger equation are obtained directly from our results as special cases

two kinds of new integrable decompositions of the gerdjikov ivanov equation the gerdjikov ivanov gi equation can be given from the first non trivial flow of the generalized kaup newell equations under a reduction condition in this paper we perform binary nonlinearization for the generalized kaup newell equations under two symmetry constraints by imposing the reduction condition on this nonlinearization we obtain two kinds of new integrable decompositions of the gi equation

efficient evaluation of oversampled functions abstract it has been known since long that an oversampled function can be represented by a generalized sinc series which converges much faster than the usual one we show that the speed of convergence is exponential and give numerical approximations to the kernel function involved in the generalized series the practical consequence is that a bandlimited function which is oversampled can be evaluated at any point with accuracy o l e l using o l operations

smoothing properties for the higher order nonlinear schrodinger equation with constant coefficients abstract we study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear schrodinger equation with constant coefficients which appears as a model for propagation of pulse in an optical fiber

hamiltonian versus energy diagrams in soliton theory parametric curves featuring hamiltonian versus energy are useful in the theory of solitons in conservative nonintegrable systems with local nonlinearities these curves can be constructed in various ways we show here that it is possible to find the hamiltonian h and energy q for solitons of non kerr law media with local nonlinearities without specific knowledge of the functional form of the soliton itself more importantly we show that the stability criterion for solitons can be formulated in terms of h and q only this allows us to derive all the essential properties of solitons based only on the concavity of the curve h vs q we give examples of these curves for various nonlinearity laws and show that they confirm the general principle we also show that solitons of an unstable branch can transform into solitons of a stable branch by emitting small amplitude waves as a result we show that simple dynamics like the transformation of a soliton of an unstable branch into a soliton of a stable branch can also be predicted from the h q diagram s1063 651x 99 09805 0

a note on the usefulness of superkernels in density estimation we consider the akaike parzen rosenblatt density estimate f nh based upon any superkernel l i e an absolutely integrable function with int l 1 whose characteristic function is 1 on lbrack 1 1 rbrack and compare it with a kernel estimate g nh based upon an arbitrary kernel k we show that for a given subclass of analytic densities inf l sup k lim sup n rightarrow infty frac inf h mathbb e int f nh f inf h mathbb e int g nh f 1 where h 0 is the smoothing factor thus asymptotically the class of superkernels is as good as any other class of kernels when certain analytic densities are estimated we also obtain exact asymptotic expressions for the expected l 1 error of the kernel estimate when superkernels are used

well posedness of the cauchy problem for the hirota equation in sobolev spaces hs abstract the well posedness of cauchy problem for the hirota equation is obtained in sobolev spaces h s local result is established for initial data in h s s 1 4 by using the fourier restriction norm method moreover the local solution is global for the initial data in h s s 1 by the generalized trilinear estimates associated with the fourier restriction norm method

nonlinear wavefields in optical fibers with finite time response and amplification effects a higher order nonlinear integro differential equation of the schrodinger type modeling the evolution of finite amplitude wavepackets in the presence of optical amplification and finite material and polarization time responses is analyzed with respect to the dynamical evolution of localized solutions nonexistence of localized travelling and stationary wave solutions is pointed out for certain parameter regimes finally the linear stage of the modulational instability of plane waves is studied and it is shown that the finite time responses produce finite bandwidth instabilities

unique continuation property for a higher order nonlinear schrodinger equation abstract we prove that if a sufficiently smooth solution u to the initial value problem associated with the equation t u i x 2 u x 3 u i u 2 u u 2 x u e u 2 x u 0 x t r is supported in a half line at two different instants of time then u 0 to prove this result we derive a new carleman type estimate by extending the method introduced by kenig et al in ann inst h poincare anal non lineaire 19 2002 191 208

pulse solutions of the cubic quintic complex ginzburg landau equation in the case of normal dispersion time localized solitary wave solutions of the one dimensional complex ginzburg landau equation cgle are analyzed for the case of normal group velocity dispersion exact soliton solutions are found for both the cubic and the quintic cgle the stability of these solutions is investigated numerically the regions in the parameter space in which stable pulselike solutions of the quintic cgle exist are numerically determined these regions contain subspaces where analytical solutions may be found an investigation of the role of group velocity dispersion changes in magnitude and sign on the spectral and temporal characteristics of the stable pulse solutions is also carried out

postfiltering versus prefiltering for signal recovery from noisy samples we consider the extension of the whittaker shannon ws reconstruction formula to the case of signals sampled in the presence of noise and which are not necessarily band limited observing that in this situation the classical sampling expansion yields inconsistent reconstruction we introduce a class of signal recovery methods with a smooth correction of the interpolation series two alternative data smoothing methods are examined based either on a global postfiltering or a local data presmoothing we assess the accuracy of the methods by the global l sub 2 error both band limited and non band limited signals are considered a general class of correlated noise processes is taken into account the weak and strong rates of convergence of the algorithms are established and their relative efficiency is discussed the influence of noise memory and its moment structure on the accuracy is thoroughly examined

truncation error bounds for the cardinal sampling expansion of band limited signals bounds are derived for the truncation error of the cardinal sampling expansion for a large class of band limited deterministic and random signals these bounds extend and improve upon the bounds available in the literature

bright and dark soliton solutions and backlund transformation for the eckhaus kundu equation with the cubic quintic nonlinearity in this paper with symbolic computation the eckhaus kundu equation which appears in the quantum field theory weakly nonlinear dispersive water waves and nonlinear optics is studied via the hirota method by virtue of the dependent variable transformation the bilinear form is obtained bilinear backlund transformation is given with the help of exchange formulae and the corresponding one soliton solution is derived bright and dark n soliton solutions are obtained propagation and interaction of the bright and dark solitons are discussed analytically and graphically interactions of the two solitons are presented bound state of the two solitons can be suppressed via the choice of parameters

on the integrability of the nonlinear schrodinger equation with randomly dependent linear potential we show that the cauchy problem for the nonlinear schrodinger equation driven by a term of the form where w t is a brownian motion can be integrated by means of the inverse scattering transform the solution is written in terms of w t and the integral of powers j 1 2 a complete set of integrals of the motion is determined we relate the probability density of the solution to the green s function of a linear partial differential operator with quadratic coefficients

higher order derivative nonlinear schrodinger equation in the critical case we consider the cauchy problem for the higher order derivative nonlinear schrodinger equation our purpose in this paper is to prove the large time asymptotic behavior of solutions under the non zero mass condition u0xdx 0

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

n soliton solutions for the 2 1 dimensional hirota maccari equation in fluids plasmas and optical fibers abstract under investigation in this paper is the hirota maccari equation which is a generalized 2 1 dimensional model in fluid dynamics plasma physics and optical fiber communication with the aid of the hirota bilinear method and symbolic computation the corresponding n soliton solutions are given and illustrated the characteristic face method and asymptotic analysis are applied to discuss the solitonic propagation and collision including the bidirectional solitons elastic interactions and inelastic interactions finally a kind of special phenomenon with the parameters varying is investigated which might provide people with useful information on the dynamics of the relevant fields

on restoring band limited signals the problem of reconstruction of band limited signals from discrete and noisy data is studied the reconstruction schemes employing cardinal expansions are proposed and their asymptotical properties are examined in particular the conditions for the convergence of the mean integrated squared error are found and the rate of convergence is evaluated the main difference between the proposed reconstruction scheme and the classical one is in treating the sampling rate and the reconstruction rate differently this distinction is necessary to ensure consistency of the reconstruction scheme in the presence of noise

the whitham equations for optical communications mathematical theory of nrz we present a model of optical communication system for high bit rate data transmission in the nonreturn to zero nrz format over transoceanic distance the system operates in a small group velocity dispersion regime and the model equation is given by the well known whitham equations describing the slow modulation of multiphase wavetrains of the defocusing nonlinear schrodinger nls equation the model equation is of hyperbolic type and nrz pulse with certain initial phase modulation develops a shock we then show how one can obtain a global solution by choosing an appropriate riemann surface on which the whitham equation is defined we also discuss the effect of third order dispersion by using an integrable hierarchy of the nls equation and we give a condition to avoid a shock formation

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

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

optical solitons in a monomode fiber we discuss the propagation of optical solitons in a monomode fiber as a model of long distance high bit rate transmission system we give several new results which did not appear in our previous papers on this subject such as 1 a derivation of the perturbed nonlinear schrodinger equation from the maxwell equation 2 on the integrability of the perturbed nonlinear schrodinger equation 3 a discussion of the soliton as a stable fixed point of certain infinite dimensional map generated by a transmission system with periodic excitations

diverging probability density functions for flat top solitary waves we investigate the statistics of flat top solitary wave parameters in the presence of weak multiplicative dissipative disorder we consider first propagation of solitary waves of the cubic quintic nonlinear schrodinger equation cqnlse in the presence of disorder in the cubic nonlinear gain we show by a perturbative analytic calculation and by monte carlo simulations that the probability density function pdf of the amplitude eta exhibits loglognormal divergence near the maximum possible amplitude eta m a behavior that is similar to the one observed earlier for disorder in the linear gain a peleg phys rev e 72 027203 2005 we relate the loglognormal divergence of the amplitude pdf to the superexponential approach of eta to eta m in the corresponding deterministic model with linear nonlinear gain furthermore for solitary waves of the derivative cqnlse with weak disorder in the linear gain both the amplitude and the group velocity beta become random we therefore study analytically and by monte carlo simulations the pdf of the parameter p where p eta 1 epsilon s beta 2 and epsilon s is the self steepening coefficient our analytic calculations and numerical simulations show that the pdf of p is loglognormally divergent near the maximum p value