the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

planning the size of survival time clinical trials with allowance for stratification approximate allocation designs for comparing two survival distributions are considered and compared with the exact solution the influence of incorporating covariates by stratification accrual and follow up duration on trial size and power is investigated the bernstein and lagakos design is suited for exponential survival due to its generality and allowance for stratification while the freedman design is convenient for distribution free survival in view of its simplicity with other specifications fixed the saving in sample size becomes minimal as the accrual period or post accrual follow up time units expand above three required sample size also increases dramatically when the number of strata exceeds three

the asymptotic properties of nonparametric tests for comparing survival distributions the asymptotic distribution under alternative hypotheses is derived for a class of statistics used to test the equality of two survival distributions in the presence of arbitrary and possibly unequal right censoring the test statistics include equivalents to the log rank statistic the modified wilcoxon statistic and the class of rank invariant test procedures introduced by peto peto when there are equal censoring distributions and the hazard functions are proportional the sample size formula for the f test used to compare exponential samples is shown to be valid for the log rank test in certain situations the power of the log rank test falls as the amount of censoring decreases

moving breathers and breather to soliton conversions for the hirota equation we find that the hirota equation admits breather to soliton conversion at special values of the solution eigenvalues this occurs for the first order as well as higher orders of breather solutions an analytic expression for the condition of the transformation is given and several examples of transformations are presented the values of these special eigenvalues depend on two free parameters that are present in the hirota equation we also find that higher order breathers generally have complicated quasi periodic oscillations along the direction of propagation various breather solutions are considered including the particular case of second order breathers of the nonlinear schrodinger equation

doubly periodic solutions of the class i infinitely extended nonlinear schrodinger equation we present doubly periodic solutions of the infinitely extended nonlinear schrodinger equation with an arbitrary number of higher order terms and corresponding free real parameters solutions have one additional free variable parameter that allows one to vary periods along the two axes the presence of infinitely many free parameters provides many possibilities in applying the solutions to nonlinear wave evolution being general this solution admits several particular cases which are also given in this article

general tensor discriminant analysis and gabor features for gait recognition traditional image representations are not suited to conventional classification methods such as the linear discriminant analysis lda because of the undersample problem usp the dimensionality of the feature space is much higher than the number of training samples motivated by the successes of the two dimensional lda 2dlda for face recognition we develop a general tensor discriminant analysis gtda as a preprocessing step for lda the benefits of gtda compared with existing preprocessing methods such as the principal components analysis pca and 2dlda include the following 1 the usp is reduced in subsequent classification by for example lda 2 the discriminative information in the training tensors is preserved and 3 gtda provides stable recognition rates because the alternating projection optimization algorithm to obtain a solution of gtda converges whereas that of 2dlda does not we use human gait recognition to validate the proposed gtda the averaged gait images are utilized for gait representation given the popularity of gabor function based image decompositions for image understanding and object recognition we develop three different gabor function based image representations 1 gabord is the sum of gabor filter responses over directions 2 gabors is the sum of gabor filter responses over scales and 3 gaborsd is the sum of gabor filter responses over scales and directions the gabord gabors and gaborsd representations are applied to the problem of recognizing people from their averaged gait images a large number of experiments were carried out to evaluate the effectiveness recognition rate of gait recognition based on first obtaining a gabor gabord gabors or gaborsd image representation then using gdta to extract features and finally using lda for classification the proposed methods achieved good performance for gait recognition based on image sequences from the university of south florida usf humanid database experimental comparisons are made with nine state of the art classification methods in gait recognition

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

nonautonomous discrete rogue wave solutions and interactions in an inhomogeneous lattice with varying coefficients abstract in this paper we analytically investigate nonautonomous discrete rogue wave solutions and their interactions in the generalized ablowitz ladik hirota lattice with variable coefficients which possess complicated wave propagation in time and differ from the usual discrete rogue waves when the amplitude of the tunnel coupling coefficient between sites decreases these nonautonomous discrete rogue wave solutions become localized in time after they propagate over some certain large critical values moreover we find that the interactions between nonautonomous discrete rogue wave solutions are elastic

semi foldon fission and fusion in the 2 1 dimensional higher order broer kaup system we obtain the general variable separation solution of the 2 1 dimensional higher order broer kaup system via the separation variable method and study semi foldon fission and fusion based on this variable separation solution

gauge transformation and symmetries of the commutative multicomponent bkp hierarchy in this paper we defined a new multi component b type kadomtsev petviashvili bkp hierarchy that takes values in a commutative subalgebra of after this we give the gauge transformation of this commutative multicomponent bkp cmbkp hierarchy meanwhile we construct a new constrained cmbkp hierarchy that contains some new integrable systems including coupled kdv equations under a certain reduction after this the quantum torus symmetry and quantum torus constraint on the tau function of the commutative multi component bkp hierarchy will be constructed

multi rogue waves for a higher order nonlinear schrodinger equation in optical fibers in this paper the rational solutions with arbitrary constants of a hirota equation are presented employing the darboux transformations then the similarity transformation allows us to relate certain class of multi rogue wave solutions of the higher order nonlinear schrodinger equation nlse to the solutions of 1 1 dimensional integrable hirota equation the multi rogue wave solutions with arbitrary constants are considered the obtained rogue wave solutions can be used to describe the possible formation mechanisms for optical and bose einstein condensates

darboux transformation and rogue wave solutions for the variable coefficients coupled hirota equations abstract we construct the lax pair and darboux transformation for the variable coefficients coupled hirota equations based on modulation instability and by taking the limit approach we derive two types of nth order rogue wave solutions with different dynamic structures in compact determinant representations the explicit first order rogue wave solution is presented prolific vector rogue wave patterns such as the dark bright composite three sister quadruple and sextuple rogue waves with multiple compression points are demonstrated in particular in contrast to the standard peregrine combs unusual vector rogue wave combs such as the dark bright and composite rogue wave combs are revealed by choosing sufficiently large periodic modulation amplitudes further some wave characteristics such as the difference between light intensity and continuous wave background and pulse energy evolution of the dark rogue wave solution that features multiple compression points are discussed in detail

rogue waves of the hirota and the maxwell bloch equations in this paper we derive a darboux transformation of the hirota and the maxwell bloch h mb system which is governed by femtosecond pulse propagation through an erbium doped fiber and further generalize it to the matrix form of the n fold darboux transformation of this system this n fold darboux transformation implies the determinant representation of n th solutions of e n p n ensuremath eta n generated from the known solution of e p ensuremath eta the determinant representation of e n p n ensuremath eta n provides soliton solutions positon solutions and breather solutions both bright and dark breathers of the h mb system from the breather solutions we also construct a bright and dark rogue wave solution for the h mb system which is currently one of the hottest topics in mathematics and physics surprisingly the rogue wave solution for p and ensuremath eta has two peaks because of the order of the numerator and denominator of them meanwhile after fixing the time and spatial parameters and changing two other unknown parameters ensuremath alpha and ensuremath beta we generate a rogue wave shape

on the continuum limit for a semidiscrete hirota equation in this paper we propose a new semidiscrete hirota equation which yields the hirota equation in the continuum limit we focus on the topic of how the discrete space step affects the simulation for the soliton solution to the hirota equation the darboux transformation and explicit solution for the semidiscrete hirota equation are constructed we show that the continuum limit for the semidiscrete hirota equation including the lax pair the darboux transformation and the explicit solution yields the corresponding results for the hirota equation as formula see text

the transmission property of the discrete heisenberg ferromagnetic spin chain we present a mechanism for displaying the transmission property of the discrete heisenberg ferromagnetic dhf spin chain via a geometric approach with the aid of a discrete nonlinear schrodinger like equation which is the discrete gauge equivalent to the dhf we show that the determination of transmitting coefficients in the transmission problem is always bistable thus a definite algorithm and general stochastic algorithms are presented a new invariant periodic phenomenon of the nontransmitting behavior for the dhf with a large probability is revealed by an adoption of various stochastic algorithms

periodic weak solutions for a classical one dimensional isotropic biquadratic heisenberg spin chain abstract in this paper we consider the model proposed by lakshmanan porsezian and daniel m lakshmanan k porsezian m daniel effect of discreteness on the continuum limit of the heisenberg spin chain phys lett a 133 1988 483 488 describing the continuum isotropic biquadratic heisenberg spin chain in one dimensional case following the results of integrability in k porsezian m daniel m lakshmanan on the integrability aspects of the one dimensional classical continuum isotropic heisenberg spin chain j math phys 33 5 1992 1807 1816 we prove the existence of periodic weak solutions to this model

deformations of third order peregrine breather solutions of the nonlinear schrodinger equation with four parameters we present a new representation of solutions of the one dimensional nonlinear focusing schr odinger equation nls as a quotient of two determinants this formulation gives in the case of the order 3 new solutions with four parameters this gives a very efficient procedure to construct families of quasirational solutions of the nls equation and to describe the apparition of multirogue waves with this method we construct analytical expressions of four parameters solutions when all these parameters are equal to 0 we recover the peregrine breather of order 3 it makes possible with this four parameters representation to generate all the types of patterns for the solutions like the triangular configurations or the ring structures

rogue waves in a resonant erbium doped fiber system with higher order effects we mainly investigate a coupled system of the generalized nonlinear schrodinger equation and the maxwell bloch equations which describes the wave propagation in an erbium doped nonlinear fiber with higher order effects including the forth order dispersion and quintic non kerr nonlinearity we derive the one fold darboux transformation of this system and construct the determinant representation of the n fold darboux transformation then the determinant representation of the nth new solutions en pn n which were generated from the known seed solutions e p is established through the n fold darboux transformation the solutions en pn n provide the bright and dark breather solutions of this system furthermore we construct the determinant representation of the nth order bright and dark rogue waves by taylor expansions and also discuss the hybrid solutions which are the nonlinear superposition of the rogue wave and breather solutions

the rogue wave and breather solution of the gerdjikov ivanov equation the gerdjikov ivanov gi system of q and r is defined by a quadratic polynomial spectral problem with 2 2 matrix coefficients each element of the matrix of n fold darboux transformation dt for this system is expressed by a ratio of n 1 n 1 determinant and n n determinant of eigenfunctions which implies the determinant representation of q n and r n generated from known solution q and r by choosing some special eigenvalues and eigenfunctions according to the reduction conditions q n r n the determinant representation of q n provides new solutions of the gi equation as examples the breather solutions and rogue wave of the gi are given explicitly by the two fold dt from a periodic seed with a constant amplitude

statistical shape analysis clustering learning and testing using a differential geometric treatment of planar shapes we present tools for 1 hierarchical clustering of imaged objects according to the shapes of their boundaries 2 learning of probability models for clusters of shapes and 3 testing of newly observed shapes under competing probability models clustering at any level of hierarchy is performed using a minimum variance type criterion and a markov process statistical means of clusters provide shapes to be clustered at the next higher level thus building a hierarchy of shapes using finite dimensional approximations of spaces tangent to the shape space at sample means we implicitly impose probability models on the shape space and results are illustrated via random sampling and classification hypothesis testing together hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval examples are presented using shapes and images from eth surrey and amcom databases

landau lifshitz and higher order nonlinear systems gauge generated from nonlinear schrodinger type equations new landau lifshitz ll and higher order nonlinear systems gauge generated from nonlinear schrodinger ns type equations are presented the consequences of gauge equivalence between different dynamical systems are discussed the gauge connections among various ll and ns equations are found and depicted through a schematic representation

on general rogue waves in the parity time symmetric nonlinear schrodinger equation abstract this article addresses the question of general rogue wave solutions in the nonlocal parity time symmetric nonlinear schrodinger equation by generalizing the previous bilinear method large classes of rogue waves are derived as gram determinants with schur polynomial elements it is shown that these rogue waves contain previously reported ones as special cases more importantly they contain many new rogue wave families it is conjectured that the rogue waves derived in this article are all rogue wave solutions in the parity time symmetric nonlinear schrodinger equation

the vortex filament in the minkowski 3 space and generalized bi schrodinger maps by extending the concept of generalized bi schrodinger maps to the case that the target manifold is a para kahler manifold we show that the third order timelike and spacelike correction models of the vortex filament in the minkowski 3 space are equivalent to the generalized bi schrodinger maps from to the hyperbolic 2 space and the de sitter 2 space respectively as a consequence all three typical second to fourth order integrable systems of the akns hierarchy are interpreted in a unified way in terms of generalized bi schrodinger maps based on this exploitation and a general discussion of the generalized bi schodinger maps from into an arbitrary riemannian surface we reveal a property of generalized bi schrodinger maps that is not admitted for schrodinger maps

akns and nls hierarchies mrw solutions pn breathers and beyond we describe a unified structure of rogue wave and multiple rogue wave solutions for all equations of the ablowitz kaup newell segur akns hierarchy and their mixed and deformed versions the definition of the akns hierarchy and its deformed versions is given in the sec ii we also consider the continuous symmetries of the related equations and the related spectral curves this work continues and summarises some of our previous studies dedicated to the rogue wave like solutions associated with akns nonlinear schrodinger and kp hierarchies the general scheme is illustrated by the examples of small rank n n 7 rational or quasi rational solutions in particular we consider rank 2 and rank 3 quasi rational solutions that can be used for prediction and modeling of the rogue wave events in fiber optics hydrodynamics and many other branches of science we describe a unified structure of rogue wave and multiple rogue wave solutions for all equations of the ablowitz kaup newell segur akns hierarchy and their mixed and deformed versions the definition of the akns hierarchy and its deformed versions is given in the sec ii we also consider the continuous symmetries of the related equations and the related spectral curves this work continues and summarises some of our previous studies dedicated to the rogue wave like solutions associated with akns nonlinear schrodinger and kp hierarchies the general scheme is illustrated by the examples of small rank n n 7 rational or quasi rational solutions in particular we consider rank 2 and rank 3 quasi rational solutions that can be used for prediction and modeling of the rogue wave events in fiber optics hydrodynamics and many other branches of science

complex form reduction and lie poisson structure for the nonlinearized spectral problem of the heisenberg hierarchy abstract in this paper the relation between the different restricted systems associated with the heisenberg magnetic hm equation is studied by the reduction procedure with the help of a lie group homomorphism of su 2 into so 3 the euler rodriguez type parameters are introduced to generate new finite dimensional integrable system it has shown that the resulting system which is the nonlinearized spectral problem of hm hierarchy on c2n is a hamiltonian system in complex form further poisson reduction and lie poisson structure are derived by the method of invariants the reduced system is found to be a hamiltonian system on the orbit space c2n tn r3n coinciding with the nonlinearized lenard spectral problem moreover the fully reduced systems on the leaves of the symplectic foliation are also given specifically the reduction extended to the common level set of the complex cones is the usual 2 2 nonlinearized spectral problem finally the integrability of the system with lie poisson structure is proven by making use of the so 3 symmetry

sample size formula for the proportional hazards regression model a formula is derived for determining the number of observations necessary to test the equality of two survival distributions when concomitant information is incorporated this formula should be useful in designing clinical trials with a heterogeneous patient population schoenfeld 1981 biometrika 68 316 319 derived the asymptotic power of a class of statistics used to test the equality of two survival distributions that result is extended to the case where concomitant information is available for each individual and where the proportional hazards model holds the loss of efficiency caused by ignoring concomitant variables is also computed

crum transformation and rational solutions of the non focusing nonlinear schrodinger equation a crum transformation for the linear problem of the non focusing nonlinear schrodinger nls equation is used to construct the sequence of rational solutions given in terms of tau functions expressed as double wronskian determinants the rational solutions of the adler kaup newell segur akns hierarchy have already been obtained in the work of sachs using the equivalence with the classical boussinesq hierarchy however we find that sachs description of the restrictions that must be placed on the akns tau functions in order to obtain nls rational solutions is not entirely correct we briefly comment on the constrained calogero moser systems associated with the nls rational solutions and indicate how the calogero moser equations can be obtained from a trilinear equation for the tau function which arises by reduction of the kadomtsev petviashvili kp hierarchy

rational solitons of wave resonant interaction models integrable models of resonant interaction of two or more waves in 1 1 dimensions are known to be of applicative interest in several areas here we consider a system of three coupled wave equations which includes as special cases the vector nonlinear schrodinger equations and the equations describing the resonant interaction of three waves the darboux dressing construction of soliton solutions is applied under the condition that the solutions have rational or mixed rational exponential dependence on coordinates our algebraic construction relies on the use of nilpotent matrices and their jordan form we systematically search for all bounded rational mixed rational exponential solutions and find a broad family of such solutions of the three wave resonant interaction equations

invariants and reduced matrix elements associated with the lie superalgebra gl m n we construct explicit formulae for the eigenvalues of certain invariants of the lie superalgebra gl m n using characteristic identities we discuss how such eigenvalues are related to reduced wigner coefficients and the reduced matrix elements of generators and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra

matching shape sequences in video with applications in human movement analysis we present an approach for comparing two sequences of deforming shapes using both parametric models and nonparametric methods in our approach kendall s definition of shape is used for feature extraction since the shape feature rests on a non euclidean manifold we propose parametric models like the autoregressive model and autoregressive moving average model on the tangent space and demonstrate the ability of these models to capture the nature of shape deformations using experiments on gait based human recognition the nonparametric model is based on dynamic time warping we suggest a modification of the dynamic time warping algorithm to include the nature of the non euclidean space in which the shape deformations take place we also show the efficacy of this algorithm by its application to gait based human recognition we exploit the shape deformations of a person s silhouette as a discriminating feature and provide recognition results using the nonparametric model our analysis leads to some interesting observations on the role of shape and kinematics in automated gait based person authentication

silhouette based gait recognition using procrustes shape analysis and elliptic fourier descriptors this paper presents a gait recognition method which combines spatio temporal motion characteristics statistical and physical parameters referred to as stm spp of a human subject for its classification by analysing shape of the subject s silhouette contours using procrustes shape analysis psa and elliptic fourier descriptors efds stm spp uses spatio temporal gait characteristics and physical parameters of human body to resolve similar dissimilarity scores between probe and gallery sequences obtained by psa a part based shape analysis using efds is also introduced to achieve robustness against carrying conditions the classification results by psa and efds are combined resolving tie in ranking using contour matching based on hu moments experimental results show stm spp outperforms several silhouette based gait recognition methods

rogue waves and rational solutions of the hirota equation the hirota equation is a modified nonlinear schrodinger equation nlse that takes into account higher order dispersion and time delay corrections to the cubic nonlinearity in describing wave propagation in the ocean and optical fibers it can be viewed as an approximation which is more accurate than the nlse we have modified the darboux transformation technique to show how to construct the hierarchy of rational solutions of the hirota equation we present explicit forms for the two lower order solutions each one is a regular nonsingular rational solution with a single maximum that can describe a rogue wave in this model numerical simulations reveal the appearance of these solutions in a chaotic field generated from a perturbed continuous wave solution

on the integrability aspects of the one dimensional classical continuum isotropic biquadratic heisenberg spin chain the integrability aspects of a classical one dimensional continuum isotropic biquadratic heisenberg spin chain in its continuum limit up to order o a4 in the lattice parameter a are studied through a differential geometric approach the dynamical equation for the spin chain is expressed in the form of a higher order generalized nonlinear schrodinger equation gnlse an integrable biquadratic chain that is a deformation of the lower order continuum isotropic spin chain is identified by carrying out a painleve singularity structure analysis on the gnlse also through gauge analysis and its properties are discussed briefly for the nonintegrable chain the perturbed soliton solution is obtained through a multiple scale analysis

recognizing gaits across views through correlated motion co clustering human gait is an important biometric feature which can be used to identify a person remotely however view change can cause significant difficulties for gait recognition because it will alter available visual features for matching substantially moreover it is observed that different parts of gait will be affected differently by view change by exploring relations between two gaits from two different views it is also observed that a part of gait in one view is more related to a typical part than any other parts of gait in another view a new method proposed in this paper considers such variance of correlations between gaits across views that is not explicitly analyzed in the other existing methods in our method a novel motion co clustering is carried out to partition the most related parts of gaits from different views into the same group in this way relationships between gaits from different views will be more precisely described based on multiple groups of the motion co clustering instead of a single correlation descriptor inside each group a linear correlation between gait information across views is further maximized through canonical correlation analysis cca consequently gait information in one view can be projected onto another view through a linear approximation under the trained cca subspaces in the end a similarity between gaits originally recorded from different views can be measured under the approximately same view comprehensive experiments based on widely adopted gait databases have shown that our method outperforms the state of the art

vector rogue waves in the manakov system diversity and compossibility we employ a nonrecursive darboux transformation formalism for obtaining a hierarchy of rogue wave solutions to the focusing vector nonlinear schrodinger equations manakov system the exact explicit rogue wave solutions up to the second order are presented typical rogue wave patterns such as peregrine type triple quadruple and sextuple vector rogue waves either bright dark or bright bright in their respective components are put forward despite the diversity there exists a universal compossibility that different rogue wave states could coexist for the same background parameters it is also shown that the higher order rogue wave hierarchy can indeed be thought of as a nonlinear superposition of a fixed well prescribed number of fundamental rogue waves these results may help understand the protean rogue wave manifestations in areas ranging from hydrodynamics to nonlinear optics

mpca multilinear principal component analysis of tensor objects this paper introduces a multilinear principal component analysis mpca framework for tensor object feature extraction objects of interest in many computer vision and pattern recognition applications such as 2d 3d images and video sequences are naturally described as tensors or multilinear arrays the proposed framework performs feature extraction by determining a multilinear projection that captures most of the original tensorial input variation the solution is iterative in nature and it proceeds by decomposing the original problem to a series of multiple projection subproblems as part of this work methods for subspace dimensionality determination are proposed and analyzed it is shown that the mpca framework discussed in this work supplants existing heterogeneous solutions such as the classical principal component analysis pca and its 2d variant 2d pca finally a tensor object recognition system is proposed with the introduction of a discriminative tensor feature selection mechanism and a novel classification strategy and applied to the problem of gait recognition results presented here indicate mpca s utility as a feature extraction tool it is shown that even without a fully optimized design an mpca based gait recognition module achieves highly competitive performance and compares favorably to the state of the art gait recognizers

generating mechanism for higher order rogue waves we introduce a mechanism for generating higher order rogue waves hrws of the nonlinear schrodinger nls equation the progressive fusion and fission of n degenerate breathers associated with a critical eigenvalue 0 creates an order n hrw by adjusting the relative phase of the breathers in the interacting area it is possible to obtain different types of hrws the value 0 is a zero point of an eigenfunction of the lax pair of the nls equation and it corresponds to the limit of the period of the breather tending to infinity by employing this mechanism we prove two conjectures regarding the total number of peaks as well as a decomposition rule in the circular pattern of an order n hrw

silhouette analysis based gait recognition for human identification human identification at a distance has recently gained growing interest from computer vision researchers gait recognition aims essentially to address this problem by identifying people based on the way they walk in this paper a simple but efficient gait recognition algorithm using spatial temporal silhouette analysis is proposed for each image sequence a background subtraction algorithm and a simple correspondence procedure are first used to segment and track the moving silhouettes of a walking figure then eigenspace transformation based on principal component analysis pca is applied to time varying distance signals derived from a sequence of silhouette images to reduce the dimensionality of the input feature space supervised pattern classification techniques are finally performed in the lower dimensional eigenspace for recognition this method implicitly captures the structural and transitional characteristics of gait extensive experimental results on outdoor image sequences demonstrate that the proposed algorithm has an encouraging recognition performance with relatively low computational cost

modulation instability the beginning abstract we discuss the early history of an important field of sturm and drang in modern theory of nonlinear waves it is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability a phenomenon resulting in a variety of nonlinear processes such as envelope solitons envelope shocks freak waves etc examples from water wave hydrodynamics electrodynamics nonlinear optics and convection theory are given

infinite hierarchy of nonlinear schrodinger equations and their solutions we study the infinite integrable nonlinear schrodinger equation hierarchy beyond the lakshmanan porsezian daniel equation which is a particular fourth order case of the hierarchy in particular we present the generalized lax pair and generalized soliton solutions plane wave solutions akhmediev breathers kuznetsov ma breathers periodic solutions and rogue wave solutions for this infinite order hierarchy we find that even order equations in the set affect phase and stretching factors in the solutions while odd order equations affect the velocities hence odd order equation solutions can be real functions while even order equation solutions are always complex

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

dynamic behaviors of interaction solutions of 3 1 dimensional shallow water wave equation abstract rational solutions and interaction solutions of the 3 1 dimensional shallow water wave equation are obtained depending on n soliton solution by the hirota bilinear method and long wave limit method and breather lump solutions and line rogue waves are obtained from two soliton solution as well as their interaction solutions are derived from four soliton solution after the collision breathers and lumps remain unchanged but the line rogue wave has a difference furthermore three soliton solution is discussed the interactions between some hybrid solutions such as hybrid solution between the stripe soliton and breather hybrid solution between stripe soliton and lump solution and so on are demonstrated in detail more interestingly stripe soliton remain unchanged but the amplitudes of the periodic line waves lump solutions and rogue waves have a great change after the collision