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

rational solutions and their interaction solutions of the 2 1 dimensional modified dispersive water wave equation abstract a bilinear form for the modified dispersive water wave mdww equation is presented by the truncated painleve series which does not lead to lump solutions in order to get lump solutions a pair of quartic linear forms for the mdww equation is constructed by selecting a suitable seed solution of the mdww equation in the truncated painleve series rational solutions are then computed by searching for positive quadratic function solutions a regular nonsingular rational solution can describe a lump in this model by combining quadratic functions with exponential functions some novel interaction solutions are founded including interaction solutions between a lump and a one kink soliton a bi lump and a one stripe soliton and a bi lump and a two stripe soliton concrete lumps and their interaction solutions are illustrated by 3d plots and contour plots

chaos in one dimensional collision complexes the classical scattering of a particle by an oscillating potential well is investigated it is shown that the singularities of the scattering map s form a cantor set for this system this result is obtained from an analysis of the particle s return map which contains a smale s horseshoe and consequently a hyperbolic invariant set with chaotic dynamics the cantor set structure of the s singularities involves arbitrarily complex scattering behaviour

painleve classification of a generalized coupled hirota system by means of singularity analysis the integrability of a system of generalized coupled hirota equations with parameter coefficients is tested it is proven that the system passes the painleve test for integrability only in two distinct cases

new exact periodic solitary wave solutions for the 3 1 dimensional generalized kp and bkp equations the extended homoclinic test approach ehta is a straightforward and efficient method to solve nonlinear partial differential equations in this paper new exact solutions including kinky periodic solitary wave solutions periodic soliton solutions and periodic kink solutions for the 3 1 dimensional generalized kadomtsev petviashvili kp and b type kadomtsev petviashvili bkp equations are obtained moreover the strangely mechanical features of these wave solutions are investigated these solutions enrich the variety of the dynamics of higher dimensional nonlinear wave field

backlund transformations and loop group actions we construct a local action of the group of rational maps from 2 to gl n on local solutions of flows of the zs akns sl n hierarchy we show that the actions of simple elements linear fractional transformations give local backlund transformations and we derive a permutability formula from different factorizations of a quadratic element we prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities however the action of the subgroup of the rational maps that satisfy the u n reality condition on the space of global rapidly decaying solutions of the flows in the u n hierarchy is global and the action of a simple element gives a global backlund transformation the actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multisolitons multibreathers we show that this theory generalizes the classical backlund theory of the sine gordon equation the group structures of backlund transformations for various hierarchies are determined by their reality conditions we identify the reality conditions the group structures for the sl n u k n k kdv kupershmidt wilson and gel fand dikii hierarchies the actions of linear fractional transformations that satisfy a reality condition modulo the center of the group of rational maps give backlund and darboux transformations for the hierarchy defined by the reality condition since the factorization cannot always be carried out under this reality condition the action is again local and backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed 2000 john wiley sons inc

rogue waves rational solutions the patterns of their zeros and integral relations the focusing nonlinear schrodinger equation which describes generic nonlinear phenomena including waves in the deep ocean and light pulses in optical fibres supports a whole hierarchy of recently discovered rational solutions we present recurrence relations for the hierarchy the pattern of zeros for each solution and a set of integral relations which characterizes them

painleve analysis of the non linear schrodinger family of equations the authors apply the painleve test to the generalised derivative non linear schrodinger equation iut uxx iauu ux ibu2ux cu3u 2 where u denotes the complex conjugate of u and a b and c are real constants to determine under what conditions the equation might be completely integrable it is shown that apart from a trivial multiplicative factor this equation possesses the painleve property for partial differential equations as formulated by weiss tabor and carnevale 1983 only if c 1 4b 2b a when then this relation holds this is equivalent under a gauge transformation to the derivative non linear schrodinger equation dnls of kaup and newell which is known to be completely integrable or else to a linear equation

rational and semi rational solutions of the y nonlocal davey stewartson i equation abstract the nonlocal davey stewartson ds i equation with a parity time symmetric potential with respect to the y direction which is called the y nonlocal ds i equation is a two dimensional analogue of the nonlocal nonlinear schrodinger nls equation the multi breather solutions for the y nonlocal ds i equation are derived by using the hirota bilinear method lump type solutions and hybrid solutions consisting of lumps sitting on periodic line waves are generated by long wave limits of the obtained soliton solutions also various types of analytical solutions for the nonlocal nls equation with negative nonlinearity including both the akhmediev breathers and the peregrine rogue waves sitting on periodic line waves can be generated with appropriate constraints on the parameters of the obtained exact solutions of the y nonlocal ds i equation particularly we show that a family of hybrid solitons describing the peregrine rogue wave that coexists with the akhmediev breather both of them sitting on a spatially periodic background can be thus obtained

resonant behavior of multiple wave solutions to a hirota bilinear equation abstract for the exponential traveling wave solutions to the hirota bilinear equations a sufficient and necessary criterion for the existence of linear superposition principle has been given motivated by this criterion we propose a new hirota bilinear equation via using a multivariate polynomial applying the linear superposition principle to this new hirota bilinear equation we finally find two types of resonant multiple wave solutions among which the resonant three wave solutions are illustrated with three dimensional plots

other 2n 2 parameters solutions of the nls equation and 2n 1 highest amplitude of the modulus of the n th order ap breather in this paper we construct new deformations of the akhmediev peregrine ap breather of order n or apn breather with real parameters other families of quasirational solutions of the nonlinear schrodinger nls equation are obtained we evaluate the highest amplitude of the modulus of the ap breather of order n we give the proof that the highest amplitude of the apn breather is equal to we get new formulas for the solutions of the nls equation which are different from these already given in previous works new solutions for the order 8 and their deformations according to the parameters are explicitly given we simultaneously get triangular configurations and isolated rings moreover the appearance for certain values of the parameters and of new configurations of concentric rings are underscored

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

few cycle optical rogue waves complex modified korteweg de vries equation in this paper we consider the complex modified korteweg de vries mkdv equation as a model of few cycle optical pulses using the lax pair we construct a generalized darboux transformation and systematically generate the first second and third order rogue wave solutions and analyze the nature of evolution of higher order rogue waves in detail based on detailed numerical and analytical investigations we classify the higher order rogue waves with respect to their intrinsic structure namely fundamental pattern triangular pattern and ring pattern we also present several new patterns of the rogue wave according to the standard and nonstandard decomposition the results of this paper explain the generalization of higher order rogue waves in terms of rational solutions we apply the contour line method to obtain the analytical formulas of the length and width of the first order rogue wave of the complex mkdv and the nonlinear schr odinger equations in nonlinear optics the higher order rogue wave solutions found here will be very useful to generate high power few cycle optical pulses which will be applicable in the area of ultrashort pulse technology

some applications of symbolic dynamics techniques to toral skew endomorphisms a class of dynamical systems of the 2 torus is considered these systems have the form of a skew product between the bernoulli endomorphism defined on the 1 torus and a translation on itself symbolic dynamics techniques allow one to single out wide classes of observables which show an exponential decay of correlations for some observables the rate of correlation decay can be explicitly estimated

degenerate determinant representation of solutions of the nonlinear schrodinger equation higher order peregrine breathers and multi rogue waves we present a new representation of solutions of the focusing nls equation as a quotient of two determinants this work is based on a recent paper in which we have constructed a multi parametric family of solutions of this equation in terms of wronskians this formulation was written in terms of a limit involving a parameter here we give a very compact formulation without presence of a limit this is a completely new result which gives a very efficient procedure to construct families of quasi rational solutions of the nls equation with this method we construct peregrine breathers of orders n 4 6 and multi rogue waves associated by deformation of parameters

abundant lump type solutions of the jimbo miwa equation in 3 1 dimensions abstract based on the hirota bilinear form of the 3 1 dimensional jimbo miwa equation ten classes of its lump type solutions are generated via maple symbolic computations whose analyticity can be easily achieved by taking special choices of the involved parameters those solutions supplement the existing lump type solutions presented previously in the literature

devil staircase behavior of dynamical invariants in chaotic scattering a crisis in chaotic scattering is characterized by the merging of two or more nonattracting chaotic saddles the fractal dimension of the resulting chaotic saddle increases through the crisis we present a rigorous analysis for the behavior of dynamical invariants associated with chaotic scattering by utilizing a representative model system that captures the essential dynamical features of crisis our analysis indicates that the fractal dimension and other dynamical invariants are a devil staircase type of function of the system parameter our results can also provide insight for similar devil staircase behaviors observed in the parametric evolution of chaotic saddles of general dissipative dynamical systems and in communicating with chaos 2000 elsevier science b v all rights reserved

classifying the hierarchy of nonlinear schrodinger equation rogue wave solutions we present a systematic classification for higher order rogue wave solutions of the nonlinear schrodinger equation constructed as the nonlinear superposition of first order breathers via the recursive darboux transformation scheme this hierarchy is subdivided into structures that exhibit varying degrees of radial symmetry all arising from independent degrees of freedom associated with physical translations of component breathers we reveal the general rules required to produce these fundamental patterns consequently we are able to extrapolate the general shape for rogue wave solutions beyond order 6 at which point accuracy limitations due to current standards of numerical generation become non negligible furthermore we indicate how a large set of irregular rogue wave solutions can be produced by hybridizing these fundamental structures

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

universality for the focusing nonlinear schrodinger equation at the gradient catastrophe point rational breathers and poles of the tritronquee solution to painleve i the semiclassical zero dispersion limit of solutions to the one dimensional focusing nonlinear schrodinger equation nls is studied in a scaling neighborhood d of a point of gradient catastrophe x 0 t 0 we consider a class of solutions specified in the text that decay as the neighborhood d contains the region of modulated plane wave with rapid phase oscillations as well as the region of fast amplitude oscillations spikes in this paper we establish the following universal behaviors of the nls solutions q near the point of gradient catastrophe i each spike has height and uniform shape of the rational breather solution to the nls scaled to the size ii the location of the spikes is determined by the poles of the tritronquee solution of the painleve i p1 equation through an explicit map between d and a region of the painleve independent variable iii if but lies away from the spikes the asymptotics of the nls solution is given by the plane wave approximation with the correction term being expressed in terms of the tritronquee solution of p1 the relation with the conjecture of dubrovin grava and klein about the behavior of solutions to the focusing nls near a point of gradient catastrophe is discussed we conjecture that the p1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point our technique is based on the nonlinear steepest descent method for matrix riemann hilbert problems and discrete schlesinger isomonodromic transformations 2013 wiley periodicals inc

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

cantor set structures in the singularities of classical potential scattering a classical mechanical system is analysed which exhibits complicated scattering behaviour in the set of all incoming asymptotes there is a fractal subset on which the scattering angle is singular though in the complement of this cantor set the deflection function is regular one can choose impact parameter intervals leading to arbitrarily complicated trajectories the authors show how the complicated scattering behaviour is caused by unstable periodic orbits having homoclinic and heteroclinic connections thereby a hyperbolic invariant set is created leading to horseshoe chaos in the flow this invariant set contains infinitely many unstable localised orbits periodic and aperiodic ones the stable manifolds of these orbits reach out into the asymptotic region and create the singularities of the scattering function

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

fractal properties of scattering singularities we discuss properties of the set of scattering singularities in regions of irregular scattering we show how a symbolic organisation of the set can be used to determine the fractal dimension and the scaling function this yields information on the distribution of lyapunov exponents of bounded orbits the specific model studied is the motion of a particle in a plane elastically reflected by three circular discs centred on the corners of an equilateral triangle

extreme superposition rogue waves of infinite order and the painleve iii hierarchy we study the fundamental rogue wave solutions of the focusing nonlinear schrodinger equation in the limit of large order using a recently proposed riemann hilbert representation of the rogue wave solution of arbitrary order k we establish the existence of a limiting profile of the rogue wave in the large k limit when the solution is viewed in appropriate rescaled variables capturing the near field region where the solution has the largest amplitude the limiting profile is a new particular solution of the focusing nonlinear schrodinger equation in the rescaled variables the rogue wave of infinite order which also satisfies ordinary differential equations with respect to space and time the spatial differential equations are identified with certain members of the painleve iii hierarchy we compute the far field asymptotic behavior of the near field limit solution and compare the asymptotic formulas with the exact solution using numerical methods for solving riemann hilbert problems in a certain transitional region for the asymptotics the near field limit function is described by a specific globally defined tritronquee solution of the painleve ii equation these properties lead us to regard the rogue wave of infinite order as a new special function

on invariant closed curves for one step methods we show that a one step method as applied to a dynamical system with a hyperbolic periodic orbit exhibits an invariant closed curve for sufficiently small step size this invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit the dynamics of the one step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression our results complement those of 2 3 where one step methods were shown to create invariant curves if the dynamical system has a periodic orbit which is stable in either time direction or if the system undergoes a hopf bifurcation

stable attracting sets in dynamical systems and in their one step discretizations we consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set of arbitrary shape under the assumption of uniform asymptotic stability of in the sense of lyapunov we show that discretized versions of the dynamical system involving one step numerical methods have nearby attracting sets h which are also uniformly asymptotically stable our proof uses the properties of a lyapunov function which characterizes the stability of

a hierarchy of non linear evolution equations its hamiltonian structure and classical integrable system abstract an isospectral problem and a new corresponding hierarchy of non linear evolution equations are proposed as a reduction a new generalized non linear schrodinger equation is obtained it is shown that the hierarchy of equations possesses a bi hamiltonian structure the eigenvalue problem is non linearized so as to be a finite dimensional completely integrable system under a constraint between the potentials and the eigenfunctions

wronskian and grammian solutions to a 3 1 dimensional generalized kp equation abstract wronskian and grammian formulations are established for a 3 1 dimensional generalized kp equation based on the plucker relation and the jacobi identity for determinants generating functions for matrix entries satisfy a linear system of partial differential equations involving a free parameter examples of wronskian and grammian solutions are computed and a few particular solutions are plotted

structural stability of flows under numerics abstract in our previous paper siam j math anal 28 1997 381 388 we showed that the qualitative properties of a morse smale gradient like flow are preserved by its discretization mapping obtained via numerical methods in this paper we extend the result to flows which satisfy axiom a and the strong transversality condition we prove that if p 2 t is a c p 1 flow on a compact manifold satisfying axiom a and the strong transversality condition and n h is a numerical method of step size h and order p then for all sufficiently small h there are a homeomorphism h h and a continuous real valued function h on m such that h h h h h x x n h h h x and h h is o h p close to the identity map on m

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

localized waves and interaction solutions to a 3 1 dimensional generalized kp equation abstract based on the hirota bilinear method and long wave limit four kinds of localized waves namely solitons lumps breathers and rogue waves are constructed for the 3 1 dimensional generalized kp equation n soliton solutions are obtained by employing bilinear method then breathers two breathers and interaction breather solutions are obtained by selecting appropriate parameters on two soliton solution and four soliton solution these breathers own different dynamic behaviors in the different planes taking a long wave limit on the two and four soliton solutions under special parameter constraints one order lumps and rogue waves two order lumps and rogue waves and interaction solutions between lumps and rogue waves are derived applying the same method on the three soliton solution interaction solutions between kink solitons with periodic solutions lumps and rogue waves are constructed respectively the influence of parameters on the solution is analyzed the propagation directions phase shifts energies and shapes for these solutions can be affected and controlled by the parameters moreover graphics are presented to demonstrate the properties of the explicit analytical localized wave solutions

general high order rogue waves and their dynamics in the nonlinear schrodinger equation general high order rogue waves in the nonlinear schrodinger equation are derived by the bilinear method these rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions it is shown that the general n th order rogue waves contain n 1 free irreducible complex parameters in addition the specific rogue waves obtained by akhmediev et al akhmediev et al 2009 phys rev e 80 026601 doi 10 1103 physreve 80 026601 correspond to special choices of these free parameters and they have the highest peak amplitudes among all rogue waves of the same order if other values of these free parameters are taken however these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns

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

numerical study of a billiard in a gravitational field abstract billiards have always been used as models for mechanical systems in this paper we describe a very simple billiard which over a range of one continous parameter only exhibits the characteristics of hamiltonian systems having two degrees of freedom and a discontinuity the relationship between this billiard and the well known one dimensional self gravitating system with n 3 is given this billiard consists of a mass point moving in a symmetric wedge of angle 2 under the influence of a constant gravitational field for 45 complete chaos is obtained suggesting k system behavior the general properties of the mapping and some numerical results obtained are discussed of special interest are invariant curves which cross a line of discontinuity and a new universality class for lyapunov numbers

two parameters wronskian representation of solutions of nonlinear schrodinger equation eighth peregrine breather and multi rogue waves in this paper we present a representation of solutions of the one dimensional focusing nonlinear schrodinger equation as a quotient of two wronskians depending on two parameters here we give the complete proof of this representation we have already constructed peregrine breathers and their two parameter deformations until order 7 with this method the construction of the explicit analytical expressions of peregrine breather of order 8 was made for the first time when parameters a or b are equal to 0 we recover the peregrine breather of order 8 we obtain multi rogue waves by deformation of parameters a and b these expressions enable us to understand the evolution of the solutions in the case of order 8 it is shown for high values of parameters a or b the appearance of peregrine breather of order 6

simple determinant representation for rogue waves of the nonlinear schrodinger equation we present a simple representation for arbitrary order rogue wave solution and a study on the trajectories of them explicitly we find that the trajectories of two valleys on whole temporal spatial distribution all look x shaped for rogue waves additionally we present different types of high order rogue wave structures which could be helpful towards realizing the complex dynamics of rogue waves

six parameters deformations of fourth order peregrine breather solutions of the nonlinear schrodinger equation we construct solutions of the focusing nonlinear schrodinger nls equation as a quotient of two determinants this formulation gives in the case of the order 4 new deformations of the peregrine breather with 6 real parameters we construct families of quasi rational solutions of the nls equation and describe the apparition of multi rogue waves with this method we construct the analytical expressions of deformations of the peregrine breather of order 4 with 6 real parameters and plot different types of rogue waves

weakly noisy chaotic scattering received 30 april 2013 published 23 september 2013 the effect of a weak source of noise on the chaotic scattering is relevant to situations of physical interest we investigate how a weak source of additive uncorrelated gaussian noise affects both the dynamics and thetopology of a paradigmatic chaotic scattering problem as the one taking place in the open nonhyperbolic regimeof the h enon heiles hamiltonian system we have found long transients for the time escape distributions forcritical values of the noise intensity for which the particles escape slower as compared with the noiseless case an analysis of the survival probability of the scattering function versus the gaussian noise intensity shows asmooth curve with one local maximum and with one local minimum which are related to those long transientsand with the basin structure in phase space on the other hand the computation of the exit basins in phase spaceshows a quadratic curve for which the basin boundaries lose their fractal like structure as noise turned on doi 10 1103 physreve 88 032914 pacs number s 05

pruning induced phase transition observed by a scattering method in hyperbolic systems transient chaos is associated with an underlying chaotic saddle in phase space the structure of the chaotic saddle of a class of piecewise linear area preserving two dimensional maps with overall constant lyapunov exponents has been observed by a scattering method the free energy obtained in this way displays a phase transition at 0 in spite of the fact that no phase transition occurs in the free energy dedcued from the spectrum of lyapunov exponents this is possible because pruning introduces a second effective scaling exponent by creating at each level of the approximation particular small pieces in the incomplete cantor set approximating the saddle the second scaling arises for a subset of values of the control parameter that is dense in the parameter interval

families of quasi rational solutions of the nls equation and multi rogue waves we construct a multi parametric family of the solutions of the focusing nonlinear schr dinger equation nls from the known results describing the multi phase almost periodic elementary solutions given in terms of riemann theta functions we give a new representation of their solutions in terms of wronskians determinants of order 2n composed of elementary trigonometric functions when we perform a special passage to the limit when all the periods tend to infinity we obtain a family of quasi rational solutions this leads to efficient representations for the peregrine breathers of orders n 1 2 3 first constructed by akhmediev and his co workers and also allows us to obtain a simpler derivation of the generic formulas corresponding the three or six rogue wave formation in the frame of the nls model first explained by v b matveev in 2010 our formulation allows us to isolate easily the second or third order peregrine breathers from generic solutions and also to compute the peregrine breathers of orders 2 and 3 easily with respect to other approaches in the cases n 2 3 we obtain the comfortable formulas to study the deformation of a higher peregrine breather of order 2 to the three rogue wave or order 3 to the six rogue wave solutions via the variation of the free parameters of our construction

algebraic construction of the darboux matrix revisited we present algebraic construction of darboux matrices for 1 1 dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case we discuss different approaches to the darboux backlund transformation based on different dependences of the darboux matrix polynomial sum of partial fractions or the transfer matrix form we derive symmetric n soliton formulae in the general case the matrix spectral parameter and dressing actions in loop groups are also discussed we describe reductions to twisted loop groups unitary reductions the matrix lax pair for the kdv equation and reductions of chiral models harmonic maps to su n and to grassmann spaces we show that in the kdv case the nilpotent darboux matrix generates the binary darboux transformation the paper is intended as a review of known results usually presented in a novel context but some new results are included as well e g general compact formulae for n soliton surfaces and linear and bilinear constraints on the nonisospectral lax pair matrices which are preserved by darboux transformations

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

the darboux transformation of the kundu eckhaus equation we construct an analytical and explicit representation of the darboux transformation dt for the kundueckhaus ke equation such solution and n fold dt tn are given in terms of determinants whose

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

hyperbolicity in a double well potential we investigate the dynamics of a particle in a double well potential there is a critical energy greater than zero beyond which no localized motion is possible for certain energies smaller than the critical energy the system can be shown to have a nonempty bounded invariant set conditions are found under which this invariant set is hyperbolic

chaotic scattering by steep repelling potentials consider a classical two dimensional scattering problem a ray is scattered by a potential composed of several tall repelling steep mountains of arbitrary shape we study when the traditional approximation of this nonlinear far from integrable problem by the corresponding simpler billiard problem of scattering by hard wall obstacles of similar shape is justified for one class of chaotic scatterers named here regular sinai scatterers the scattering properties of the smooth system indeed limit to those of the billiards for another class the singular sinai scatterers these two scattering problems have essential differences though the invariant set of such singular scatterers is hyperbolic possibly with singularities that of the smooth flow may have stable periodic orbits even when the potential is arbitrarily steep it follows that the fractal dimension of the scattering function of the smooth flow may be significantly altered by changing the ratio between the steepness parameter and a parameter which measures the billiards deviation from a singular scatterer thus even in this singular case the billiard scattering problem is utilized as a skeleton for studying the properties of the smooth flow finally we see that corners have nontrivial and significant impact on the scattering functions

chaotic scattering modelled by an inclined billiard abstract chaotic scattering as observed in satellite encounters and in other problems is modelled by a simple billiard a point particle bounces on two disks and in addition is subjected to a constant acceleration in an appropriate limit this problem reduces to a piecewise linear mapping we consider a one parameter family of orbits obtained by starting the particle at rest from a constant height we find that this family exhibits chaotic scattering it has an infinite number of discontinuities forming a self similar cantor set with the help of symbolic dynamics the structure of the family can be completely elucidated

novel interaction phenomena of localized waves in the generalized 3 1 dimensional kp equation abstract based on hirota bilinear method the n soliton solution of the 3 1 dimensional generalized kadomtsev petviashvili equation is derived explicitly from which some localized waves such as soliton breather lump and their interactions are obtained by the approach of long wave limit especially by selecting particular parameter constraints in the n soliton solutions the one breather or one lump can be obtained from two soliton the elastic interaction solutions between one bell shaped soliton and one breather or between one bell shaped soliton and one lump can be obtained from three soliton the elastic interaction solutions among two bell shaped solitons and one breather among two bell shaped solitons and one lump between two breathers or between two lumps can be obtained from four soliton the elastic interactions solutions among one bell shaped soliton and two breathers among one breather and three bell shaped solitons among one lump and three bell shaped solitons among one bell shaped soliton and two breathers or among one breather one lump and one bell shaped soliton can be obtained from five soliton detailed behaviors of such interaction phenomena are illustrated analytically and graphically the results obtained in this paper may be helpful for understanding the evolution of nonlinear localized waves in shallow water

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

new periodic wave cross kink wave and the interaction phenomenon for the jimbo miwa like equation abstract by means of symbolic computation with the help of maple diverse of exact solutions for the jimbo miwa like equation are successfully derived based on the generalized bilinear method these new solutions named the periodic wave and cross kink wave have greatly enriched the existing literature on the jimbo miwa like equation particularly we have obtained the interaction solutions between the lump solution and an arbitrary function for the jimbo miwa like equation as long as the researcher needs it he can choose an arbitrary function to observe the collision phenomenon which he wants between the lump and the solitary waves via the three dimensional images and density images with the help of maple the physical characteristics of these waves are described very well that will be widely used to describe many interesting physical phenomena in the fields of gas plasma optics acoustics heat transfer fluid dynamics classical mechanics and so on

the n soliton solution and localized wave interaction solutions of the 2 1 dimensional generalized hirota satsuma ito equation abstract in this paper the n soliton solution is constructed for the 2 1 dimensional generalized hirota satsuma ito equation from which some localized waves such as line solitons lumps periodic solitons and their interactions are obtained by choosing special parameters especially by selecting appropriate parameters on the multi soliton solutions the two soliton can reduce to a periodic soliton or a lump soliton the three soliton can reduce to the elastic interaction solution between a line soliton and a periodic soliton or the elastic interaction between a line soliton and a lump soliton while the four soliton can reduce to elastic interaction solutions among two line solitons and a periodic soliton or the elastic interaction ones between two periodic solitons detailed behaviours of such solutions are illustrated analytically and graphically by analysing the influence of parameters finally an inelastic interaction solution between a lump soliton and a line soliton is constructed via the ansatz method and the relevant interaction and propagation characteristics are discussed graphically the results obtained in this paper may be helpful for understanding the interaction phenomena of localized nonlinear waves in two dimensional nonlinear wave equations

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

the lump solutions and the backlund transformation for the three dimensional three wave resonant interaction a backlund transformation is found for the three dimensional three wave resonant interaction and from it n lump exact solutions may be constructed the one lump solution is analyzed in detail and it is shown that it describes such effects as pulse decay upconversion and explosive instabilities all in three dimensions

fractal properties of escape from a two dimensional potential abstract this paper summarizes a numerical investigation of the escape of particles from the two dimensional potential v x y 1 2 x 2 1 2 y 2 x 2 y 2 for variable but fixed energy h 0 12 for 1 1 4h 2 08 escape is impossible but for 1 particles will escape for such larger the final outcome as characterized by the time and direction in which particles escape is a smooth function of initial conditions in certain phase space regions however in other regions the evolution evidences an extremely sensitive dependence on initial conditions indicative of a complex microscopic evolution for sufficiently large values of 2 4 90 0 01 these sensitive regions appear to exhibit a fractal structure with simple scaling properties implying that at a coarse grained level the late time behavior is quite regular specifically the evolution is asymptotically markovian in the sense that for particles that have not yet escaped at late times the coarse grained escape probabilities per unit time tend toward time independent values p 0 independent of the initial conditions the asymptotic values p and the rate of convergence towards these values were examined as a function of and the scale of the coarse graining and were observed to exhibit simple scaling behaviour at least for values of 5 7 p 2 with a critical exponent 0 49 0 05 for a coarse graining corresponding to a linear scale r 0 05 the time t required for convergence towards this value scales as t 2 with 0 39 0 14 0 06 the difference 0 09 0 04 0 02 the value p appears independent of r but the convergence time scales as t r r with 0 08 0 03 seemingly independent of to within statistical uncertainties 0

n solitons breathers lumps and rogue wave solutions to a 3 1 dimensional nonlinear evolution equation abstract based on hirota bilinear method n solitons breathers lumps and rogue waves as exact solutions of the 3 1 dimensional nonlinear evolution equation are obtained the impacts of the parameters on these solutions are analyzed the parameters can influence and control the phase shifts propagation directions shapes and energies for these solutions the single kink soliton solution and interactions of two and three kink soliton overtaking collisions of the hirota bilinear equation are investigated in different planes the breathers in three dimensions possess different dynamics in different planes via a long wave limit of breathers with indefinitely large periods rogue waves are obtained and localized in time it is shown that the rogue wave possess a growing and decaying line profile that arises from a nonconstant background and then retreat back to the same nonconstant background again the results can be used to illustrate the interactions of water waves in shallow water moreover figures are given out to show the properties of the explicit analytic solutions

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

akhmediev breathers ma solitons and general breathers from rogue waves a case study in the manakov system we present explicit forms of general breather gb akhmediev breather ab ma soliton ms and rogue wave rw solutions of the two component nonlinear schr odinger nls equation namely manakov equation we derive these solutions through two different routes in the forward route we first construct a suitable periodic envelope soliton solution to this model from which we derive gb ab ms and rw solutions we then consider the rw solution as the starting point and derive ab ms and gb in the reverse direction the second approach has not been illustrated so far for the two component nls equation our results show that the above rational solutions of the manakov system can be derived from the standard scalar nonlinear schr odinger equation with a modified nonlinearity parameter through this two way approach we establish a broader understanding of these rational solutions which will be of interest in a variety of situations

on structural stability of ordinary differential equations with respect to discretization methods on compact m dimensional discs morse smale differential systems having no periodic orbits are considered the main result is that they are correctly reproduced by one step discretization methods for methods of order p and stepsize h sufficiently small the time h map of the induced local flow and the h discretized system are joined by a conjugacy o h p near to the identity the paper fits well in the rapidly growing list of results stating that hyperbolic transversal structures are preserved by discretization the proof relies heavily on techniques elaborated by robbin 1971 in establishing his structural stability theorem on self diffeomorphisms of compact c infty manifolds

triangular rogue wave cascades by numerically applying the recursive darboux transformation technique we study high order rational solutions of the nonlinear schr odinger equation that appear spatiotemporally as triangular arrays of peregrine solitons these can be considered as rogue wave cascades and complement previously discovered circular cluster forms in this analysis we reveal a general parametric restriction for their existence and investigate the interplay between cascade and cluster forms as a result we demonstrate how to generate many more hybrid rogue wave solutions including semicircular clusters that resemble claws

on the numerical approximation of phase portraits near stationary points we show that the phase portrait of a dynamical system near a stationary hyperbolic point is reproduced correctly by numerical methods such as one step methods or multi step methods satisfying a strong root condition this means that any continuous trajectory can be approximated by an appropriate discrete trajectory and vice versa to the correct order of convergence and uniformly on arbitrarily large time intervals in particular the stable and unstable manifolds of the discretization converge to their continuous counterparts

resonances and transport in the sawtooth map we study transport in a completely chaotic hamiltonian system the hyperbolic sawtooth map analytical expressions are obtained for its cantori and resonances we show that resonances give a complete partition of phase space the flux leaking out of a resonance is given by its turnstiles whose form and areas are obtained analytically when the total flux out of a resonance becomes one third the area of an island the topology of the turnstiles changes at the same parameter value a horseshoe is formed corresponding to the orbits trapped within the resonance based on this a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained the partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles we calculate the survival probability within a resonance using the markov model the results are compared with results obtained numerically and from periodic orbit theory

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

generation of higher order rogue waves from multibreathers by double degeneracy in an optical fiber in this paper we construct a special kind of breather solution of the nonlinear schrodinger nls equation the so called breather positon b positon for short which can be obtained by taking the limit j 1 of the lax pair eigenvalues in the order n periodic solution which is generated by the n fold darboux transformation from a special seed solution plane wave further an order n b positon gives an order n rogue wave under a limit 1 0 here 0 is a special eigenvalue in a breather of the nls equation such that its period goes to infinity several analytical plots of order 2 breather confirm visually this double degeneration the last limit in this double degeneration can be realized approximately in an optical fiber governed by the nls equation in which an injected initial ideal pulse is created by a frequency comb system and a programable optical filter wave shaper according to the profile of an analytical form of the b positon at a certain position z 0 we also suggest a new way to observe higher order rogue waves generation in an optical fiber namely measure the patterns at the central region of the higher order b positon generated by above ideal initial pulses when 1 is very close to the 0 the excellent agreement between the numerical solutions generated from initial ideal inputs with a low signal to noise ratio and analytical solutions of order 2 b positon supports strongly this way in a realistic optical fiber system our results also show the validity of the generating mechanism of a higher order rogue waves from a multibreathers through the double degeneration

unstable manifolds for rfdes under discretization the euler method abstract a generalization of the well known unstable manifold theorem near hyperbolic equilibrium points of functional differential equations is formulated the result says that the unstable manifold of the functional differential equation is close to its discretized counterpart if the stepsize of the discretization is sufficiently small

persistence of invariant sets for dissipative evolution equations we show that results concerning the persistence of invariant sets of ordinary differential equations under perturbation may be applied directly to a certain class of partial differential equations our framework is particularly well suited to encompass numerical approximations of these partial differential equations specifically we show that for a class of pdes with ac 1 inertial form certain natural numerical approximations possess an inertial form close to that of the underlying pde in thec 1 norm

on spurious fixed points of runge kutta methods in this paper we investigate the onset of spurious fixed points when runge kutta methods are applied to study the dynamics of differential equations it is shown computationally that the spurious equilibria of griffithset al 14 are connected at infinity with fixed points inherited from the differential equation we introduce and study the concept of b regularity which is in connection to the concept of regularity introduced by iserles principium cuius hinc nobis exordia sumet nullam rem e nilo gigni diuinitus umquam nam si de nilo fierent ex omnibus rebus omne genus nasci posset nil semine egeret de rerum natura liber primus 149 lucrecio

second order nonlinear schrodinger equation breather solutions in the degenerate and rogue wave limits we present an explicit analytic form for the two breather solution of the nonlinear schr odinger equation with imaginary eigenvalues it describes various nonlinear combinations of akhmediev breathers and kuznetsov ma solitons the degenerate case when the two eigenvalues coincide is quite involved the standard inverse scattering technique does not generally provide an answer to this scenario we show here that the solution can still be found as a special limit of the general second order expression and appears as a mixture of polynomials with trigonometric and hyperbolic functions a further restriction of this particular case where the two eigenvalues are equal to i produces the second order rogue wave with two free parameters considered as differential shifts the illustrations reveal a precarious dependence of wave profile on the degenerate eigenvalues and differential shifts thus we establish a hierarchy of second order solutions revealing the interrelated nature of the general case the rogue wave and the degenerate breathers

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

hirota bilinear equations with linear subspaces of solutions we explore when hirota bilinear equations possess linear subspaces of solutions first we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to hirota bilinear equations second we show that multivariate polynomials whose zeros form a vector space can generate the desired hirota bilinear equations with given linear subspaces of solutions and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero third applying an algorithm using weights we present parameterizations of wave numbers and frequencies achieved by using one parameter to compute the desired hirota bilinear equations

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach

diversity of interaction solutions to the 2 1 dimensional ito equation abstract we aim to show the diversity of interaction solutions to the 2 1 dimensional ito equation based on its hirota bilinear form the proof is given through maple symbolic computations an interesting characteristic in the resulting interaction solutions is the involvement of an arbitrary function special cases lead to lump solutions lump soliton solutions and lump kink solutions two illustrative examples of the resulting solutions are displayed by three dimensional plots and contour plots

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