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

squared wavefunctions approach to periodic solutions of vector nonlinear schrodinger equation we develop a method of squared wavefunctions for the vector nonlinear schrodinger equation the squared wavefunctions of the octet representation of su 3 group give periodic solutions in terms of weierstrass elliptic functions specific limits of the obtained solution are the plane wave the soliton and cnoidal waves which were previously obtained using the ansatz of stationary motion

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

divisibility of reduction in groups of rational numbers given a multiplicative group of nonzero rational numbers and a positive integer m we consider the problem of determining the density of the set of primes p for which the order of the reduction modulo p of the group is divisible by m in the case when the group is finitely generated the density is explicitly computed some examples of groups with infinite rank are considered

a characterization of all elliptic algebro geometric solutions of the akns hierarchy here 9 x p x wl w3 denotes the elliptic weierstrass function with fundamental periods 2wl and 2w3 im w3 wl 0 in the special case where wl is real and w3 is purely imaginary the potential q x in 1 1 is real valued and ince s striking result 51 in modern spectral theoretic terminology yields that the spectrum of the unique self adjoint operator associated with the differential expression l2 d2 dx 2 q x in l2 r exhibits finitely many bands and gaps respectively that is

the calculation of novel class of solutions of a non linear fourth order evolution equation by the weierstras transform method abstract in this paper the weierstras transform method is applied to a non linear partial differential equation npde of fourth order to obtain new exact class of solutions in terms of the weierstras elliptic function g 2 g 3 the major aspect of this paper however is the fact that we are able to calculate distinctive class of solutions of a less studied non linear evolution equation explicitly given by u tt u xxxt 3 u xx u xt 0 the derived class of solutions cannot be found in the current literature more precisely the mathematical structure of these class of solutions are quite different from those derived by classical ansatz methods in other words using this special algebraic procedure the solution manifold is augmented to further class of solutions including special functions they are connected in a complicate way and allow a more general representation as found in previous time simultaneously we would like to stress the necessity of such sophisticated methods since a general theory of npdes does not exist in addition this paper is a logical completion of our recent studies where we have applied successfully the weierstras transform method to the important kdv mkdv equation

a generalized henon heiles system and related integrable newton equations a detailed description is given of integrable cases of the generalized henon heiles systems which differs from the standard h h ones by the term q22 their connection with fifth order one component soliton equations is discussed lax representations are constructed and the bi hamiltonian formulation of dynamics is given it is also shown that the gh h system can be mapped onto another system of newton equations with a nonstandard hamiltonian structure

some general periodic solutions for optical cascading explicit travelling wave periodic solutions of the pair of coupled quadratically non linear equations of optical cascading are obtained in terms of the weierstrass elliptic function the solutions obtained efficiently generalize some solutions previously known in terms of jacobian elliptic functions and some solitary wave solutions moreover new bounded periodic solutions are found

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

exact energy bands and fermi surfaces of separable abelian potentials we present a general theory for multidimensional schrodinger equations with separable abelian potentials with an arbitrary number of gaps in the spectrum in particular we derive general equations which allow one to express the energy and the wavevectors in the brillouin zone as a function of the spectral parameters by using the solutions of these equations we show how to construct the energy bands and the fermi surfaces in the first brillouin zone of the reciprocal lattice as illustrative examples we consider the case of two dimensional separable potentials with one two and three gaps in the spectrum the method can be applied to crystals with a cubic or a rectangular parallelogram wigner seitz cell in arbitrary dimensions the possibility to generalize the theory to other crystal symmetries is also briefly discussed

consecutive rosochatius deformations of the neumann system consecutive rosochatius deformations of the neumann system are investigated it is first shown that different realizations of a classical sl 2 gaudin magnet model yield different integrable hamiltonian systems then an algorithm of constructing infinitely many symplectic realizations of sl 2 algebra from a known one is presented and thus the neumann system can be deformed consecutively the second rosochatius deformation of the neumann system is taken as an illustrative example to show that the deformed systems admit separations of variables and may be linearized on the jacobi variety

solutions and laws of conservation for coupled nonlinear schrodinger equations lie group analysis a set of two coupled nonlinear schr odinger equations is systematically analyzed by means of lie group technique the physical situations under consideration include nonlinear propagation in strongly birefringent and multimode optical fibers the most general lie group of point symmetries its lie algebra and a group of adjoint representations that correspond to the lie algebra are identified as a result a complete list of group invariant exact solutions is obtained and compared with earlier results the corresponding laws of conservation are derived employing noether s theorem

treibich verdier potentials and the stationary m kdv hierarchy here x x col co3 denotes the elliptic weierstrass function with fundamental periods f p 2a l 2co3 im co3 wl 5 0 see 1 ch 18 in the special case where col is real and co3 is purely imaginary the potential q is real valued and ince s striking result 31 in modern spectral theoretic terminology yields the fact that the self adjoint operator l associated with the differential expression d 2 q in l2 ir has a finite gap spectrum of the type

elliptic solitons and grobner bases we consider the solution of spectral problems with elliptic coefficients in the framework of the hermite ansatz we show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the grobner bases method and others new integrable potentials and corresponding solutions of the sawada kotera kaup kupershmidt boussinesq equations and others are found

picard potentials and hill s equation on a torus hill s equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness of particular importance in the last 20 years is its connection with the kdv hierarchy and hence with integrable systems we show in this paper that regarding the independent variable as a complex variable yields a breakthrough for the problem of an efficient characterization of all elliptic finitegap potentials a major open problem in the field specifically we show that elliptic finitegap potentials of hill s equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variable complementing a classical theorem of picard the intimate connection between picard s theorem and elliptic finite gap solutions of completely integrable systems is established in this paper for the first time in addition we construct the hyperelliptic riemann surface associated with a finitegap potential not necessarily elliptic i e determine its branch and singular points from a comparison of the geometric and algebraic multiplicities of eigenvalues of certain auxiliary operators associated with hill s equation these multiplicities are intimately correlated with the pole structure of the diagonal green s function of the operator h d2 dx2 q x in l2 r our construction is new in the present general complex valued periodic finite gap case before describing our approach in some detail we shall give a brief account of the history of the problem involved this theme dates back to a 1940 paper of ince 43 who studied what is presently called the lam6 ince potential

linearity inside nonlinearity exact solutions to the complex ginzburg landau equation abstract systems of two linear partial differential equations pdes with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear pdes for the one dimensional complex ginzburg landau equation the usual representations of the complex field re a or a are multivalued which makes difficult the search for exact solutions in this manner fortunately the painleve analysis naturally introduces an elementary single valued like representation of a by two fields z grad respectively complex abd real uniquely defined by an explicit expression for and a ze i another important feature is the invariance by parity on a which increases the class of expected solutions this allows to retrieve quite easily the four famous solutions of nozaki and bekki represented by the constant coefficients of two linear partial differential equations and a finite set of constants

the nahm equations finite gap potentials and lame functions it is shown that the lame equation which is the simplest example of a finite gap hill s equation can be written in terms of the composition of two first order matrix operators the coefficients of which satisfy nahm s equation the characteristic eigenvalues of the lame equation emerge from the representation theory of the lie algebra so 3 a special case is that of reflectionless potentials and their associated bound states

exact travelling wave solutions of nonlinear evolution equation of surface waves in a convecting fluid exact travelling wave solutions in terms of the weierstrass p function of the equation eta t a0 eta x a1 eta eta x a2 eta xxx b0 eta xx b 1 eta eta x x b2 eta xxxx 0 are obtained these solutions may describe particularly both the evolution of a bounded conoidal wave and kink shaped waves

on the two gap locus for the elliptic calogero moser model we give an analytical description of the locus of the two gap elliptic potentials associated with the corresponding flow of the calogero moser system we start with the description of treibich verdier two gap elliptic potentials the explicit formulae for the covers wavefunctions and lame polynomials are derived together with a new lax representation for the particle dynamics on the locus we then consider more general potentials within the weierstrass reduction theory of theta functions to lower genera the reduction conditions in the moduli space of the genus 2 algebraic curves are given this is a subvariety of the humbert surface which can be singled out by the condition of the vanishing of some theta constants

saturation and castelnuovo mumford regularity abstract let i be a homogeneous ideal of the polynomial ring k x 0 x n where k is an arbitrary field avoiding the construction of a minimal graded free resolution of i we provide effective methods for computing the castelnuovo mumford regularity of i that also compute other cohomological invariants of k x 0 x n i

multicomponent cnoidal waves in cascade quasisynchronous frequency conversion it is shown that cascade quasisynchronous frequency conversion due to quadratic nonlinearity can be described in terms of an effective cubic nonlinearity this enables one to reduce a four mode interaction problem to solving a system of two coupled nonlinear schrodinger equations for the amplitudes of the waves participating in both nonlinear processes exact analytic solutions of the corresponding system are found in the form of multicomponent cnoidal waves with components expressed through a sum and a difference of two similar fundamental solutions of the lame equation with shifted arguments it is shown that solutions obtained in such a way enable one to optimize the conversion efficiency because of full coverage of the range of possible boundary conditions

how to compute the stanley depth of a monomial ideal abstract let j i be monomial ideals we show that the stanley depth of i j can be computed in a finite number of steps we also introduce the fdepth of a monomial ideal which is defined in terms of prime filtrations and show that it can also be computed in a finite number of steps in both cases it is shown that these invariants can be determined by considering partitions of suitable finite posets into intervals

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

quasi periodic solutions of mixed akns equations abstract a hierarchy of new nonlinear evolution equations which are composed of the positive and negative akns flows is proposed on the basis of the theory of algebraic curves the corresponding flows are straightened using the abel jacobi coordinates the meromorphic function the baker akhiezer vector and the hyperelliptic curve k n are introduced and by using these quasi periodic solutions of the first three nonlinear evolution equations in the hierarchy are constructed according to the asymptotic properties and the algebro geometric characters of and k n

on hill s equation with a singular complex valued potential 1 introduction the differential equation y00 qy ey is called hill s equation if q is a realvalued periodic locally integrable function of a real variable it is very well known that the spectrum of the operator associated with this equation consists of a countably in nite number of compact intervals of the real line these intervals are called spectral bands and are generally separated by open intervals called gaps in some instances however all but a nite number of gaps are empty resulting in a spectrum consisting of a nite number of compact bands and one closed interval extending to negative in nity in this case one calls q a nite band or nite gap potential the relationship of nite band potentials with the hierarchy of korteweg de vries equations has been the reason for a heightened interest in them during the past twenty years for recent accounts see for instance the monographs by dickey 8 and belokolos et al 3 the rst non trivial example of a nite band potential was given by ince in 1940 18 consider the lame potentials q x yg g 1 x c jq q0 where g 2n and jq q0 denotes weierstrass s elliptic function with fundamental half periods q and q0 if q is real q0 is pure imaginary and c q0 then q is realvalued periodic and continuous ince s result when translated to the present language shows that q then is a nite band potential with g non empty gaps between the bands this however raises immediately the question of what happens when c 6 q0 or when the half periods are more general in these cases one encounters potentials which are complex valued but continuous as well as potentials which have singularities of the form yg g 1 x and are realor complex valued before giving an overview of the content of the paper i want to give a brief account of the history of the subject let q be a real locally integrable periodic function with period p l the differential expression d dx q and h the self adjoint operator in l r associated with l then consider the following four statements a there are precisely 2g 1 real numbers e2g e0 such that the spectrum of h is given by

exact periodic solutions of the complex ginzburg landau equation three new exact periodic solutions of the complex ginzburg landau equation are obtained in terms of the weierstrass elliptic function furthermore the new periodic solutions and other shock solutions appear as their bounded limits along the real axis for particular relationships between the coefficients in the equation in the general case bounded limits are nothing but the already known pulse hole and shock solutions it is also shown that the shapes of the solutions are quite different from the shape of the usual envelope wave solution in particular the spatial structure of the new bounded periodic solutions varies in time while the pulse solution may exhibit breather like behavior

quasiperiodic and periodic solutions for vector nonlinear schrodinger equations we consider quasiperiodic and periodic cnoidal wave solutions of a set of n component vector nonlinear schrodinger equations vnlses in a biased photorefractive crystal with a drift mechanism of nonlinear response and kerr type nonlinearity n component nonlinear schrodinger equations can be used to model self trapped mutually incoherent wave packets these equations also model pulse pulse interactions in wavelength division multiplexed channels of optical fiber transmission systems quasiperiodic wave solutions for the vnlses in terms of n dimensional kleinian functions are presented periodic solutions in terms of hermite polynomials and generalized hermite polynomials for n component nonlinear schrodinger equations are found

quasi periodic and periodic solutions for coupled nonlinear schrodinger equations of manakov type we consider travelling periodic and quasi periodic wave solutions in coupled nonlinear schrodinger equations in fibre optics these equations can be used to model single mode fibres with strong birefringence and two mode optical fibres recently these equations appear as a model describing pulse pulse interactions in wavelength division multiplexed channels of optical fibre transmission systems in some cases this model reduces to the integrable manakov system ims two phase quasi periodic solutions for the ims are given in terms of two dimensional kleinian functions the reduction of quasi periodic solutions to elliptic functions is discussed new solutions are found in terms of generalized hermite polynomials which are associated with two gap treibich verdier potentials

the expansions of spectral function and the corresponding finite dimensional integrable systems abstract in this paper taking the lax pairs of kdv equation as an illustrative example with the help of a transformation of the spectral problem another form of spectral function is used to give finite dimensional integrable systems these integrable systems are generated by two different kinds of polynomial expansion of spectral function on spectral parameter further the root variables of the spectral function are introduced to study the canonical equations of hamilton finally based on the hamilton jacobi theory the action angle variables are built and the soliton solutions of kdv equation are obtained by inversion

symmetry properties and exact patterns in birefringent optical fibers we carry out a group theoretical study of the pair of nonlinear schr o dinger equations describing the propagation of waves in nonlinear birefringent optical fibers we exploit the symmetry algebra associated with these equations to provide examples of specific exact solutions among them we obtain the soliton profile which is related to the coordinate translations and the constant change of phase

on the r rank artin conjecture we assume the generalized riemann hypothesis and prove an asymptotic formula for the number of primes for which f p can be generated by r given multiplicatively independent numbers in the case when the r given numbers are primes we express the density as an euler product and apply this to a conjecture of brown zassenhaus j number theory 3 1971 306 309 finally in some examples we compare the densities approximated with the natural densities calculated with primes up to 9 10 4

lame equation sl 2 algebra and isospectral deformations it is proven that the n zone lame equation is equivalent to a spectral problem for a quadratic element of a universal enveloping sl 2 algebra the riemannian surface corresponding to the eigenstates in a parametric space forms the 2n 1 sheet surface which splits in four subsurfaces three of them contain the same number of sheets

integrable rosochatius deformations of the restricted soliton flows a method to construct integrable rosochatius deformations of the restricted soliton flows in the setup of lax formulation is presented the integrable rosochatius deformations of the restricted soliton flows such as the restricted ablowitz kaup newell segur flow the restricted tu meng flow the restricted tu flow with neumann type constraints and the restricted modified korteweg de vries flow together with their lax representations are presented in addition a lax representation of the jacobi rosochatius system is obtained

elliptic solutions and blow up in an integrable henon heiles system we consider an integrable case of the henon heiles system and use an isomorphism with the two gap kdv flow to construct families of real elliptic trajectories which are associated with two gap elliptic solitons of the kdv equation some of these solutions exhibit blow up in finite time

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

how to construct finite dimensional bi hamiltonian systems from soliton equations jacobi integrable potentials a systematic method of constructing finite dimensional integrable systems starting from a bi hamiltonian hierarchy of soliton equations is introduced the existence of two hamiltonian structures of the hierarchy leads to a bi hamiltonian formulation of the resulting finite dimensional systems the case of coupled kdv hierarchies is studied in detail a surprising connection with separable jacobi potentials is uncovered and described

a schauder and riesz basis criterion for non self adjoint schrodinger operators with periodic and antiperiodic boundary conditions under the assumption that v l2 0 dx we derive necessary and sufficient conditions in terms of spectral data for non self adjoint schrodinger operators d2 dx2 v in l2 0 dx with periodic and antiperiodic boundary conditions to possess a riesz basis of root vectors i e eigenvectors and generalized eigenvectors spanning the range of the riesz projection associated with the corresponding periodic and antiperiodic eigenvalues we also discuss the case of a schauder basis for periodic and antiperiodic schrodinger operators d2 dx2 v in lp 0 dx p 1

quasiperiodic solutions for matrix nonlinear schrodinger equations the adler kostant symes theorem yields isospectral hamiltonian flows on the dual erm g of a lie subalgebra erm g of a loop algebra erm g a general approach relating the method of integration of krichever novikov and dubrovin to such flows is used to obtain finite gap solutions of matrix nonlinear schrodinger equations in terms of quotients of functions

exact solutions for a class of integrable henon heiles type systems we study the exact solutions of a class of integrable henon heiles type systems according to the analysis of bountis et al phys rev a 25 1257 1982 these solutions are expressed in terms of two dimensional kleinian functions special periodic solutions are expressed in terms of the well known weierstrass function we extend some of our results to a generalized henon heiles type system with n 1 degrees of freedom

quasi bi hamiltonian systems and separability two quasi bi hamiltonian systems with three and four degrees of freedom are presented these systems are shown to be separable in terms of nijenhuis coordinates moreover the most general pfaffian quasi bi hamiltonian system with an arbitrary number of degrees of freedom is constructed in terms of nijenhuis coordinates and its separability is proved

on computing some special values of multivariate hypergeometric functions abstract this paper obtains several evaluations of multivariate hypergeometric functions for particular parameter values and at special algebraic points they have a high interest not only on their own but also in the light of the remarkable implications for both pure mathematics and mathematical physics following our research started in 30 and 31 we provide some contribution to such functions computability inside and outside their disks of convergences in the first part we provide some new results in the spirit of theorem 3 1 of 31 obtaining formulae for the values of multivariate hypergeometric functions by generalizing a well known identity of kummer 23 to the hypergeometric functions of two or more variable like those of appell and lauricella denoted f d n in the second part using some reduction schemes of hyperelliptic integrals due to goursat 16 hermite 18 19 we evaluate appell and lauricella s f d n hypergeometric functions and their analytic continuations at some particular locations finally by exploiting reductions of hyperelliptic integrals to elliptic due to belokolos et al 5 eilbeck and enol skii 11 enol skii and kostov 12 and by maier 27 we obtain further links from multivariate hypergeometric functions to complete elliptic integrals and to we thus reach a conceptual settlement of the piece of research started by us in 30 and 31

the generalized kupershmidt deformation for the fifth order coupled kdv equations hierarchy abstract based on the kupershmidt deformation we propose the generalized kupershmidt deformation gkd to construct new systems from integrable bi hamiltonian system as applications the generalized kupershmidt deformation of the fifth order coupled kdv equations hierarchy with self consistent sources and its lax representation are presented

stationary modulated amplitude waves in the 1d complex ginzburg landau equation we reformulate the one dimensional complex ginzburg landau equation as a fourth order ordinary dierential equation in order to nd stationary spatiallyperiodic solutions using this formalism we prove the existence and stability of stationary modulated amplitude wave solutions approximate analytic expressions and a comparison with numerics are given

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

the hyperelliptic ζ function and the integrable massive thirring model we provide a treatment of algebrogeometric solutions of the classical massive thirring system in the framework of the weierstrassklein theory of hyperelliptic functions we show that the equations

on periodic solutions and their modulations of the manakov system periodic solutions of the manakov system are studied with the aim to establish links between the recent results of wright wright 2013 physica d 264 1 and kamchatnov kamchatnov 2013 europhys lett 103 60003 where one phase solutions have been obtained by different methods and with different parameterizations two types of wave which in the context of the dynamics of two component bose einstein condensates can be called density and polarization waves are determined as particular solutions of the manakov system whitham equations describing modulations of these two types of waves are obtained

an explicit characterization of calogero moser systems combining theorems of halphen floquet and picard and a frobenius type analysis we characterize rational meromorphic simply periodic and elliptic kdv potentials in particular we explicitly describe the proper extension of the airault mckean moser locus associated with these three classes of algebro geometric solutions of the kdv hierarchy with special emphasis on the case of multiple collisions between the poles of solutions this solves a problem left open since the mid 1970s

a class of exact periodic solutions of nonlinear envelope equations a class of periodic solutions of nonlinear envelope equations e g the nonlinear schrodinger equation nls is expressed in terms of rational functions of elliptic functions the hirota bilinear transformation and theta functions are used to extend and generalize this class of solutions first reported for nls earlier in the literature in particular a higher order nls and the davey stewartson ds equations are treated doubly periodic standing waves solutions are obtained for both the dsi and dsii equations a symbolic manipulation software is used to confirm the validity of the solutions independently

bi hamiltonian structure of an integrable henon heiles system by making use of the system of coordinates in which the separation of the variables in the hamilton jacobi equation takes place we find a bi hamiltonian stmclure of a two degrees of freedom hamiltmian system corresponding to the integrable henon

darboux coordinates and liouville arnold integration in loop algebras darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part tilde mathfrak g of loop algebras these are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves defined within a given matrix representation a liouville generating function is obtained in completely separated form and shown through the liouvile arnold integration method to lead to the abel map linearization of all hamiltonian flows induced by the spectral invariants as illustrative examples the caseg sl 2 together with its real forms is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics with the liouville generating function expressed in hyperellipsoidal coordinates forg sl 3 the method is applied to the computation of quasi periodic solutions of the two component coupled nonlinear schrodinger equation a case which requires further symplectic constraints in order to deal with singularities in the spectral data at

elliptic baker akhiezer functions and an application to an integrable dynamical system an ansatz for elliptic baker akhiezer functions is used to find new solutions for various spectral problems the sturm liouville problem with four and five gap lame potentials and two third order problems one of the third order problems is the classical halphen equation with a spectral parameter the eigenfunctions are used to construct elliptic solutions for integrable cases of a generalized henon heiles system

the full positive flows of manakov hierarchy hamiltonian structures and conservation laws based on four sets of lenard recursion sequences and zero curvature equation we derive the full positive flows of the manakov hierarchy associated with a 3x3 matrix spectral problem from which some new nonlinear evolution equations are proposed with the help of the darboux transformation soliton solutions of two new nonlinear evolution equations in the manakov hierarchy are constructed as two special reductions the full positive flows of the coupled modified korteweg de vries hierarchy and the sasa satsuma hierarchy are deduced in which some new nonlinear evolution equations are included and then we construct the hamiltonian structures of the manakov hierarchy and infinite conservation laws of several nonlinear evolution equations

effective integration of ultra elliptic solutions of the focusing nonlinear schrodinger equation abstract an effective integration method based on the classical solution of the jacobi inversion problem using kleinian ultra elliptic functions and riemann theta functions is presented for the quasi periodic two phase solutions of the focusing cubic nonlinear schrodinger equation each two phase solution with real quasi periods forms a two real dimensional torus modulo a circle of complex phase factors expressed as a ratio of theta functions associated with the riemann surface of the invariant spectral curve the initial conditions of the dirichlet eigenvalues satisfy reality conditions which are explicitly parametrized by two physically meaningful real variables the squared modulus and a scalar multiple of the wavenumber simple new formulas for the maximum modulus and the minimum modulus are obtained in terms of the imaginary parts of the branch points of the riemann surface

on a theorem of halphen and its application to integrable systems abstract we extend halphen s theorem which characterizes solutions of certain nth order differential equations with rational coefficients and meromorphic fundamental systems to a first order n n system of differential equations as an application of this circle of ideas we consider stationary rational algebro geometric solutions of the kdv hierarchy and illustrate some of the connections with completely integrable models of the calogero moser type in particular our treatment recovers the complete characterization of the isospectral class of such rational kdv solutions in terms of a precise description of the airault mckean moser locus of their poles

weierstrass semi rational expansion method and new doubly periodic solutions of the generalized hirota satsuma coupled kdv system in the paper with the aid of symbolic computation we investigate the generalized hirota satsuma coupled kdv system via our weierstrass semi rational expansion method presented recently using the rational expansion of weierstrass elliptic function and its first order derivative as a consequence three families of new weierstrass elliptic function solutions via weierstrass elliptic function p g2 g3 and its first order derivative p g2 g3 moreover the corresponding new jacobi elliptic function solutions and solitary wave solutions are also presented and when these solitary wave solutions approach to some constants

nonlinear analysis of the eckhaus instability modulated amplitude waves and phase chaos with nonzero average phase gradient abstract we analyze the eckhaus instability of plane waves in the one dimensional complex ginzburg landau equation cgle and describe the nonlinear effects arising in the eckhaus unstable regime modulated amplitude waves maws are quasi periodic solutions of the cgle that emerge near the eckhaus instability of plane waves and cease to exist due to saddle node sn bifurcations these maws can be characterized by their average phase gradient and by the spatial period p of the periodic amplitude modulation a numerical bifurcation analysis reveals the existence and stability properties of maws with arbitrary and p maws are found to be stable for large enough and intermediate values of p for different parameter values they are unstable to splitting and attractive interaction between subsequent extrema of the amplitude defects form from perturbed plane waves for parameter values above the sn of the corresponding maws the break down of phase chaos with average phase gradient 0 wound up phase chaos is thus related to these sns a lower bound for the break down of wound up phase chaos is given by the necessary presence of sns and an upper bound by the absence of the splitting instability of maws

hyperelliptic solutions of modified korteweg de vries equation of genus g essentials of the miura transformation explicit hyperelliptic solutions of the modified korteweg de vries equations without any ambiguous parameters were constructed in terms of only the hyperelliptic al functions over the non degenerate hyperelliptic curve y2 f x of arbitrary genus g in the derivation any functions or baker akhiezer functions were not essentially used then the miura transformation naturally appears as the connection between the hyperelliptic functions and hyperelliptic al functions

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

error correction of linear codes via colon ideals abstract we show that errors in data transmitted through linear codes can be thought of as codewords of minimum weight of new linear codes to determine errors we can then use methods specific to finding such special codewords one of these methods consists of finding the primary decomposition of the saturation of a certain homogeneous ideal when good words i e vectors with a unique nearest neighbor are error corrected the saturated ideal is just the prime ideal of a point so the primary decomposition is superfluously determined we show that this ideal can be computed by coloning the original homogeneous ideal with a power of a certain variable we then determine the smallest such power for any linear code

on the integrability of stationary and restricted flows of the kdv hierarchy a bi hamiltonian formulation for stationary flows of the kdv hierarchy is derived in an extended phase space a map between stationary flows and restricted flows is constructed in one case it connects an integrable henon heiles system and the garnier system moreover a new integrability scheme for hamiltonian systems is proposed that holds in the standard phase space

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

on elliptic solutions of a coupled nonlinear schrodinger system abstract an explicit formula is obtained for single phase bounded elliptic solutions of the manakov system of integrable coupled nonlinear schrodinger equations in terms of the weierstrass sigma function with a real quasiperiod the parametrization is effective in the sense that the reality conditions are completely characterized for each of the three possible couplings focusing focusing defocusing defocusing and focusing defocusing the manakov soliton is recovered in the soliton limit and the small wave modulation limit is shown to satisfy the linearized dispersion relation of planewave solutions

bifurcations of invariant manifolds in the generalized henon heiles system abstract the generalized henon heiles hamiltonian h 1 2 q dot 1 2 q dot 2 2 aq 1 2 bq 2 2 q 1 2 q 2 e 3 q 2 3 in the integrable case e 6 is studied from topological point of view the topology of the real level sets for all generic values of the constants of the motion and all generic bifurcations of the liouville tori and cylinders are described by making use of the reach algebraic structure of the problem

exact front soliton and hole solutions for a modified complex ginzburg landau equation from the harmonic balance method abstract a novel approach of using harmonic balance hb method is presented to find front soliton and hole solutions of a modified complex ginzburg landau equation three families of exact solutions are obtained one of which contains two parameters while the others one parameter the hb method is an efficient technique in finding limit cycles of dynamical systems in this paper the method is extended to obtain homoclinic heteroclinic orbits and then coherent structures it provides a systematic approach as various methods may be needed to obtain these families of solutions as limit cycles with arbitrary value of bifurcation parameter can be found through parametric continuation this approach can be extended further to find analytic solution of complex quintic ginzburg landau equation in terms of fourier series