fock space representations of affine lie algebras and integral representations in the wess zumino witten models fock space representations of affine lie algebras are studied explicit forms of correction terms adding to the currentsf i z are determined it is proved that the sugawara energy momentum tensor on the fock spaces is quadratic in free bosons furthermore screening operators are constructed this implies the existence of generalized hypergeometric integrals satisfying the knizhnik zamolodchikov equation

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

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

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

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

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

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

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

k theoretic duality for shifts of finite type we will study the stable and unstable ruelle algebras associated to a hyperbolic homeomorphism of a compact space to do this we will describe a notion of k theoretic duality for algebras which generalizes spanier whitehead duality in topology a criterion for checking that it holds is presented as an application it is shown that the ruelle algebras which are associated to subshifts of finite type and agree with cuntz krieger algebras in this case satisfy this criterion and hence are dual

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

realization of the space of conformal blocks in lie algebra modules abstract using integrable irreducible representations of generalized twisted affine lie algebra modules we give a realization of the space of conformal blocks of conformal field theory on a stable algebraic curve many basic properties of the conformal blocks such as finite dimensionality of the space invariance of the conformal blocks under suitable formal neighborhood changes and the property of propagation of vacua are discussed finally a relative local 1 form around a fixed point of the order two automorphism of the curve is given

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

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

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

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

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

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

conformal blocks on elliptic curves and the knizhnik zamolodchikov bernard equations we give an explicit description of the vector bundle of wzw conformal blocks on elliptic curves with marked points as a subbundle of a vector bundle of weyl group invariant vector valued theta functions on a cartan subalgebra we give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes in some cases explicit calculations are possible and confirm the conjecture the friedan shenker flat connection is calculated and it is shown that horizontal sections are solutions of bernard s generalization of the knizhnik zamolodchikov equation

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