a distinguisher for high rate mceliece cryptosystems the goppa code distinguishing gd problem consists in distinguishing the matrix of a goppa code from a random matrix the hardness of this problem is an assumption to prove the security of code based cryptographic primitives such as mceliece s cryptosystem up to now it is widely believed that the gd problem is a hard decision problem we present the first method allowing to distinguish alternant and goppa codes over any field our technique can solve the gd problem in polynomial time provided that the codes have sufficiently large rates the key ingredient is an algebraic characterization of the key recovery problem the idea is to consider the rank of a linear system which is obtained by linearizing a particular polynomial system describing a key recovery attack it appears that this dimension depends on the type of code considered explicit formulas derived from extensive experimentations for the rank are provided for generic random alternant and goppa codes over any field finally we give theoretical explanations of these formulas in the case of random codes alternant codes over any field of characteristic two and binary goppa codes

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

first order compatibility for a 2 1 dimensional diffusion equation a class of first order partial differential equations is obtained compatible with a 2 1 dimensional linear diffusion equation with a nonlinear source term we will show that if the source term is quasilinear then compatible equations are quasilinear furthermore we obtain diffusion equations with source terms that admit non quasilinear compatible equations

new exact solutions of the broer kaup equations in 2 1 dimensional spaces abstract with the aid of maple several new kinds of exact solutions for the broer kaup equations in 2 1 dimensional spaces are obtained by using a new ansatz this approach can also be applied to other nonlinear evolution equations

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

singularity structure analysis and hirota s bilinearisation of the davey stewartson equation the singularity structure aspects of the davey stewartson equation is investigated and it is shown that the system possesses the generalised painleve property in the sense of weiss tabor and carnevale 1983 further the associated backlund transformation in constructed and hirota s bilinearisation is obtained straightforwardly from the painleve analysis

seiberg witten equations in r4 lie symmetries particular solutions integrability it is shown that the 11 parameter automorphism group of lie algebra of four dimensional euclidean group is a maximal lie symmetries group of the seiberg witten equations in r4 particular explicit solutions which are invariant under so 3 subgroups of the maximal lie symmetries group are constructed it is established that seiberg witten equations do not possess the painleve property nevertheless so 3 invariant solutions obtained are turned to admit a characteristic singularity structure

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

new multiple soliton like solutions to the generalized 2 1 dimensional kp equation a new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations pdes in a unified way the main idea of this method is to take full advantage of the riccati equation which has more new solutions more new multiple soliton like solutions are obtained for the the generalized 2 1 dimensional kp equation

nonclassical analysis and painleve property for the kupershmidt equations by means of the nonclassical lie approach and the direct method all the similarity reductions of the kupershmidt equations are found two reduction equations which can also be obtained by the classical lie approach are the painleve type ii and iv respectively the other three reduction equations which cannot be obtained by the classical lie approach are only linear or riccati type

a new algorithm for solving ring lpn with a reducible polynomial the learning parity with noise lpn problem has recently proved to be of great importance in cryptology a special and very useful case is the ring lpn problem which typically provides improved efficiency in the constructed cryptographic primitive we present a new algorithm for solving the ring lpn problem in the case when the polynomial used is reducible it greatly outperforms the previous algorithms for solving this problem using the algorithm we can break the lapin authentication protocol for the proposed instance using a reducible polynomial in sim 2 71 bit operations

comparison of detrending methods for fluctuation analysis abstract we examine several recently suggested methods for the detection of long range correlations in data series based on similar ideas as the well established detrended fluctuation analysis dfa in particular we present a detailed comparison between the regular dfa and two recently suggested methods the centered moving average cma method and a modified detrended fluctuation analysis mdfa we find that cma performs the same as dfa in long data with weak trends and is slightly superior to dfa in short data with weak trends when comparing standard dfa to mdfa we observe that dfa performs slightly better in almost all examples we studied we also discuss how several types of trends affect different types of dfa for weak trends in the data the new methods are comparable with dfa in these respects however if the functional form of the trend in data is not a priori known dfa remains the method of choice only a comparison of dfa results using different detrending polynomials yields full recognition of the trends a comparison with independent methods is recommended for proving long range correlations

painleve property and geometry abstract the painleve property for discrete hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times the augmented manifold is constructed by expanding the hamilton jacobi equation a complete classification of the types of poles allowed in complex time is given for hamiltonians which separate into the direct product of hyperelliptic curves for such systems bounds on the degrees of the polynomial separating variable change and the other integrals in involution can be found from the pole series and the hamilton jacobi equation it is shown how branching can arise naturally in a painleve system

exact soliton solutions of the one dimensional complex swift hohenberg equation abstract using painleve analysis the hirota multi linear method and a direct ansatz technique we study analytic solutions of the 1 1 dimensional complex cubic and quintic swift hohenberg equations we consider both standard and generalized versions of these equations we have found that a number of exact solutions exist to each of these equations provided that the coefficients are constrained by certain relations the set of solutions include particular types of solitary wave solutions hole dark soliton solutions and periodic solutions in terms of elliptic jacobi functions and the weierstrass function although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic swift hohenberg equations those presented here are the first examples of exact analytic solutions found thus far

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

towards classification of dimensional integrable equations integrability conditions i in this paper we attempt to extend the symmetry approach well developed in the case of 1 1 dimensional equations to the 2 1 dimensional case presence of nonlocal terms in symmetries and conservation laws is the main feature of integrable 2 1 dimensional equations we have introduced a concept of quasi local functions to characterize nonlocalities we have found a few first integrability conditions for a class of scalar equations in terms of quasi local functions and have demonstrated that they are suitable for testing integrability

painleve test of coupled gross pitaevskii equations a painleve p test of the coupled gross pitaevskii equations has been carried out with the result that the coupled equations pass the p test only if a special relation containing system parameters masses and scattering lengths is satisfied computer algebra is applied to evaluate the j 4 compatibility condition for admissible external potentials the appearance of an arbitrary real potential embedded in the external potentials is shown to be the consequence of the coupling the connection with recent experiments related to the stability of two component bose einstein condensates of rb atoms is discussed

one or two frequencies the empirical mode decomposition answers this paper investigates how the empirical mode decomposition emd a fully data driven technique recently introduced for decomposing any oscillatory waveform into zero mean components behaves in the case of a composite two tones signal essentially two regimes are shown to exist depending on whether the amplitude ratio of the tones is greater or smaller than unity and the corresponding resolution properties of the emd turn out to be in good agreement with intuition and physical interpretation a refined analysis is provided for quantifying the observed behaviors and theoretical claims are supported by numerical experiments the analysis is then extended to a nonlinear model where the same two regimes are shown to exist and the resolution properties of the emd are assessed

huygens principle and integrable systems abstract the theory of integrable systems in particular of the generalised calogero moser problem is applied to constructing and investigating examples of the hyperbolic equations satisfying huygens principle in hadamard s sense

complete and exact solutions of a class of nonlinear diffusion equations and problem of velocity selection in this rapid communication complete and exact solutions of a class of nonlinear diffusion equations have been presented the exact solutions give a tutorial explanation about the mechanism of velocity selection the marginal stability hypothesis is extended to predict velocity selection for the equations it has been shown that this class of equations can be transformed to the heat equation via nonlinear transformations numerical experiments have been performed to test the theoretical prediction s1063 651x 97 51411 5

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

on the integrability of nonlinear dirac equations the integrability of nonlinear dirac equations is discussed applying recent results in soliton theory using the lie point transformation groups of the nonlinear dirac equations we reduce these partial differential equations to systems of ordinary differential equations and study whether these systems are integrable we also discuss whether lie backlund vector fields exist we conclude that the nonlinear dirac equations are not integrable

new applications of the singular manifold method to the 2 1 dimensional burgers equations new applications of the singular manifold method are presented to a 2 1 dimensional burgers equation and a higher order burgers equation by choosing different seed solutions auto backlund transformations cole hopf transformations and general functional separation solutions containing three arbitrary functions are obtained for the equations in question localized coherent structures are for the first time expressed by jacobi elliptic functions not infinite periodic limit and their interaction properties are numerically studied

on traveling wave solutions of fisher s equation in two spatial dimensions it is shown for the quadratic fisher equation in two spatial dimensions that along with a plane wave there exist several other traveling waves with nontrivial front geometry some of the solutions are found in explicit form others are constructed approximately the dispersion relationship and velocity curvature dependence generated by these solutions are studied

the inner structure of empirical mode decomposition abstract the empirical mode decomposition emd is a nonlinear method that is truly adaptive with good localization property in the time domain for analyzing non stationary complex data the emd has been proven useful in a wide range of applications however due to the nonlinear and complex nature of the sifting process the most essential step of the emd a firm mathematical foundation or a transparent physical description are still lacked for emd here we embark on constructing a mathematical theory of the sifting operator we first show that the sifting operator can be expressed as the data plus the sum of the responses to the impulses multiplied by the data value at the extrema such an expression of the sifting operator is then used to investigate the adaptive nature and the localizing effect of the emd alternatively the sifting operator can also be represented by a sifting matrix which depends nonlinearly on the extrema distribution based on the eigen decomposition of the sifting matrix the transfer function of the sifting process is analyzed finally we answer what an intrinsic mode function imf is from the wave perspective by exploring the physical basis of the imfs

a new truncation in painleve analysis the author observes that when the expansion variable in painleve analysis satisfies a system of riccati equations truncation at a level higher than constant level is allowed this extends the range of exact solutions to nonlinear partial differential equations that one is able to obtain using truncated painleve expansions

formal variable separation approach for nonintegrable models using a formal variable separation approach a nonlinear partial differential equation can be solved by solving ordinary different equations or even algebraic equations taking the kdv burgers and modified kdv burgers equations with background interaction as simple examples some explicit solitary wave solutions which are induced by background source and nonlinearity or dispersion are obtained

on the complexity of the rank syndrome decoding problem in this paper we propose two new generic attacks on the rank syndrome decoding rsd problem let c be a random n k rank code over gf q m and let y x e be a received word such that x in c and rank e r the first attack the support attack is combinatorial and permits to recover an error e of rank weight r in min o n k 3 m 3 q r lfloor km n rfloor o n k 3 m 3 q r 1 lfloor k 1 m n rfloor operations on gf q this new attack improves the exponent for the best generic attack for the rsd problem in the case n m by introducing the ratio m n in the exponential coefficient of the previously best known attacks the second attack the annulator polynomial attack is an algebraic attack based on the theory of q polynomials introduced by ore we propose a new algebraic setting for the rsd problem that permits to consider equations and unknowns in the extension field gf q m rather than in gf q as it is usually the case we consider two approaches to solve the problem in this new setting the linearization technique shows that if n ge k 1 r 1 1 the rsd problem can be solved in polynomial time more generally we prove that if lceil r 1 k 1 n 1 r rceil le k the rsd problem can be solved with an average complexity of o r 3 k 3 q r lceil r 1 k 1 n 1 r rceil operations in the base field gf q we also consider solving with grobner bases for which we discuss theoretical complexity we also consider hybrid solving with grobner bases on practical parameters as an example of application we use our new attacks on all recent cryptosystems parameters which repair the gpt cryptosystem we break all examples of published proposed parameters and some parameters are broken in less than 1 s in certain cases

non integrability of su 2 yang mills and yang mills higgs systems the spherically symmetric time dependent su 2 yang mills equations and yang mills higgs equations are shown to be non integrable by using the weiss tabor and carnevale 1983 method of painleve analysis reduced equations corresponding to these systems are also found to be non integrable

generalized solitary and periodic wave solutions to a 2 1 dimensional zakharov kuznetsov equation abstract in this paper the exp function method is employed to the zakharov kuznetsov equation as a 2 1 dimensional model for nonlinear rossby waves the observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient the obtained results may be useful for better understanding the properties of two dimensional coherent structures such as atmospheric blocking events

factorization of the classical boussinesq system the scalar equation equivalent to the classical boussinesq or broer kaup system is shown to factorize into a differential operator acting on burgers equation the linearizability of this latter equation provides a very simple explanation for the recently found property of fusion and fission of a class of solitons for this system

efficient encryption from random quasi cyclic codes we propose a framework for constructing efficient code based encryption schemes that do not hide any structure in their public matrix the framework is in the spirit of the schemes first proposed by alekhnovich in 2003 and based on the difficulty of decoding random linear codes from random errors of low weight we depart somewhat from alekhnovich s approach and propose an encryption scheme based on the difficulty of decoding random quasi cyclic codes we propose two new cryptosystems instantiated within our framework the hamming quasi cyclic cryptosystem hqc based on the hamming metric and the rank quasi cyclic cryptosystem rqc based on the rank metric we give a security proof which reduces the indistinguishability under chosen plaintext attack security of our systems to a decision version of the well known problem of decoding random families of quasi cyclic codes for the hamming and rank metrics the respective mathsf qcsd and mathsf rqcsd problems we also provide an analysis of the decryption failure probability of our scheme in the hamming metric case for the rank metric there is no decryption failure our schemes benefit from a very fast decryption algorithm together with small key sizes of only a few thousand bits the cryptosystems are very efficient for low encryption rates and are very well suited to key exchange and authentication asymptotically for lambda the security parameter the public key sizes are respectively in mathcal o lambda 2 for hqc and in mathcal o left lambda frac 4 3 right for rqc practical parameter compares well to the systems based on ring learning parity with noise or the recent moderate density parity check codes system

darboux transformation and solutions for an equation in 2 1 dimensions painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2 1 dimensions this procedure has allowed us to obtain the lax pair darboux transformation and functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively

painleve singularity structure analysis of three component gross pitaevskii type equations in this paper we have studied the integrability nature of a system of three coupled gross pitaevskii type nonlinear evolution equations arising in the context of spinor bose einstein condensates by applying the painleve singularity structure analysis we show that only for two sets of parametric choices corresponding to the known integrable cases the system passes the painleve test

the cavalcante tenenblat equation does the equation admits physical significance the main purpose of the given paper is to analyze a less studied third order non linear partial differential equation the so called cavalcante tenenblat equation cte in the following form u t u x 1 2 x x u x 3 2 since general class of solutions are of basic interest a complete characterization of the group properties is given the traveling wave ansatz restricts the solution manifold to special class of solutions and hence a generalize algorithm is necessary we determine the lie point symmetry vector fields and calculate similarity ansatze further we also derive a few non linear transformations and some similarity solutions are obtained explicitly due to the complexity of some similarity solutions a numerical procedure is of advantage moreover the non classical case potential symmetries is studied to the first time and further we show how the cte leads to approximate symmetries and we apply the method to the first time we call the disturbed equation the cte e equation and we show how to derive new class of solutions finally the equation does not pass the painleve test and is therefore not soluble by the inverse scattering transform method ist hence suitable alternative algebraic approaches are necessary to derive class of solutions explicitly

the action angle transformation for soliton equations abstract for third order completely integrable equations in 1 1 dimensions canonical transformations which map in the multi soliton case gradients of action variables to gradients of angle variables are constructed furthermore this action angle map is shown always to be an infinitesimal symmetry group generator of the interacting soliton equation

symmetry analysis of the two dimensional diffusion equation with a source term a symmetry analysis is performed on a 2 1 dimensional linear diffusion equation with a nonlinear source term involving the dependent variable and its spatial derivatives in the first part of the paper we use the classical method to classify source terms where the original equation admits a nontrivial symmetry in the second part of the paper we use the nonclassical method and show that we simply recover the classical symmetries

application of exp function method to the whitham broer kaup shallow water model using symbolic computation in this letter the exp function method is applied to the whitham broer kaup shallow water model with the help of symbolic computation several kinds of new solitary wave solutions are formally derived

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

on the hardness of the decoding and the minimum distance problems for rank codes we give a randomized reduction for the rank syndrome decoding problem and rank minimum distance problem for rank codes over extension fields our results are based on embedding linear codes in the hamming space into linear codes over an extension field equipped with the rank metric we prove that if any of the previous problems for the rank metric is in zpp rp cap corp then we would have np zpp we also give complexity results for the respective rank metric approximation problems

the painleve property a modified boussinesq equation and a modified kadomtsev petviashvili equation abstract it is shown that the modified boussinesq equation and the modified kadomtsev petviashvili equation both of which are known to be completely integrable possess the painlev e property for differential equations as defined by weiss tabor and carnevale

symmetry properties and solutions of nonlinear dispersive thin film equations in three dimensions the lie point symmetries are obtained for each one of two nonlinear dispersive thin film equations in 2 1 variables by means of the invariants of the one and two dimensional subgroups all reductions to ordinary differential equations and to partial differential equations in two variables are obtained solutions are provided for a number of the reduced equations it is also ascertained that neither one of the two considered thin film equations is integrable in some cases solitary and periodic wave solutions having curved wavecrests are obtained

analytic solutions of a microstructure pde and the kdv and kadomtsev petviashvili equations by invariant painleve analysis and generalized hirota techniques abstract truncated painleve expansions invariant painleve analysis and generalized hirota expansions are employed in combination to solve partially reduce to quadrature the integrable kdv and kp equations and a nonintegrable generalized microstructure gms equation although the multisolitons of the kdv and kp equations are very well known the solutions obtained here for all the three nlpdes are novel and non trivial the solutions obtained via invariant painleve analysis are all complicated rational functions with arguments which themselves are confluent hypergeometric kdv or trigonometric gms functions of various distinct non traveling kdv and traveling wave variables in some cases this is slightly reminiscent of doubly periodic elliptic function solutions when nonlinear ode systems are reduced to quadratures the solutions obtained by the use of recently generalized hirota type expansions in the truncated painleve expansions are closer in functional form to conventional hyperbolic secant solutions although with non trivial traveling wave arguments which are distinct for the three nlpdes considered here

on the integrability of the generalized fisher type nonlinear diffusion equations in this paper the geometric integrability and lax integrability of the generalized fisher type nonlinear diffusion equations with modified diffusion in 1 1 and 2 1 dimensions are studied by the pseudo spherical surface geometry method and prolongation technique it is shown that the 1 1 dimensional fisher type nonlinear diffusion equation is geometrically integrable in the sense of describing a pseudo spherical surface of constant curvature 1 only for m 2 and the generalized fisher type nonlinear diffusion equations in 1 1 and 2 1 dimensions are lax integrable only for m 2 this paper extends the results in bindu et al 2001 j phys a math gen 34 l689 and further provides the integrability information of 1 1 and 2 1 dimensional fisher type nonlinear diffusion equations for m 2

a new approach to sparse decomposition of nonstationary signals with multiple scale structures using self consistent nonlinear waves abstract a new method for signal decomposition is proposed and tested based on self consistent nonlinear wave equations with self sustaining physical mechanisms in mind the new method is adaptive and particularly effective for dealing with synthetic signals consisting of components of multiple time scales by formulating the method into an optimization problem and developing the corresponding algorithm and tool we have proved its usefulness not only for analyzing simulated signals but more importantly also for real clinical data

singularity structure symmetries and integrability of generalized fisher type nonlinear diffusion equation in this letter the integrability aspects of a generalized fisher type equation with modified diffusion in 1 1 and 2 1 dimensions are studied by carrying out a singularity structure and symmetry analysis it is shown that the painleve property exists only for a special choice of the parameter m 2 a backlund transformation is shown to give rise to the linearizing transformation to the linear heat equation for this case m 2 a lie symmetry analysis also picks out the same case m 2 as the only system among this class having a nontrivial infinite dimensional lie algebra of symmetries and that the similarity variables and similarity reductions lead in a natural way to the linearizing transformation and physically important classes of solutions including known ones in the literature thereby giving a group theoretical understanding of the system for nonintegrable cases in 2 1 dimensions associated lie symmetries and similarity reductions are indicated

new contiguity relation of the sixth painleve equation from a truncation for the master painleve equation p6 u we define a consistent method adapted from the weiss truncation for partial differential equations which allows us to obtain the first degree birational transformation of okamoto two new features are implemented to achieve this result the first one is the homography between the derivative of the solution u and a riccati pseudopotential the second one is an improvement of a conjecture by fokas and ablowitz on the structure of this birational transformation we then build the contiguity relation of p6 which yields one new second order non autonomous discrete equation

polynomial time attack on wild mceliece over quadratic extensions we present a polynomial time structural attack against the mceliece system based on wild goppa codes defined over a quadratic finite field extension we show that such codes can be efficiently distinguished from random codes the attack uses this property to compute a filtration that is to say a family of nested subcodes which will reveal their secret algebraic description

on the computational complexity of the empirical mode decomposition algorithm abstract it has been claimed that the empirical mode decomposition emd and its improved version the ensemble emd eemd are computation intensive in this study we will prove that the time complexity of the emd eemd which has never been analyzed before is actually equivalent to that of the fourier transform numerical examples are presented to verify that emd eemd is in fact a computationally efficient method

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

travelling waves and static structures in a two dimensional exactly solvable reaction diffusion system exact travelling wave solutions of a fisher type reaction diffusion equation in two spatial dimensions are obtained the waves have nontrivial geometries and possess velocities smaller and larger than that of a plane wave the dispersion relationship and velocity curvature dependence generated by these solutions are characterized static structures with and without circular symmetry are constructed

n soliton solutions to a dimensional integrable equation we give explicitly n soliton solutions of a 2c 1 dimensional equation xtc xxxz 4cxxzcxxz 2c 1 x zzz 4 d 0 this equation is obtained by unifying two directional generalizations of the potential kdv equation the closed ring with the potential kp equation and the calogero bogoyavlenskij schiff equation this equation is also a reduction of the kp hierarchy we also find the miura transformation which yields the same ring of the corresponding modified equations the study of higher dimensional integrable systems is one of the central themes in integrable systems a typical example of a higher dimensional integrable system is obtained by modifying the lax operators of a basic equation the potential kdv p kdv equation in this paper the lax pair of the p kdv equation have the form l x t d 2 xcx x t 1 t x t d l x t 3 2

integrable nonlocal hirota equations we construct several new integrable systems corresponding to nonlocal versions of the hirota equation which is a particular example of higher order nonlinear schrodinger equations the integrability of the new models is established by providing their explicit forms of lax pairs or zero curvature conditions the two compatibility equations arising in this construction are found to be related to each other either by a parity transformation p by a time reversal t or a p t transformation possibly combined with a conjugation we construct explicit multisoliton solutions for these models by employing hirota s direct method as well as darboux crum transformations the nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behavior the multisoliton solutions are of varied type being for instance nonlocal in space nonlocal in time of time crystal type regular with local structures either in time space or of rogue wave type we construct several new integrable systems corresponding to nonlocal versions of the hirota equation which is a particular example of higher order nonlinear schrodinger equations the integrability of the new models is established by providing their explicit forms of lax pairs or zero curvature conditions the two compatibility equations arising in this construction are found to be related to each other either by a parity transformation p by a time reversal t or a p t transformation possibly combined with a conjugation we construct explicit multisoliton solutions for these models by employing hirota s direct method as well as darboux crum transformations the nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behavior the multisoliton solutions are of varied type being for instance nonlocal in space nonlocal in time of time crystal type regular

an exp function method for new n soliton solutions with arbitrary functions of a 2 1 dimensional vcbk system in this paper the exp function method is generalized to construct n soliton solutions of the 2 1 dimensional variable coefficient broer kaup vcbk system as a result new and more general 1 soliton 2 soliton and 3 soliton solutions with arbitrary functions are obtained from which the uniform formulae of n soliton solutions are derived based on one of the obtained 2 soliton solution the fission and fusion phenomena between the peakons and solitary waves are investigated it is shown that the exp function method may provide us with a straightforward effective and alternative mathematical tool for generating n soliton solutions of nonlinear evolution equations in mathematical physics

on nonlinear population waves we discuss a model system of partial differential equations for description of the spatio temporal dynamics of interacting populations we are interested in the waves caused by migration of the populations we assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations for the particular case of one population and one spatial dimension the general model is reduced to analytically tractable pde with polynomial nonlinearity up to 4th order we investigate this particular case and obtain two kinds of solutions i approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and ii exact solutions which describe nonlinear kink and solitary waves in an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node finally we derive conditions for the asymptotic stability of the obtained solutions

a new integrable modified korteweg de vries equation with one half degree of nonlinearity an alternative generalized equation which incorporates the modified korteweg de vries mkdv and kdv equations is proposed and from it a new integrable system of the mkdv type is found

a study of the characteristics of white noise using the empirical mode decomposition method based on numerical experiments on white noise using the empirical mode decomposition emd method we find empirically that the emd is effectively a dyadic filter the intrinsic mode function imf components are all normally distributed and the fourier spectra of the imf components are all identical and cover the same area on a semi logarithmic period scale expanding from these empirical findings we further deduce that the product of the energy density of imf and its corresponding averaged period is a constant and that the energy density function is chi squared distributed furthermore we derive the energy density spread function of the imf components through these results we establish a method of assigning statistical significance of information content for imf components from any noisy data southern oscillation index data are used to illustrate the methodology developed here

equivalence and characterizations of linear rank metric codes based on invariants abstract we show that the sequence of dimensions of the linear spaces generated by a given rank metric code together with itself under several applications of a field automorphism is an invariant for the whole equivalence class of the code the same property is proven for the sequence of dimensions of the intersections of itself under several applications of a field automorphism these invariants give rise to easily computable criteria to check if two codes are inequivalent we derive some concrete values and bounds for these dimension sequences for some known families of rank metric codes namely gabidulin and generalized twisted gabidulin codes we then derive conditions on the length of the codes with respect to the field extension degree such that codes from different families cannot be equivalent furthermore we derive upper and lower bounds on the number of equivalence classes of gabidulin codes and twisted gabidulin codes improving a result of schmidt and zhou for a wider range of parameters in the end we use the aforementioned sequences to determine a characterization result for gabidulin codes

nonclassical symmetries as special solutions of heir equations in phys d 78 1994 124 we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the given equation in the present paper we show that special solutions of the right order heir equation correspond to classical and nonclassical symmetries of the original equations an infinite number of nonlinear equations which possess nonclassical symmetries are derived

non isospectral problem in 2 1 dimensions in this paper the singular manifold method allows us to obtain a non isospectral lax pair darboux transformations and miura transformations for an equation in 2 1 dimensions and its modified version in this way we can iteratively build different kinds of solutions with solitonic behaviour

painleve analysis lie symmetries and invariant solutions of potential kadomstev petviashvili equation with time dependent coefficients in this paper the painleve analysis of 2 1 dimensional variable coefficients potential kadomstev petviashvili vcpkp equation s x email protected t s email protected x email protected t s x x x email protected t s y y 0 has been performed to study the painleve properties the vcpkp equation is also reduced to a 1 1 dimensional partial differential equation via classical lie group method and reduced equations are further studied to obtain certain general solutions

non classical symmetries and the singular manifold method a further two examples this paper discusses two equations with the conditional painleve property the usefulness of the singular manifold method as a tool for determining the non classical symmetries that reduce the equations to ordinary differential equations with the painleve property is confirmed once more the examples considered in this paper are particularly interesting because they have recently been proposed by other authors as counterexamples of the conjecture made by the authors that the singular manifold method allows us to identify non classical symmetries we demonstrate here that the conjecture still holds for these two cases as well a detailed study of the way of solving this apparent contradiction is offered

symmetries and a hierarchy of the general kdv equation two groups old and new of symmetries and their lie algebra properties for the kdv and cylindrical kdv equations are unified and extended to the general kdv equation

traveling waves to a burgers korteweg de vries type equation with higher order nonlinearities abstract in this paper first we survey some recent advances in the study of traveling wave solutions to the burgers korteweg de vries equation and some comments are given then we study a burgers korteweg de vries type equation with higher order nonlinearities a qualitative analysis to a two dimensional autonomous system which is equivalent to the burgers kdv type equation is presented and indicates that under certain conditions the burgers korteweg de vries type equation has neither nontrivial bell profile solitary waves nor periodic waves finally a solitary wave solution is obtained by means of the first integral method which is based on the ring theory of commutative algebra

kdv extensions with painleve property by means of the conformal invariance mobious transformation invariance the well known kdv equation is extended to i a 1 1 dimensional space time symmetric form ii two 2 1 dimensional space isotropic forms and iii general 3 1 dimensional and n dimensional forms the extensions are proven to be integrable under the meaning that they possess the painleve property

on a coupled system of equations describing pulse propagation in quadratic media in the slowly varying envelope approximation we derive the basic equations that describe the propagation of ultrashort pulses in quadratically nonlinear media in which a wave at a fundamental frequency interacts with its second harmonic in the governing equations we keep linear terms that account for both second and third order dispersion and nonlinear terms describing both nonlinear dispersion and self steepening of the pulse edge we then perform the painleve singularity structure analysis of the most general system of coupled partial differential equations we derived in a specific case when third order dispersion is negligible by using a hirota like method we found zero and one parameter families of bright fundamental frequency and dark second harmonic solitary waves which travel at a locked group velocity