on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

a new auxiliary function method for solving the generalized coupled hirota satsuma kdv system abstract in this letter a new auxiliary function method is presented for constructing exact travelling wave solutions of nonlinear partial differential equations the main idea of this method is to take full advantage of the solutions of the elliptic equation to construct exact travelling wave solutions of nonlinear partial differential equations more new exact travelling wave solutions are obtained for the generalized coupled hirota satsuma kdv system

exactly soluble random field ising models in one dimension the authors solve exactly the one dimensional random field ising model for two classes of magnetic field distributions symmetric exponential model i and non symmetric exponential model ii for both models expressions for the free energy at all finite temperatures are presented the low temperature region is examined in more detail they obtain the zero temperature energy and entropy in closed form it is shown that the free energy of both models has an expansion in integer powers of temperature model i has a non vanishing zero point entropy for all values of the parameters as soon as randomness is diluted in model ii the zero point entropy is zero except for a discrete sequence of values of one parameter in some cases the zero temperature magnetisation is positive whereas the average magnetic field is negative the magnetisation may also change sign as a function of temperature

supersymmetric quantum mechanics with levy disorder in one dimension we consider the schrodinger equation with a random potential of the form open image in new window where w is a levy noise we focus on the problem of computing the so called complex lyapunov exponent open image in new window where n is the integrated density of states of the system and is the lyapunov exponent in the case where the levy process is non decreasing we show that the calculation of reduces to a stieltjes moment problem we ascertain the low energy behaviour of the density of states in some generality and relate it to the distributional properties of the levy process we review the known solvable cases where can be expressed in terms of special functions and discover a new one

orbits and phase transitions in the multifractal spectrum we consider the one dimensional classical ising model in a symmetric dichotomous random field the problem is reduced to a random iterated function system rifs for an effective field the dq spectrum of the invariant measure of this effective field exhibits a sharp drop of all dq with q 0 at some critical strength of the random field we introduce the concept of orbits which naturally group the points of the support of the invariant measure we then show that the pointwise dimension at all points of an orbit has the same value and calculate it for a class of periodic orbits and their so called offshoots as well as for generic orbits in the non overlapping case the sharp drop in the dq spectrum is analytically explained by a drastic change of the scaling properties of the measure near the points of a certain periodic orbit at a critical strength of the random field which is explicitly given a similar drastic change near the points of a special family of periodic orbits explains a second hitherto unnoticed transition in the dq spectrum as it turns out a decisive role in this mechanism is played by a specific offshoot we furthermore give rigorous upper and or lower bounds on all dq in a wide parameter range in most cases the numerically obtained dq coincide with either the upper or the lower bound the results in this paper are relevant for the understanding of rifss in the case of moderate overlap in which periodic orbits with weak singularity can play a decisive role

new exact traveling wave solutions for the modified form of degasperis procesi equation abstract with the aid of symbolic computation a generalized auxiliary equation method is proposed to construct more general exact solutions to some types of nlpdes in this letter using the auxiliary equation method some new solitary and traveling wave solutions of such nonlinear mdp equations given by u t u xxt 4 u 2 u x 3 u x u xx uu xxx are obtained in a very general form several interesting special cases of these general solutions are also discussed

einstein maxwell solitons the application of the inverse scattering method to the einstein maxwell equations for stationary axisymmetric exterior fields leads the authors to an explicit formula for the ernst and electromagnetic potentials of new exact solutions generated from an arbitrary seed solution

darboux transforms deep reductions and solitons explicit formalisms for deep reductions of matrix differential equations and darboux covariance properties are presented the matrix 2 2 spectral problem of the second order is considered this problem with the reduction constraints being imposed on the potentials is the first equation of the lax representation of the hirota satsuma system 1981 the two reduction cases are treated with the help of the bilinear delta forms the covariance of these forms with respect to the darboux transforms under restrictions gives rise to explicit formulas of n soliton solutions in particular the two parameter soliton solutions of the hirota satsuma system are obtained the specific feature of such solutions evolution is that the singularity appears in some parameter region the yajima oikawa system 1976 is given as an example of the technique application to a 3 3 spectral problem

multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system compared with most of the existing tanh methods the jacobi elliptic function method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters the solutions obtained in this paper include a kink shaped and bell shaped soliton solutions b rational solutions c triangular periodic solutions and d jacobi and weierstrass doubly periodic wave solutions among them the jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition the efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper kdv mkdv ito s fifth mkdv hirota nizhnik novikov veselov broer kaup generalized coupled hirota satsuma coupled schrodinger kdv 2 1 dimensional dispersive long wave 2 1 dimensional davey stewartson equations in addition as an illustrative sample the properties of the soliton solutions and jacobi doubly periodic solutions for the hirota equation are shown by some figures the links among our proposed method the tanh method extended tanh method and the jacobi elliptic function method are clarified generally

singular behaviour of the free energy for random ising chains the ising chains in a random external field or with random interactions are studied the equivalence of these models is shown and the functional equation for the quenched free energy is derived for the system in a random external field the authors prove rigorously the non analytical behaviour of the free energy in the limit of large constant interactions f j approximately exp 2j alpha with positive derrida hilhorst roots alpha 1 determined by exp 2h alpha 1 the generalization to quasi one dimensional systems is given the system with random interactions the fraction of which is infinitely large is solved exactly for large but finite interactions they prove again that the singular behaviour of f can be found depending on the magnitude of the external magnetic field and on the probability distribution of interactions

hard variants of stable marriage the stable marriage problem and its many variants have been widely studied in the literature gusfield and irving the stable marriage problem structure and algorithms mit press cambridge ma 1989 roth and sotomayor two sided matching a study in game theoretic modeling and analysis econometric society monographs vol 18 cambridge university press cambridge 1990 knuth stable marriage and its relation to other combinatorial problems crm proceedings and lecture notes vol 10 american mathematical society providence ri 1997 partly because of the inherent appeal of the problem partly because of the elegance of the associated structures and algorithms and partly because of important practical applications such as the national resident matching program roth j political economy 92 6 1984 991 and similar large scale matching schemes here we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered we show that under surprisingly restrictive assumptions a number of these variants are hard and hard to approximate the key observation is that in contrast to the case where preference lists are complete or strictly ordered or both a given problem instance may admit stable matchings of different sizes in this setting examples of problems that are hard are finding a stable matching of maximum or minimum size determining whether a given pair is stableeven if the indifference takes the form of ties on one side only the ties are at the tails of lists there is at most one tie per list and each tie is of length 2 and finding or approximating both an egalitarian and a minimum regret stable matching however we give a 2 approximation algorithm for the problems of finding a stable matching of maximum or minimum size we also discuss the significant implications of our results for practical matching schemes

convolution of multifractals and the local magnetization in a random field ising chain the local magnetization in the one dimensional random field ising model is essentially the sum of two effective fields with multifractal probability measure the probability measure of the local magnetization is thus the convolution of two multifractals in this paper we prove relations between the multifractal properties of two measures and the multifractal properties of their convolution the pointwise dimension at the boundary of the support of the convolution is the sum of the pointwise dimensions at the boundary of the support of the convoluted measures and the generalized box dimensions of the convolution are bounded from above by the sum of the generalized box dimensions of the convoluted measures the generalized box dimensions of the convolution of cantor sets with weights can be calculated analytically for certain parameter ranges and illustrate effects we also encounter in the case of the measure of the local magnetization returning to the study of this measure we apply the general inequalities and present numerical approximations of the dq spectrum for the first time we are able to obtain results on multifractal properties of a physical quantity in the one dimensional random field ising model which in principle could be measured experimentally the numerically generated probability densities for the local magnetization show impressively the gradual transition from a monomodal to a bimodal distribution for growing random field strength h

approximability results for stable marriage problems with ties we consider instances of the classical stable marriage problem in which persons may include ties in their preference lists we show that in such a setting strong lower bounds hold for the approximability of each of the problems of finding an egalitarian minimum regret and sex equal stable matching we also consider stable marriage instances in which persons may express unacceptable partners in addition to ties in this setting we prove that there are constants such that each of the problems of approximating a maximum and minimum cardinality stable matching within factors of respectively is np hard under strong restrictions we also give an approximation algorithm for both problems that has a performance guarantee expressible in terms of the number of lists with ties this significantly improves on the best known previous performance guarantee for the case that the ties are sparse our results have applications to large scale centralized matching schemes

an algebro geometric solution for a hamiltonian system with application to dispersive long wave equation by using an iterative algebraic method we derive from a spectral problem a hierarchy of nonlinear evolution equations associated with dispersive long wave equation it is shown that the hierarchy is integrable in liouville sense and possesses bi hamiltonian structure two commutators with zero curvature and lax representations for the hierarchy are constructed respectively by using two different systematic methods under a bargmann constraint the spectral is nonlinearized to a completely integrable finite dimensional hamiltonian system by introducing the abel jacobi coordinates an algebro geometric solution for the dispersive long wave equation is derived by resorting to the riemann theta function

hyperelliptic functions solutions of some nonlinear partial differential equations using the direct method in this paper we will use a simple and direct method to obtain particular solutions of some 2 1 dimensional nonlinear partial differential equations expressed in terms of the kleinian hyperelliptic functions for a given curve y 2 f x whose genus is two we observe that this method generalizes the auxiliary method

strongly coupled ising chain under a weak random field abstract we present here an analytical method based on the transfer matrix m to study quenched disorder in one dimensional spin systems in the limit of strong couplings and weak disorder the procedure is formulated for the random field ising chain of finite length l and its properties represented as functions of m satisfy a differential equation of the fokker planck type this equation describes evolution with l and a central limit theorem of a novel kind provides the equation and its solutions with universal character we obtain analytical expressions for the moments of the magnetization of the infinite length chain and study the approach to the infinite coupling limit we find that the random field free energy f h and the edwards anderson order parameter m 2 satisfy a simple relation we discuss our results in connection to previous work by luck and nieuwenhuizen

a new general algebraic method and its applications to the 2 1 dimensional broer kaup kupershmidt equations abstract using a computerized symbolic computation technique a new method named the repeated general algebraic method rgam is established in this study in order to find exact solutions of nonlinear partial differential equations nlpdes the new method is validated based on the 2 1 dimensional broer kaup kupershmidt bkk equations by using the rgam in various conditions a number of exact solutions of nlpdes have been obtained showing potential importance in future physical applications also it is anticipated that the rgam can be applied to other nonlinear evolution equations in mathematical physics to produce some interesting outcomes

solitary wave and periodic wave solutions for the thermally forced gravity waves in atmosphere by introducing a new transformation a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system which extends fan s direct algebraic method to the case when r 4 the solutions of a first order nonlinear ordinary differential equation with a higher degree nonlinear term and fan s direct algebraic method of obtaining exact solutions to nonlinear partial differential equations are applied to the combined kdv mkdv gkdv equation which is derived from a simple incompressible non hydrostatic boussinesq equation with the influence of thermal forcing and is applied to investigate internal gravity waves in the atmosphere as a result by taking advantage of the new first order nonlinear ordinary differential equation with a fifth degree nonlinear term and an eighth degree nonlinear term periodic wave solutions associated with the jacobin elliptic function and the bell and kink profile solitary wave solutions are obtained under the effect of thermal forcing most importantly the mechanism of propagation and generation of the periodic waves and the solitary waves is analysed in detail according to the values of the heating parameter which show that the effect of heating in atmosphere helps to excite westerly or easterly propagating periodic internal gravity waves and internal solitary waves in atmosphere which are affected by the local excitation structures in atmosphere in addition as an illustrative sample the properties of the solitary wave solution and jacobin periodic solution are shown by some figures under the consideration of heating interaction

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

some applications of the g g expansion method to non linear partial differential equations in the present paper we construct the traveling wave solutions involving parameters of the 2 1 dimensional higher order broer kaup equations the 2 1 dimensional breaking soliton equations the 2 1 dimensional asymmetric nizhnik novikov vesselov equations and the 2 1 dimensional bkp equations in terms of the hyperbolic functions trigonometric functions and the rational functions by using a new approach namely the g g expansion method where g g x satisfies a second order linear ordinary differential equation when the parameters are taken special values the solitary waves are derived from the traveling waves the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions

link between solitary waves and projective riccati equations many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective hence linearizable riccati system from that property the authors deduce a method for building these solutions by determining only a finite number of coefficients this method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation they handle several examples for the henon heiles hamiltonian system they obtain several exact solutions one of them defines a new solitary wave solution for a coupled system of boussinesq and nonlinear schrodinger equations for a third order dispersive equation with two monomial nonlinearities they isolate all cases where the general solution is single valued

reduction of dispersionless coupled korteweg de vries equations to the euler darboux equation a quasilinear hyperbolic system of two first order equations is introduced the system is linearized by means of the hodograph transformation combined with riemann s method of characteristics in the process of linearization the main step is to explicitly express the characteristic velocities in terms of the riemann invariants the procedure is shown to be performed by quadrature only for specific combinations of the parameters in the system we then apply the method developed here to the dispersionless versions of the typical coupled korteweg de vries ckdv equations including the broer kaup ito hirota satsuma and bogoyavlenskii equations and show that these equations are transformed into the classical euler darboux equation a more general quasilinear system of equations is also considered with application to the dispersionless ckdv equations for the jaulent miodek and nutku oguz equations

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

semianalytical solution of the random product problem of matrices and discrete time random evolution we consider the product of a large number of matrices chosen randomly with some correlation as a or b we introduce an exact functional equation whose solution gives the limiting distribution and the lyapunov exponent for the product of large number of matrices thus we suggest new semianalytical method to solve the random product problem of correlated matrices the method can be easily generalized to high order matrices as we deal with a larger systems of functional equations we have provided a known numerical algorithm for the solution of the matrix product problem we have also applied our method for a solution of discrete time random evolution our functional equation method provides exact transition point

the distribution of the reflection phase of disordered conductors the reflection coefficients of one dimensional disordered conductors is asymptotically a pure phase when the sample length is much larger than the localisation length the authors study the stationary distribution of this reflection phase for tight binding chains with diagonal disorder i e random site potentials they derive the dyson integral equation obeyed by this distribution and relate it to the lyapunov exponent and the integrated density of states this general approach allows them to consider then more specifically two types of disorder in the case of a binary alloy where the potentials take two values the distribution of the reflection phase is shown to have generically an infinity of power law singularities related to the occurrence of periodic patterns of potentials this phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy dependent threshold they also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials this non trivial solvable model allows them to examine analytically various properties of the phase distribution such as the anomalies which occur at weak disorder

two types of generalized integrable decompositions and new solitary wave solutions for the modified kadomtsev petviashvili equation with symbolic computation the modified kadomtsev petviashvili mkp equation is shown in this paper to be decomposable into the first two soliton equations of the 2n coupled chen lee liu and kaup newell hierarchies by respectively nonlinearizing two sets of symmetry lax pairs in these two cases the decomposed 1 1 dimensional nonlinear systems both have a couple of different lax representations which means that there are two linear systems associated with the mkp equation under the same constraint between the potential and eigenfunctions for each lax representation of the decomposed 1 1 dimensional nonlinear systems the corresponding darboux transformation is further constructed such that a series of explicit solutions of the mkp equation can be recursively generated with the assistance of symbolic computation in illustration four new families of solitary wave solutions are presented and the relevant stability is analyzed

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

korteweg de vries modified korteweg de vries systems and darboux transforms in 1 1 and 2 1 dimensions the jaulent miodek hierarchy m jaulent and i miodek lett math phys 1 243 1976 and its two dimensional analog are considered from the point of view of darboux transforms structures a new form of those transforms as well as old fashioned forms are presented the determinant form of solitons is obtained the combination of different transforms in the 2 1 case allows the buildup of localized solutions in both spatial dimensions including the two dimensional korteweg de vries kdv equation

generalized tanh method with symbolic computation and generalized shallow water wave equation abstract we report that symbolic computation with the generalized tanh method leads to new soliton like solutions for a 2 1 dimensional generalization of the shallow water wave equations

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

universal distribution of lyapunov exponents for products of ginibre matrices starting from exact analytical results on singular values and complex eigenvalues of products of independent gaussian complex random n n matrices also called the ginibre ensemble we rederive the lyapunov exponents for an infinite product we show that for a large number t of product matrices the distribution of each lyapunov exponent is normal and we compute its t dependent variance as well as corrections in a large t expansion originally lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order we discuss a general mechanism for 2 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large t limit we thereby rederive newman s triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits and on the global scale and on the local scale we show as a mathematical byproduct that a particular asymptotic expansion of a meijer g function with large index leads to a gaussian

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

correlations in random ising chains at zero temperature we present a general method to calculate the connected correlation function of random ising chains at zero temperature this quantity is shown to relate to the survival probability of some one dimensional adsorbing random walker on a finite interval the size of which is controlled by the strength of the randomness the correlation length is exactly calculated for different random field and random bond distributions

a generalized hirota equation in 2 1 dimensions using an asymptotically exact reduction method based on fourier expansion and spatiotemporal rescaling a new integrable and nonlinear partial differential equation pde in 2 1 dimensions is obtained starting from the kadomtsev petviashvili equation we apply the reduction technique to the lax pair of the kadomtsev petviashvili equation and demonstrate the integrability property of the new equation because we obtain the corresponding lax pair the new equation reduces to the hirota equation in the 1 1 dimensional limit

some new integrable systems constructed from the bi hamiltonian systems with pure differential hamiltonian operators abstract when both hamiltonian operators of a bi hamiltonian system are pure differential operators we show that the generalized kupershmidt deformation gkd developed from the kupershmidt deformation in 10 offers an useful way to construct new integrable system starting from the bi hamiltonian system we construct some new integrable systems by means of the generalized kupershmidt deformation in the cases of harry dym hierarchy classical boussinesq hierarchy and coupled kdv hierarchy we show that the gkd of harry dym equation gkd of classical boussinesq equation and gkd of coupled kdv equation are equivalent to the new integrable rosochatius deformations of these soliton equations with self consistent sources we present the lax pair for these new systems therefore the generalized kupershmidt deformation provides a new way to construct new integrable systems from bi hamiltonian systems and also offers a new approach to obtain the rosochatius deformation of soliton equation with self consistent sources

quasi periodic solution of a new 2 1 dimensional coupled soliton equation a new 2 1 dimensional integrable soliton equation is proposed which has a close connection with the levi soliton hierarchy through the nonlinearization of the levi eigenvalue problems we obtain a finite dimensional integrable system the abel jacobi coordinates are constructed to straighten out the hamiltonian flows by which the solutions of both the 1 1 and 2 1 levi equations are obtained through linear superpositions an inversion procedure gives the quasi periodic solution in the original coordinates in terms of the riemann theta functions

symbolic analysis and exact travelling wave solutions to a new modified novikov equation in this paper a new modified novikov equation is introduced multiple exact travelling wave solutions are obtained by using the extended tanh function method furthermore based on the homogeneous balance method an auto backlund transformation is derived and some solitary wave solutions to this new equation are established

an algebraic method to construct the darboux matrix we present an effective procedure to construct the 1 soliton darboux matrix our approach based on the zakharov shabat mikhailov s dressing method is especially useful in the case of non canonical normalization and for non isospectral linear problems the construction is divided into two steps first we represent a given linear problem as a system of some algebraic constraints on two matrices in this context we introduce and discuss invariants of the darboux matrix second we derive the darboux matrix demanding that it preserves the algebraic constraints in particular we consider in details the restrictions imposed by various reduction groups on the form of the darboux matrix

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

exact solutions for ising chains in a random field the authors present an exact solution for the one dimensional ising model in a random field the distribution of field strengths is site independent symmetric and has a three peak structure the free energy is obtained for all finite temperatures the low temperature behaviour is studied in detail they find an expansion for the free energy in integer powers of temperature the ground state energy and the zero point entropy are calculated explicitly the specific heat is linear for low temperatures in contradiction with mean field theory the origin of an additional exponential parameter is discussed

two dimensional ising model with annealed random fields we investigate the influence of annealed random fields on the phase diagram of the ferromagnetic ising model on the square lattice we find that the gaussian random field makes the ferromagnetic ground state unstable and we have a super antiferromagnetic state at low temperatures for the binary random field in which the random field takes h with probability p and h with probability 1 p there is a critical field hc and the ferromagnetic ground state is stable for h hc but unstable for h 62 hc when the ferromagnetic ground state is unstable we have a re entrant phase transition

symmetry algebras of the potential nizhnik novikov veselov model two sets of generalized symmetries of the potential nizhnik novikov veselov equation are given in a simple formula by developing a generalized formal series method for searching symmetries of one type of generalized partial differential equations these two sets of symmetries constitute an infinite dimensional lie algebra which contains three types of virasoro algebras and three w type algebras as its special cases

hodograph solutions of the dispersionless coupled kdv hierarchies critical points and the euler poisson darboux equation it is shown that the hodograph solutions of the dispersionless coupled kdv dckdv hierarchies describe critical and degenerate critical points of a scalar function which obeys the euler poisson darboux equation singular sectors of each dckdv hierarchy are found to be described by solutions of higher genus dckdv hierarchies concrete solutions exhibiting shock type singularities are presented

an efficient algorithm for the optimal stable marriage in an instance of size n of the stable marriage problem each of n men and n women ranks the members of the opposite sex in order of preference a stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching it is well known 2 that at least one stable matching exists for every stable marriage instance however the classical gale shapley algorithm produces a marriage that greatly favors the men at the expense of the women or vice versa the problem arises of finding a stable matching that is optimal under some more equitable or egalitarian criterion of optimality this problem was posed by knuth 6 and has remained unsolved for some time here the objective of maximizing the average or equivalently the total satisfaction of all people is used this objective is achieved when a person s satisfaction is measured by the position of his her partner in his her preference list by exploiting the structure of the set of all stable matchings and using graph theoretic methods an o n 4 algorithm for this problem is derived

constructing of new exact solutions to the gkdv mkdv equation with any order nonlinear terms by g g expansion method abstract the g g expansion method is introduced to construct more general exact travelling wave solutions to the generalized kdv mkdv equation with any order nonlinear terms as a result hyperbolic function solution trigonometric function solution and rational solution with parameters are obtained when the parameters take up special values the solitary waves are also derived from the travelling wave solutions the g g expansion method presents a wider applicability for handling nonlinear wave equations

quasi rational function solutions of an elliptic equation and its application to solve some nonlinear evolution equations abstract using the differential transformation method and the homogeneous balance method some new solutions of an auxiliary elliptic equation are obtained these solutions possess the forms of rational functions in terms of trigonometric functions hyperbolic functions exponential functions power functions elliptic functions and their operation and composite functions and so on which are so called quasi rational function solutions based on these new quasi rational functions solutions a direct method is proposed to construct the exact solutions of some nonlinear evolution equations with the aid of symbolic computation the coupled kdv mkdv equation and broer kaup equations are chosen to illustrate the effectiveness and convenience of the suggested method for obtaining quasi rational function solutions of nonlinear evolution equations

a hierarchy of coupled korteweg de vries equations and the normalisation conditions of the hilbert riemann problem the authors discuss the normalisation conditions of the hilbert riemann problem which is of importance for the construction of the ladder of soliton solutions for a given spectral problem they provide the prescription for constructing the appropriate normalisation condition all this is applied to a new hierarchy of coupled korteweg de vries equations to find the soliton solution which displays some peculiar phenomena

exact electronic spectra and inverse localization lengths in one dimensional random systems i random alloy liquid metal and liquid alloy analytic continuations into the complex energy plane of dyson schmidt type of equations for the calculation of the density of states are constructed for a random alloy model a liquid metal and for a liquid alloy in all these models the characteristic function follows from the solution of this equation its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions the equations have been solved exactly for some distributions of the random variables in the random alloy case the strengths of the delta potentials have an exponential distribution they may also have finite exponentially distributed values with probability 0 p 1 and be infinite with probability q 1 p in the liquid metal the liquid particles are assumed to behave like hard rods this implies an exponential distribution of the distances between the particles the common electronic potential may be arbitrary but is assumed to vanish outside the rods in the one dimensional liquid alloy there is apart from positional randomness of the liquid particles a distribution of the strengths of the electronic delta potentials for cauchy distributions an argument of lloyd is extended to obtain the characteristic function from the one in the model with equal strengths for the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above for several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed new results are found exponential decay of the density of states near special energies in the random alloy and liquid metal divergence of the density of states at certain energies with non classical exponent 13 in the random alloy if the average of the potential strengths vanishes exponentially small broadening of the bound state levels for low concentrations of the liquid particles peak in the localization length at the bound state energies which becomes exponentially narrow for low concentrations different exponent in the decay of the inverse localization length at large energies for delta potentials and square well potentials further an expression for the grand potential is given involving a sum over the characteristic function at certain points and divergence of the zero point energy is found for cauchy distributions of the delta potential strengths

towards a unified method for exact solutions of evolution equations an application to reaction diffusion equations with finite memory transport we present a brief report on the different methods for finding exact solutions of nonlinear evolution equations explicit exact traveling wave solutions are the most amenable besides implicit and parametric ones it is shown that most of methods that exist in the literature are equivalent to the generalized mapping method that unifies them by using this method a class of formal exact solutions for reaction diffusion equations with finite memory transport is obtained attention is focused to the finite memory transport fisher and nagumo equations

the hyperbolic allen cahn equation exact solutions using the first integral method a general set of analytical solutions is obtained for the hyperbolic allen cahn equation the solutions are presented by i the class of continual solutions described by profiles for traveling waves of the order parameter and ii the class of singular solutions which exhibit unbounded discontinuity in the profile of the order parameter at the origin of the coordinate system it is shown that the solutions include the previous analytical results for the parabolic allen cahn equation as a limited class of functions in which the inertial effects are omitted

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

a generalized coupled korteweg de vries hierarchy bi hamiltonian structure and darboux transformation a new generalized coupled korteweg de vries kdv hierarchy is presented starting from a 4 4 matrix spectral problem with four potentials its generalized bi hamiltonian structure is also investigated by using the trace identity most importantly a new generalized coupled kdv equation is produced moreover a darboux transformation for the generalized coupled kdv equation is established with the aid of the gauge transformation between the corresponding 4 4 matrix spectral problems by which some explicit solutions of the generalized coupled kdv equation are obtained as a reduction a darboux transformation of the kdv equation and its explicit solutions are given

exact solutions for spectra and green s functions in random one dimensional systems for chains of harmonic oscillators with random masses a set of equations is derived which determine the spatial fourier components of the average one particle green s function these equations are valid for complex values of the frequency a relation between the spectral density and functions introduced by schmidt is discussed exact solutions for this green s function and the less complicated characteristics function the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions are derived for exponential distributions of the masses for some cases the characteristic function is calculated numerically for gamma distributions the equations are cast in the form of ordinary higher order differential equations these have been solved numerically for determining the characteristic function for arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the green s function are discussed the method is also applied to chains where the spring constants and or the masses have random values also for these systems exact solutions are discussed for exponential distributions e g of both masses and spring constants the characteristic function is expressed in bessel functions the relation with certain random relaxation models is shown finally x y hamiltonians with random exchange constants and or magnetic fields or equivalently tight binding electron models with diagonal and or off diagonal disorder are considered here the green s function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin new solutions for the characteristic function and green s function are derived for a number of cases including exponentially distributed magnetic fields and power law distributed exchange constants

delta dressing and exact solutions for the 2 1 dimensional harry dym equation exact implicit solutions with functional parameters of the 2 1 dimensional harry dym hd equation are constructed by the delta dressing method the interrelation between the hd and modified kadomtsev petviashvili mkp equations is established within the framework of the delta dressing method knowledge of the mkp eigenfunctions allows the solutions of the 2 1 dimensional hd equation to be represented in a simple parametric form

the inhomogeneous toda lattice its hierarchy and darboux backlund transformations the authors show how one can construct hierarchies of nonlinear differential difference equations with n dependent coefficients among these equations they present explicitly a set of inhomogeneous toda lattice equations which are associated with a discrete schrodinger spectral problem whose potentials diverge asymptotically then they derive a new darboux transformation which allows them to get bounded solutions for the equations presented before and apply it in a specially simple case when the solution turns out to be expressed in terms of hermite polynomials

second harmonic generation hamiltonian structures and particular solutions for the equations describing second harmonic generation shg we find a linear system different from one found by kaup but connected with it by a gauge transformation which enables us to consider the shg equations as related to the first negative flow in the coupled kdv hierarchy and to find therefore the hamiltonian structures and the recursion operator with the help of the recursion operator we write down the simplest commuting flows and consider their stationary points in order to find particular solutions of the shg equations we also investigate some particular invariant solutions of the shg equations that can be of physical interest and find their connections with invariant solutions of the tzitzeica equation the latter are either connected with painleve iii or solvable in elliptic functions

the g g expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics i the present paper we construct the traveling wave solutions involving parameters of the combined korteweg de vries modified korteweg de vries equation the reaction diffusion equation the compound kdv burgers equation and the generalized shallow water wave equation by using a new approach namely the g g expansion method where g g satisfies a second order linear ordinary differential equation when the parameters take special values the solitary waves are derived from the traveling waves the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

the entropy of a binary hidden markov process the entropy of a binary symmetric hidden markov process is calculated as an expansion in the noise parameter e we map the problem onto a one dimensional ising model in a large field of random signs and calculate the expansion coefficients up to second order in e using a conjecture we extend the calculation to 11th order and discuss the convergence of the resulting series

the multi component dispersive long wave equation hierarchy its integrable couplings and their hamiltonian structures abstract an isospectral problem is established based on loop algebra x by making use of the generalized tu scheme the multi component liouville integrable dispersive long wave dlw equation hierarchy is obtained then two expanding loop algebra f m and y are presented which devoted to working out two integrable couplings of the multi component dlw equation hierarchy finally the hamiltonian structures of the multi component dlw equation hierarchy and the second integrable couplings are obtained by employing the trace variational identity

a new approach to the problem of disordered harmonic chains abstract in the problem of harmonic chains with random masses the exponential growth rate and the spectral distribution function are shown to be the real and imaginary part of dyson s characteristic function along its cut in the complex plane a new function satisfying a certain integral equation is introduced from which the characteristic function follows immediately the functions introduced by dyson schmidt and matsuda and ishii are recovered as special cases and it is shown that schmidt s distributions w n converge as n relations between dyson s and schmidt s functions are obtained finally the same method is applied to the problem of harmonic chains with random variables k n m n and k n 1 m n and to the tight binding electron model with random site energies

c neumann and bargmann systems associated with the coupled kdv soliton hierarchy under two different constraints between the potentials and the eigenfunctions the eigenvalue problem associated with the coupled kdv hierarchy is nonlinearised to be a completely integrable c neumann system on the tangent bundle of sphere tsn 1 with the hamiltonian h 1 2 lambda q p 1 2 q q p p 1 2 q p 2 1 2 q p lambda p p and a completely integrable bargmann system r2n dpdq h with the hamiltonian h 1 2 lambda q p 1 2 p p q p 1 2 q q respectively the involutive solutions of the coupled kdv equation associated with the two systems are given

one dimensional random ising models a study of several one dimensional random ising chains and strips is presented the free energy is expressed in terms of a probability distribution the support of which is studied with use of a recurrence relation the shape of the support is shown to vary from a connected set to a cantor set depending on the type of randomness monte carlo calculations and analytical approximations are presented

correlation function of random field ising chains is it lorentzian or not the two point connected correlation function or wavevector dependent susceptibility of the ferromagnetic ising chain in a random field is calculated exactly at any temperature for a two parameter family of diluted symmetric exponential distributions of the magnetic fields thermodynamic properties of this model have been derived in a previous work by the authors besides the correlation function itself the solution provides exact results for the usual susceptibility the correlation length and the edwards anderson parameter the low temperature regime is examined in full detail the authors obtain in closed form the limit value at zero temperature of various quantities and the first correction to this limit which behaves linearly with temperature the correlation length is discontinuous at zero temperature they also derive the scaling form of the correlation function for small p where the dilution p is the probability for a spin to be subjected to a non zero random field even in this limit the correlation function is more complicated than the simple lorentzian predicted e g by mean field theory

a generalized expansion method for nonlinear wave equations a generalized jacobian exponential expansion method for finding the exact traveling wave solutions of a nonlinear partial differential equation is discussed we use this method to construct many new previously undiscovered exact solutions for the boussinesq and modified kdv equations we also apply it to the shallow long wave approximate equations new solutions are deduced for this system of partial differential equations

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

efficient algorithms for generalized stable marriage and roommates problems we consider a generalization of the stable roommates problem sr in which preference lists may be partially ordered and forbidden pairs may be present denoted by srpf this includes as a special case a corresponding generalization of the classical stable marriage problem sm denoted by smpf by extending previous work of feder we give a two step reduction from srpf to 2 sat this has many consequences including fast algorithms for a range of problems associated with finding optimal stable matchings and listing all solutions given variants of sr and sm for example given an smpf instance i we show that there exists an o m succinct certificate for the unsolvability of i an o m algorithm for finding all the super stable pairs in i an o m kn algorithm for listing all the super stable matchings in i an o m 1 5 algorithm for finding an egalitarian super stable matching in i and an o m algorithm for finding a minimum regret super stable matching in i where n is the number of men m is the total length of the preference lists and k is the number of super stable matchings in i analogous results apply in the case of srpf

exact travelling wave solutions of the generalized bretherton equation we search for exact travelling wave solutions of the generalized bretherton equation for integer greater than one values of the exponent m of the nonlinear term by two methods the truncated painleve expansion method and an algebraic method we find periodic solutions for m 2 and m 5 to add to those already known for m 3 in all three cases these solutions exist for finite intervals of the wave velocity we also find a kink shaped solitary wave for m 5 and a family of elementary unbounded solutions for arbitrary m

low temperature thermodynamics of random field ising chains exact results the authors consider the random field ferromagnetic ising chain for a class of continuous symmetric probability distributions of the random magnetic fields of a diluted power times exponential type this class of distribution is exactly solvable in the sense that the disorder can be integrated explicitly at any temperature a detailed analysis of the low temperature thermodynamics is presented exact expressions are obtained for the ground state energy the zero temperature entropy and the amplitude of the specific heat which vanishes linearly at low temperature the diluted symmetric binary distribution where the magnetic fields can only assume the values or hb or zero can be viewed as a limiting case of the class of exactly solvable distributions the low temperature physics of this discrete model is investigated in detail including the exponential fall off of the specific heat

rational solutions of the classical boussinesq hierarchy the classical boussinesq system and its associated higher order flows in the bilinear form are considered an infinite family of homogeneous rational solutions of this hierarchy is verified by the use of the wronskian technique developed by freeman and nimmo 1983

elliptic solutions of the quintic complex one dimensional ginzburg landau equation the conte musette method has been modified for the search of only elliptic solutions to systems of differential equations a key idea of this a prior restriction is to simplify calculations by means of the use of a few laurent series solutions instead of one and the use of the residue theorem the application of this approach to the quintic complex one dimensional ginzburg landau equation cgle5 allows us to find elliptic solutions in the wave form restrictions on coefficients which are necessary conditions for the existence of elliptic solutions for the cgle5 have been found as well we demonstrate that to find elliptic solutions the analysis of a system of differential equations is preferable to the analysis of the equivalent single differential equation

a family of completely integrable multi hamiltonian systems explicitly related to some celebrated equations by introducing a spectral problem with an arbitrary parameter we derive a kaup newell type hierarchy of nonlinear evolution equations which is explicitly related to many important equations such as the kundu equation the kaup newell kn equation the chen lee liu cll equation the gerdjikov ivanov gi equation the burgers equation the modified korteweg devries mkdv equation and the sharma tasso olver equation it is shown that the hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure under the bargann constraint between the potentials and the eigenfunctions the spectral problem is nonlinearized as a finite dimensional completely integrable hamiltonian system the involutive representation of the solutions for the kaup newell type hierarchy is also presented in addition an n fold darboux transformation of the kundu equation is constructed with the help of its lax pairs and a reduction technique according to the darboux transformation the solutions of the ku

multi soliton and pfaffian solutions of a 2 1 dimensional nonlinear evolution equation via the jaulent miodek hierarchy abstract under investigation in this paper is a 2 1 dimensional nonlinear evolution equation generated by the jaulent miodek hierarchy via the hirota bilinear method multi soliton solutions of such equation are obtained soliton propagation and interactions are analyzed and both the elastic and inelastic interactions are seen with some parameters selected furthermore it is proved that the equation possesses the n soliton solutions expressed in terms of gramm type pfaffian and can be written in the form of a pfaffian identity

symbolic computation and a uniform direct ansatze method for constructing travelling wave solutions of nonlinear evolution equations on the basis of the computer symbolic system maple and the tanh method the riccati equation method as well as all kinds of improved versions of these methods we present a further uniform direct ansatze method for constructing travelling wave solutions of nonlinear evolution equations compared with the existing methods the presented method can be used to construct more new general solutions and we give some examples to illustrate the key step of our method

three fast algorithms for four problems in stable marriage the stable marriage problem is a well known problem of matching n men to n women to achieve a certain type of stability the o n 2 time gale shapley gs algorithm for finding two particular but extreme stable marriages out of a possibly exponential number of stable marriages is also well known in this paper we consider four problems concerned with finding information about the set of all stable marriages and with finding stable marriages other than those obtained by the gale shapley algorithm in particular we give an o n 2 time algorithm which for any problem instance of n men and n women finds every man woman pair that is contained in at least one stable marriage we show that the same algorithm finds all the rotations for the problem instance in o n 2 time we give an o n 2 n s time and o n 2 space bounded algorithm which is time and space optimal to enumerate all stable marriages where s is the set of them and we give an o n 2 time algorithm to find the mini

lie transformations nonlinear evolution equations and painleve forms we present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one parameter continuous lie group of transformations it is shown that in general the corresponding invariant variables the subclass of which is the usual similarity variables lead to ordinary differential equations of painleve type in the case of inverse scattering transform solvable equations as conjectured by ablowitz ramani and segur this is found to be also true for certain higher spatial dimensional versions such as the kadomtsev petviashivilli two dimensional sine gordon and ernst equations for the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points