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

frozen states of a disordered globular heteropolymer the disordered heteropolymer chain with a random sequence of links a simple model of globular protein is investigated using the replica approach in the mean field approximation it is shown that for low dimensions of space d 1 it follows that the set of equilibrium states is ultrametric with an anomalously large number of weakly overlapping states

painleve classification of coupled korteweg de vries systems in this work we give a classification of coupled korteweg de vries equations we found new systems of equations that are completely integrable in the sense of painleve

a coupled korteweg de vries equation with dispersion a dispersive system describing a vector multiplet interacting with the korteweg de vries field is shown to be a number of a bi hamiltonian integrable hierarchy

statistical mechanics of a single particle in a multiscale random potential parisi landscapes in finite dimensional euclidean spaces we construct an n dimensional gaussian landscape with multiscale translation invariant logarithmic correlations and investigate the statistical mechanics of a single particle in this environment in the limit of high dimension n the free energy of the system and overlap function are calculated exactly using the replica trick and parisi s hierarchical ansatz in the thermodynamic limit we recover the most general version of the derrida s generalized random energy model grem the low temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape if the latter consists of k discrete values the system is characterized by a k step replica symmetry breaking solution we argue that our construction is in fact valid in any finite spatial dimensions n 1 we discuss the implications of our results for the singularity spectrum describing multifractality of the associated boltzmann gibbs measure finally we discuss several generalizations and open problems such as the dynamics in such a landscape and the construction of a generalized multifractal random walk

bifurcation of traveling wave solutions of 2 1 dimensional konopelchenko dubrovsky equations abstract the bifurcation theory method of planar dynamical systems is efficiently employed to find the bounded traveling wave solutions of the 2 1 dimensional konopelchenko dubrovsky equations the bifurcation parameter sets and the corresponding phase portraits are given under different parameter conditions the exact explicit parametric representations of solitary wave solutions kink anti kink wave solutions and periodic wave solutions are obtained

on toy ageing we consider the dynamics of a simple one dimensional model and we discuss the phenomenon of ageing i e the strong dependence of the dynamical correlation functions over the waiting time our model is the so called random random walk the toy model of a directed polymer evolving in a random medium

two component integrable generalizations of burgers equations with nondiagonal linearity two component second and third order burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries new integrable systems are obtained master symmetries of the obtained symmetry integrable systems and bi poisson structures of those that also possess conservation laws are given

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

polymers with long range self repulsion a variational approach the authors use a variational approach to study self interacting polymers with long range repulsion decaying for large distances as r lambda for 2 or lambda or 4 and lambda d where d is the spatial dimension they find that the exponent for the end to end distance of the chain is v 2 lambda with logarithmic corrections for lambda 2 or lambda 4 in agreement with renormalization group computations and recent numerical simulations they discuss the entire lambda d plane using a renormalization group near lambda 4 d 4

completeness of multiseparable superintegrability on the complex 2 sphere the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability here we demonstrate for nondegenerate potentials how to establish a complete list of such potentials that are superintegrable on the complex 2 sphere using essentially algebraic means we classify all such potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters the method of proof generalizes to other spaces and dimensions we show for the 2 sphere that all these superintegrable systems possess the remarkable property that they correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

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

toward a classification of semidegenerate 3d superintegrable systems superintegrable systems of 2nd order in 3 dimensions with exactly 3 parameter potentials are intriguing objects next to the nondegenerate 4 parameter potential systems they admit the maximum number of symmetry operators but their symmetry algebras do not close under commutation and not enough is known about their structure to give a complete classification some examples are known for which the 3 parameter system can be extended to a 4th order superintegrable system with a 4 parameter potential and 6 linearly independent symmetry generators in this paper we use bocher contractions of the conformal lie algebra to itself to generate a large family of 3 parameter systems with 4th order extensions on a variety of manifolds all from bocher contractions of a single generic system on the 3 sphere we give a contraction scheme relating these systems the results have myriad applications for finding explicit solutions for both quantum and classical systems

symbolic computation and families of jacobi elliptic function solutions of the 2 1 dimensional nizhnik novikov veselov equation in this paper with the aid of the symbolic computation we have improved the extended f expansion method in chaos solitons and fractals 22 2004 111 and proposed the further improved f expansion method by this method we have gotten many new exact solutions which we have never seen before within our knowledge of the 2 1 dimensional nizhnik novikov veselov equations in addition the solutions we get are more general than the solutions the extended f expansion method gets at the same time our method is more convenient than the method in chaos solitons and fractals 18 2003 299 309 and the solutions we get are more abundant than the solutions in h q zh appl math e notes 1 2001 139 in other word the solutions in h q zh appl math e notes 1 2001 139 are part of our solutions our method can also apply to other partial differential equations and also get many new exact solutions

construction of non travelling wave solutions for the generalized variable coefficient gardner equation with the aid of two first order nonlinear ordinary differential equations the new generalized algebraic method is used to study the generalized variable coefficient gardner equation the main idea of the method is to take full advantage of solutions to the first order ordinary differential equations more new exact non travelling solutions which include soliton solutions combined soliton solutions triangular periodic solutions jacobi elliptic function solutions and combined jacobi elliptic function solutions for the gardner equation are obtained the method used here can be also extended to many other nonlinear partial differential equations

variable coefficient jacobi elliptic function expansion method for 2 1 dimensional nizhnik novikov vesselov equations abstract in this paper a variable coefficient jacobi elliptic function expansion method is proposed to seek more general exact solutions of nonlinear partial differential equations being concise and straightforward this method is applied to the 2 1 dimensional nizhnik novikov vesselov equations as a result many new and more general exact non travelling wave and coefficient function solutions are obtained including jacobi elliptic function solutions soliton like solutions and trigonometric function solutions to give more physical insights to the obtained solutions we present graphically their representative structures by setting the arbitrary functions in the solutions as specific functions

travelling wave solutions for the generalized zakharov kuznetsov equation with higher order nonlinear terms abstract with the aid of symbolic computation and two auxiliary ordinary differential equations the new generalized algebraic method is extended to the generalized zakharov kuznetsov gzk equation with higher order nonlinear terms for constructing a series of new and more general travelling wave solutions because of the higher order nonlinear terms the equation can not be directly dealt with by the method and require some kinds of techniques by means of two proper transformations we transform the gzk equation to an ordinary differential equation that is easy to solve and find a rich variety of new exact travelling wave solutions for the gzk equation which include soliton solutions combined soliton solutions triangular periodic solutions jacobi elliptic function solutions combined jacobi elliptic function solutions and rational function solutions the method used here can be also extended to other nonlinear partial differential equations with higher order nonlinear terms

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

the sine gordon equations complete and partial integrability the sine gordon equation in one space one time dimension is known to possess the painleve property and to be completely integrable it is shown how the method of singular manifold analysis obtains the backlund transform and the lax pair for this equation a connection with the sequence of higher order kdv equations is found the modified sine gordon equations are defined in terms of the singular manifold these equations are shown to be identically painleve also certain rational solutions are constructed iteratively the double sine gordon equation is shown not to possess the painleve property however if the singular manifold defines an affine minimal surface then the equation has integrable solutions this restriction is termed partial integrability the sine gordon equation in n 1 variables n space 1 time where n is greater than one is shown not to possess the painleve property the condition of partial integrability requires the singular manifold to be an einstein space with null scalar curvature the known integrable solutions satisfy this constraint in a trivial manner finally the coupled kdv or hirota satsuma equations possess the painleve property the associated modified equations are derived and from these the lax pair is found

exact solutions to the two dimensional korteweg de vries burgers equation two methods are described for obtaining exact travelling monotone shock wave solutions to the two dimensional korteweg de vries burgers equation they are compared with two methods that have been reported recently by other authors

second order superintegrable systems in conformally flat spaces ii the classical two dimensional stackel transform this paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in conformally flat spaces here we study the stackel transform or coupling constant metamorphosis as an invertible mapping between classical superintegrable systems on different spaces through the use of this tool we derive and classify for the first time all two dimensional 2d superintegrable systems the underlying spaces are exactly those derived by koenigs in his remarkable paper giving all 2d manifolds with zero potential that admit at least three second order symmetries our derivation is very simple and quite distinct we also show that every superintegrable system is the stackel transform of a superintegrable system on a constant curvature space

quasiperiodic waves and asymptotic behavior for bogoyavlenskii s breaking soliton equation in 2 1 dimensions based on a multidimensional riemann theta function the hirota bilinear method is extended to explicitly construct multiperiodic quasiperiodic wave solutions for the 2 1 dimensional bogoyavlenskii breaking soliton equation among these periodic waves the one periodic waves are well known cnoidal waves their surface pattern is one dimensional and often they are used as one dimensional models of periodic waves in shallow water the two periodic biperiodic waves are a direct generalization of one periodic waves their surface pattern is two dimensional that is they have two independent spatial periods in two independent horizontal directions the two periodic waves may be considered to represent periodic waves in shallow water without the assumption of one dimensionality a limiting procedure is presented to analyze asymptotic behavior of the one and two periodic waves in details the exact relations between the periodic wave solutions and the well known soliton solutions are established it is rigorously shown that the periodic wave solutions tend to the soliton solutions under a small amplitude limit

numerical integration of a coupled korteweg de vries system abstract we introduce a numerical method for general coupled korteweg de vries systems the scheme is valid for solving cauchy problems for an arbitrary number of equations with arbitrary constant coefficients the numerical scheme takes its legality by proving its stability and convergence which gives the conditions and the appropriate choice of the grid sizes the method is applied to the hirota satsuma hs system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy we also illustrate the method to show the effects of constants with a transition to nonintegrable cases

universality classes for extreme value statistics the equilibrium low temperature physics of disordered systems is governed by the statistics of extremely low energy states it is thus relevant to discuss the possible universality classes for extreme value statistics we compare the usual probabilistic classification to the results of the replica approach we show in detail for several problems including the random energy model and the decaying burgers turbulence that one class of independent variables corresponds exactly to the so called one step replica symmetry breaking solution in the replica language we argue that this universality class holds if the correlations are sufficiently weak and propose a conjecture on the level of correlations which leads to different universality classes

replica symmetry breaking condition exposed by random matrix calculation of landscape complexity we start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random n 1 dimensional gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface this is used to show that for a generic smooth concave confining potentials the condition of the zero temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in n for such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically different behaviour reported in our earlier short communication fyodorov et al in jetp lett 85 261 2007 was due to non analyticity of the hard wall confinement potential finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points in particular we show that in the vicinity of critical confinement the saddle points with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the hessian

completeness of superintegrability in two dimensional constant curvature spaces we classify the hamiltonians h px2 py2 v x y of all classical superintegrable systems in two dimensional complex euclidean space with two additional second order constants of the motion we similarly classify the superintegrable hamiltonians h j12 j22 j32 v x y z on the complex two sphere where x2 y2 z2 1 this is achieved in all generality using properties of the complex euclidean group and the complex orthogonal group

ultrametric matrices and representation theory the consequences of replica symmetry breaking on the structure of ultrametric matrices appearing in the theory of disordered systems is investigated with the help of representation theory and the results are compared with those obtained by temesvari de dominicis and kondor

a strange recursion operator demystified we show that a new integrable two component system of kdv type studied by karasu kalkanli et al 2004 acta appl math 83 85 94 is bi hamiltonian and its recursion operator which has a highly unusual structure of nonlocal terms can be written as a ratio of two compatible hamiltonian operators found by us using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution and the evolution systems corresponding to these symmetries are bi hamiltonian as well we also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form with the integration operator d 1x appearing in each term at most once

random field p spin glass model on regular random graphs we investigate in detail the phase diagrams of the p body j ising model with and without random fields on random graphs with fixed connectivity one of our most interesting findings is that a thermodynamic spin glass phase is present in the three body purely ferromagnetic model in random fields unlike for the canonical two body interaction random field ising model we also discuss the location of the phase boundary between the paramagnetic and spin glass phases that does not depend on the change of the ferromagnetic bias the independence of the ferromagnetic bias has also been observed from the perspective of the distribution of the partition function zeros on the complex field plane this behavior is explained by a gauge transformation which shows that gauge invariant properties generically do not depend on the strength of the ferromagnetic bias for the j ising model on regular random graphs

second order superintegrable systems in conformally flat spaces i two dimensional classical structure theory this paper is part of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in real or complex conformally flat spaces here we consider classical superintegrable systems with nondegenerate potentials in three dimensions we show that there exists a standard structure for such systems based on the algebra of 3 3 symmetric matrices and that the quadratic algebra always closes at order 6 we show that the spaces of truly second third fourth and sixth order constants of the motion are of dimension 6 4 21 and 56 respectively and we construct explicit bases for the fourth and sixth order constants in terms of products of the second order constants

interfaces in a random medium and replica symmetry breaking the authors develop a variational approach for studying interfaces and other manifolds in a disordered quenched medium the method may be applied to problems which range from directed polymers to the interface of an ising model in a random magnetic field they find that replica symmetry is spontaneously broken and the results of the flory approximation are recovered in a simple way corrections to this approximation may be computed in a systematic way

unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold in this paper we prove that the two dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the poisson algebra of the integrals of motion there are six general fundamental classes of superintegrable systems analytic formulas for the involved integrals are calculated in all the cases all the known superintegrable systems are classified as special cases of these six general classes

invariant classification of orthogonally separable hamiltonian systems in euclidean space the problem of the invariant classification of the orthogonal coordinate webs defined in euclidean space is solved within the framework of felix klein s erlangen program the results are applied to the problem of integrability of the calogero moser model

a class of nonautonomous coupled kdv systems a class of nonautonomous coupled korteweg de vries kdv systems in 1 1 dimensions are considered for integrability classification integrability of the systems is associated with the existence of a certain recursion operator some new integrable nonautonomous two component kdv systems are found

completeness of multiseparable superintegrability in e2 c the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability examples of this type of system are well known in this paper we demonstrate how to establish a complete list of such potentials using essentially algebraic means our approach is to classify all nondegenerate potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters this is carried out for two dimensional complex euclidean space though the method generalizes to other spaces and dimensions we show that all these superintegrable systems correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

how super rough is the glassy phase of a crystalline surface with a disordered substrate we discuss the behaviour of a crystalline surface with a disordered substrate we focus on the possible existence of a super rough glassy phase with height height correlation functions which vary as the square logarithm of the distance with numerical simulations we establish the presence of such a behaviour that does not seem to be connected to finite size effects we comment on the variational approach and suggest that a more general extension of the method could be needed to fully explain the behaviour of the model

towards the complete classification of homogeneous two component integrable equations in this article we suggest an improved method for classifying general two component integrable evolution equations homogeneous in a given weighting scheme this method relies on linear changes of variables and on an appropriate splitting of the solution space to illustrate the method we implement the classification of coupled kdv type equations we show that there are five nontriangular systems possessing higher order generalized symmetries one of these systems is previously unknown this seems to be the first classification of coupled integrable equations homogeneous in a given weighting scheme without any restrictions on the form of the main matrix

from integrable to superintegrable it is shown that the presence of multiple eigenvalue parameters may result in the superintegrability of the restricted soliton flows the restricted akns flow the neumann system and the restricted mkdv flow with an eigenvalue parameter whose multiplicity is greater than two are demonstrated to be superintegrable

invariant classification of second order conformally flat superintegrable systems in this paper we continue the work of kalnins et al in classifying all second order conformally superintegrable laplace type systems over conformally flat spaces using tools from algebraic geometry and classical invariant theory the results obtained show through stackel equivalence that the list of known nondegenerate superintegrable systems over three dimensional conformally flat spaces is complete in particular a seven dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems this manifold is foliated by the nonlinear action of the conformal group in three dimensions two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation this foliation is explicitly described using algebraic varieties formed from representations of the conformal group the proof of these results rely heavily on grobner basis calculations using the computer algebra software packages maple and singular

factorization technique and new exact solutions for the modified camassa holm and degasperis procesi equations abstract in this paper the recent factorization technique is applied to the modified camassa holm and degasperis procesi equations and two first order ordinary differential equations are obtained respectively subsequently some new exact solitary wave solutions for the two equations are proposed the figures for the bell type and peakon type solutions of the modified camassa holm are plotted to describe the properties of the solutions

elliptic equation s new solutions and their applications to two nonlinear partial differential equations abstract in the present paper with the aid of symbolic computation families of new nontrivial solutions of first order elliptic equation 2 a 0 a 1 a 2 2 a 3 3 a 4 4 where d d x are obtained to our knowledge these nontrivial solutions can not be found in chaos solitons fract 26 2005 785 794 and phys lett a 336 2005 463 476 by yomba and other existent papers until now by using these nontrivial solutions a direct algebraic method is described to construct several kinds of exact non travelling wave solutions for the 2 1 dimensional breaking soliton equations and the 2 1 dimensional asymmetric nizhnik novikov vesselov equation by using this method many other physically important nonlinear partial differential equations nlpdes can be investigated and new non travelling wave solutions can be explicitly obtained with the aid of symbolic computation system maple

classification of polynomial integrable systems of mixed scalar and vector evolution equations i we perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries we consider polynomial systems that are homogeneous under a suitable weighting of variables this paper deals with the kdv weighting the burgers or potential kdv or modified kdv weighting the ibragimov shabat weighting and two unfamiliar weightings the case of other weightings will be studied in a subsequent paper making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems we give the complete lists of second order systems with a third order or a fourth order symmetry and third order systems with a fifth order symmetry for all but a few systems in the lists we show that the system or at least a subsystem of it admits either a lax representation or a linearizing transformation a thorough comparison with recent work of foursov and olver is made

a new general algebraic method with symbolic computation to construct new travelling wave solution for the 1 1 dimensional dispersive long wave equation with the aid of symbolic computation a new algebraic method named riccati equation rational expansion rere method is devised for constructing multiple travelling wave solutions for nonlinear evolution equations nees compared with most existing tanh methods for finding travelling wave solutions the proposed method not only recover the results by most known algebraic methods but also provides new and more general solutions with the aid of symbolic computation we choose 1 1 dimensional dispersive long wave equation dlwe to illustrate our method as a result we obtain many types of solutions including rational form solitary wave solutions triangular periodic wave solutions and rational wave solutions the properties of the new solitary wave solutions are shown by some figures

a variational approach to directed polymers the authors develop a variational approach for directed polymers in random media in the framework of the replica formalism they derive the functional expression for the free energy in terms of the probability distribution of the probability of finding a polymer at a given point they compare these results with those coming from the broken replica symmetry approach and present some new computations in this framework

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

nondegenerate three dimensional complex euclidean superintegrable systems and algebraic varieties a classical or quantum second order superintegrable system is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent second order constants of the motion polynomial in the momenta the maximum possible such systems have remarkable properties multi integrability and multiseparability an algebra of higher order symmetries whose representation theory yields spectral information about the schrodinger operator deep connections with special functions and with quasiexactly solvable systems here we announce a complete classification of nondegenerate i e four parameter potentials for complex euclidean 3 space we characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints the euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation there are exactly ten nondegenerat

novel soliton like and multi solitary wave solutions of 3 1 dimensional burgers equation abstract in this paper the 3 1 dimensional burgers equation is studied by means of painleve truncation expansion extended riccati equation method and wang s multiple riccati equations method in wang and li d s wang h li symbolic computation and non travelling wave solutions of 2 1 dimensional non linear evolution equations chaos soliton fract 38 2008 383 390 as a result some novel soliton like complexiton solutions and multi solitary wave solutions are derived it is shown that the process to derive them is very direct subsequently an n 1 dimensional burgers type equation is presented and its solutions are given

non travelling wave solutions of two 2 1 dimensional nonlinear mathematical physics equations the overall objective of this paper is to find and analyze the new nonlinear travelling wave solutions of two nonlinear mathematical physics equations the 2 1 dimensional perturbed kdv equation and the 2 1 dimensional breaking soliton equation in this paper based on symbolic computation and the further improved f expansion method many new non travelling wave solutions of the two equations are obtained and their properties are studied by analyzing their figures

stability of hierarchical interfaces in a random field model we study a hierarchical model for interfaces in a random field ferromagnet we prove that in dimensiond 3 at low temperatures and for weak disorder such interfaces are rigid our proof uses renormalization group transformations for stochastic sequences

darboux covariant lax pairs and infinite conservation laws of the 2 1 dimensional breaking soliton equation in this paper the binary bell polynomials are applied to succinctly construct bilinear formulism bilinear backlund transformations lax pairs and darboux covariant lax pairs for the 2 1 dimensional breaking soliton equation an extra auxiliary variable is introduced to get the bilinear formulism the infinitely local conservation laws of the equation are found by virtue of its lax equation and a generalized miura transformation all conserved densities and fluxes are given with explicit recursion formulas

backlund transformations for the nizhnik novikov veselov equation the backlund transformations for the nizhnik novikov veselov equation are presented it is shown that these transformations can be iterated and that the resulting sequence can be described by the volterra equations the relationships between the nizhnik novikov veselov equation and the volterra hierarchy are discussed

block diagonalizing ultrametric matrices the problem of diagonalizing a class of complicated matrices to be called ultrametric matrices is investigated these matrices appear at various stages in the description of disordered systems with many equilibrium phases by the technique of replica symmetry breaking the residual symmetry remaining after the breaking of permutation symmetry between replicas allows us to bring all ultrametric matrices to a block diagonal form by a common similarity transformation a large number of these blocks are in fact of size 1 1 i e in a vast sector the transformation actually diagonalizes the matrix in the other sectors we end up with blocks of size r 1 r 1 where r is the number of replica symmetry breaking steps these blocks cannot be further reduced without giving more information in addition to ultrametric symmetry about the matrix similar results for the inverse of a generic ultrametric matrix are also derived

directed paths in a complex random potential effects of long range correlations of the disorder the authors apply the replica method supplemented by a variational approach to directed paths which acquire random amplitudes and random phases this problem is pertinent to the hopping of electrons in disordered materials in the variable range hopping regime here they consider the long range case in which both the amplitude and phase correlations of the disorder are long ranged and a mixed case in which amplitude correlations are long ranged and phase correlations are short ranged the variational approach is expected to become exact in the limit of infinite dimensions and it yields the flory exponents for the transverse fluctuations stationary solutions which extremize the free energy and are characterized by non trivial patterns of replica symmetry breaking are obtained both analytically and numerically the mixed case has a non trivial phase diagram marked by abrupt changes in the pattern of replica symmetry breaking in this case the imaginary part of the random potential plays an important role

second order superintegrable systems in conformally flat spaces iv the classical 3d stackel transform and 3d classification theory this article is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in conformally flat spaces in the first part of the article we study the stackel transform or coupling constant metamorphosis as an invertible mapping between classical superintegrable systems on different three dimensional spaces we show first that all superintegrable systems with nondegenerate potentials are multiseparable and then that each such system on any conformally flat space is stackel equivalent to a system on a constant curvature space in the second part of the article we classify all the superintegrable systems that admit separation in generic coordinates we find that there are eight families of these systems

multi component bi super hamiltonian kdv systems it is shown that a new class of classical multi component super kdv equations is bi super hamiltonian by extending the method for the verification of graded jacobi identity the multi component extension of super mkdv equations is obtained by using the super miura transformation

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series

fine structure for 3d second order superintegrable systems three parameter potentials a classical or quantum superintegrable system of second order is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent constants of the motion quadratic in the momenta the maximum possible for n 3 on conformally flat spaces with nondegenerate i e four parameter potentials the extreme case we have worked out the structure and classified most of the possible spaces and potentials here we extend the analysis to a more degenerate class of functionally linearly independent superintegrable systems the three parameter potential case we show that for true three parameter potentials the algebra of constants of the motion no longer closes at order 6 but still all such systems are stackel transforms of systems on complex euclidean space or the complex 3 sphere this is a significant step towards the complete structure analysis of all types of second order superintegrable systems