the mckean vlasov equation in finite volume we study the mckean vlasov equation on the finite tori of length scale l in d dimensions we derive the necessary and sufficient conditions for the existence of a phase transition which are based on the criteria first uncovered in gates and penrose commun math phys 17 194 209 1970 and kirkwood and monroe j chem phys 9 514 526 1941 therein and in subsequent works one finds indications pointing to critical transitions at a particular model dependent value of the interaction parameter we show that the uniform density which may be interpreted as the liquid phase is dynamically stable for and prove abstractly that a critical transition must occur at however for this system we show that under generic conditions l large d 2 and isotropic interactions the phase transition is in fact discontinuous and occurs at some theta text t theta sharp finally for h stable bounded interactions with discontinuous transitions we show that with suitable scaling the theta text t l tend to a definitive non trivial limit as l

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

stability in the energy space of the sum of n peakons for a modified camassa holm equation with higher order nonlinearity considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons

a comparison principle for hamilton jacobi equations related to controlled gradient flows in infinite dimensions we develop new comparison principles for viscosity solutions of hamilton jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures our examples are optimal control of ginzburg landau and fokker planck equations they arise in limit considerations of externally forced non equilibrium statistical mechanics models or through the large deviation principle for interacting particle systems our approach is based on two key ingredients an appropriate choice of geometric structure defining the gradient flow and a free energy inequality resulting from such gradient flow structure the approach allows us to handle hamiltonians with singular state dependency in the nonlinear term as well as hamiltonians with a state space which does not satisfy the radon nikodym property in the case where the state space is a hilbert space the method simplifies existing theories by avoiding the perturbed optimization principle

a method of linearization for painleve equations painleve iv v abstract a method to linearize the initial value problem of the painleve equations iv v is given the procedure involves formulating a riemann hilbert boundary value problem on intersecting lines for the inverse monodromy problem this boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space schlesinger transformations allow one to completely cover the parameter space special solutions are constructed from special cases of the riemann problem as well

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

from a large deviations principle to the wasserstein gradient flow a new micro macro passage we study the connection between a system of many independent brownian particles on one hand and the deterministic diffusion equation on the other for a fixed time step h 0 a large deviations rate functional j h characterizes the behaviour of the particle system at t h in terms of the initial distribution at t 0 for the diffusion equation a single step in the time discretized entropy wasserstein gradient flow is characterized by the minimization of a functional k h we establish a new connection between these systems by proving that j h and k h are equal up to second order in h as h 0 this result gives a microscopic explanation of the origin of the entropy wasserstein gradient flow formulation of the diffusion equation simultaneously the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow

the ideal membership problem and polynomial identity testing given a monomial ideal i where m i are monomials and a polynomial f by an arithmetic circuit the ideal membership problem is to test if email protected i we study this problem and show the following results a when the ideal i for a constant k we can test whether email protected i in randomized polynomial time this result holds even for f given by a black box when f is of small degree b when i for a constantkandf is computed by a email protected s circuit with output gate of bounded fanin we can test whether email protected i in deterministic polynomial time this generalizes the kayal saxena result 11 of deterministic polynomial time identity testing for email protected s circuits with bounded fanin output gate c when k is not constant the problem is conp hard we also show that the problem is upper bounded by coma p p over the field of rationals and by conp m o d p p over finite fields d finally we discuss identity testing for certain restricted depth 4 arithmetic circuits for ideals i where each f email protected f x 1 x k is an arbitrary polynomial but k is a constant we show similar results as a and b above

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

a fully supersymmetric akns theory we construct a fully supersymmetric bihamiltonian theory in four superfields admitting zero curvature and lax formulation this theory is an extension of the classical akns which can be recovered as a reduction other supersymmetric theories are obtained as reductions of the susy akns namely a nonlinear schrodinger a modified kdv and the manin radul kdv the susy nonlinear schrodinger hierarchy is related to the one of roelofs and kersten we determine its bihamiltonian and lax formulation finally we show that the susy kdv s mentioned before are related through a susy miura map

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

blow up phenomena and global existence for a two component camassa holm system with an arbitrary smooth function abstract in the paper we mainly investigate blow up phenomena and global existence of strong solutions to a two component camassa holm systems with an arbitrary smooth function h for three types of smooth functions h by using a conservation law and the sign preserving property of strong solutions we obtain two new blow up results and a new global existence result for the two component system our obtained results generalize and cover the recent results in yan et al 2015 zhang and yin 2015 2016

on the solvability of the discrete second painleve equation the inverse monodromy method for studying the riemann hilbert problem associated with classical painlev e equations is applied to the discrete second painlev e equation

the primes contain arbitrarily long polynomial progressions we establish the existence of infinitely many polynomial progressions in the primes more precisely given any integer valued polynomials p 1 p k z m in one unknown m with p 1 0 p k 0 0 and given any e 0 we show that there are infinitely many integers x and m with 1 leqslant m leqslant x varepsilon such that x p 1 m x p k m are simultaneously prime the arguments are based on those in 18 which treated the linear case p j j 1 m and e 1 the main new features are a localization of the shift parameters and the attendant gowers norm objects to both coarse and fine scales the use of pet induction to linearize the polynomial averaging and some elementary estimates for the number of points over finite fields in certain algebraic varieties

multiple recurrence and convergence along the primes let e subset bbb z be a set of positive upper density suppose that p 1 p 2 ldots p k in bbb z x are polynomials having zero constant terms we show that the set e cap e p 1 p 1 cap cdots cap e p k p 1 is non empty for some prime number p furthermore we prove convergence in l 2 of polynomial multiple averages along the primes

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

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

on a class of first order hamilton jacobi equations in metric spaces abstract we establish well posedness of a class of first order hamilton jacobi equation in geodesic metric spaces the result is then applied to solve a hamilton jacobi equation in the wasserstein space of probability measures which arises from the variational formulation of a compressible euler equation

second order second degree painleve equations related with painleve i vi equations and fuchsian type transformations one to one correspondence between the painleve i vi equations and certain second order second degree equations of painleve type is investigated the transformation between the painleve equations and second order second degree equations is the one involving the fuchsian type equation

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

kac s program in kinetic theory this paper is devoted to the study of propagation of chaos and mean field limits for systems of indistinguishable particles undergoing collision processes the prime examples we will consider are the many particle jump processes of kac and mckean kac in proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 mckean in j comb theory 2 358 382 1967 giving rise to the boltzmann equation we solve the conjecture raised by kac proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 motivating his program on the rigorous connection between the long time behavior of a collisional many particle system and the one of its mean field limit for bounded as well as unbounded collision rates

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

global solutions to a special case of the generalized weakly dissipative periodic two component camassa holm system abstract we establish global well posedness to the generalized weakly dissipative periodic two component camassa holm system in some special case

a hardy field extension of szemeredi s theorem abstract in 1975 szemeredi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions bergelson and leibman showed in 1996 that the common difference of the arithmetic progression can be a square a cube or more generally of the form p n where p n is any integer polynomial with zero constant term we produce a variety of new results of this type related to sequences that are not polynomial we show that the common difference of the progression in szemeredi s theorem can be of the form n where is any positive real number and x denotes the integer part of x more generally the common difference can be of the form a n where a x is any function that is a member of a hardy field and satisfies a x x k and a x x k 1 0 for some non negative integer k the proof combines a new structural result for hardy sequences techniques from ergodic theory and some recent equidistribution results of sequences on nilmanifolds

periodic solutions of the camassa holm equation based on the bilinear form since the famous work of camassa and holm various analytic techniques have been used to find the exact solutions of camassa holm ch equation and many interesting results have been found by the different methods in this paper we first give an extended bilinear form of the ch equation which contains four more free parameters than those in the results proposed in parkers finding for soliton solutions thus we construct the smooth periodic solution of the ch equation in a parametric form based on the previous work of nakumura we also show that the periodic solution degenerates into the smooth solitary wave solution by limiting behavior of is the parameter appeared from the riemann theta function

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

on the solvability of the painleve vi equation a rigorous method was introduced by fokas and zhou 1992 for studying the riemann hilbert problem associated with the painleve ii and iv equations the same methodology has been applied to the painleve i iii and v equations in this paper we will apply the same methodology to the painleve vi equation we will show that the cauchy problem for the painleve vi equation admits in general a global meromorphic solution in t furthermore the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the riemann hilbert problem

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

chazy s second degree painleve equations we examine two sets of second degree painlev equations derived by chazy in 1909 denoted by systems ii and iii the last member of each set is a second degree version of the painlev vi equation and there are no other second order second degree painlev equations in the polynomial class with this property we map the last member of system ii into the fokas yortsos equation and demonstrate how both schlesinger and okamoto transformations for painlev vi can be read off the chazy equation the 24 fundamental schlesinger transformations were known to garnier in 1943 while the 64 okamoto transformations date from 1987 in an appendix we gather together the solutions of the five members of system ii system iii is better known being equivalent to jimbo and miwa s equations for the logarithmic derivatives of the tau functions of the six painlev transcendents the last member known to painlev in 1906 was written in a manifestly symmetric form by jimbo and miwa suggesting many induced symmetries for painlev vi in particular schlesinger and okamoto transformations for painlev vi can be read off immediately

on inverse recursion operator and tri hamiltonian formulation for a kaup newell system of dnls equations an inverse of the recursion operator is computed and a clear presentation of a tri hamiltonian formulation is provided for a kaup newell system of derivative nonlinear schrodinger dnls equations therefore all kaup newell systems in the whole hierarchy of dnls equations are tri hamiltonian and have an inverse hierarchy of common commuting symmetries

an integrable semi discretization of the camassa holm equation and its determinant solution an integrable semi discretization of the camassa holm ch equation is presented the keys of its construction are bilinear forms and determinant structure of solutions of the ch equation determinant formulas of n soliton solutions of the continuous and semi discrete camassa holm equations are presented based on determinant formulas we can generate multi soliton multi cuspon and multi soliton cuspon solutions numerical computations using the integrable semi discrete camassa holm equation are performed it is shown that the integrable semi discrete camassa holm equation gives very accurate numerical results even in the cases of cuspon cuspon and soliton cuspon interactions the numerical computation for an initial value condition which is not an exact solution is also presented

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

the multiscale expansions of difference equations in the small lattice spacing regime and a vicinity and integrability test i we propose an algorithmic procedure i to study the distance between an integrable pde and any discretization of it in the small lattice spacing regime and at the same time ii to test the asymptotic integrability properties of such discretization this method should provide in particular useful and concrete information on how good is any numerical scheme used to integrate a given integrable pde the procedure illustrated on a fairly general ten parameter family of discretizations of the nonlinear schr dinger equation consists of the following three steps i the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in following degasperis et al 1997 physica d 100 187 211 ii the application to such an expansion of the degasperis procesi dp integrability test degasperis a and procesi m 1999 asymptotic integrability symmetry and perturbation theory spt98 ed a degasperis and g gaeta singapore world scientific pp 23 37 degasperis a 2001 multiscale expansion and integrability of dispersive wave equations lectures given at the euro summer school what is integrability isaac newton institute cambridge uk 13 24 august integrability lecture notes in physics vol 767 ed a mikhailov berlin springer to test the asymptotic integrability properties of the discrete system and its distance from its continuous limit iii the use of the main output of the dp test to construct infinitely many approximate symmetries and constants of motion of the discrete system through novel and simple formulas

combinatorial theorems in sparse random sets we develop a new technique that allows us to show in a unified way that many well known combinatorial theorems including turan s theorem szemeredi s theorem and ramsey s theorem hold almost surely inside sparse random sets for instance we extend turan s theorem to the random setting by showing that for every 0 and every positive integer t 3 there exists a constant c such that if g is a random graph on n vertices where each edge is chosen independently with probability at least cn 2 t 1 then with probability tending to 1 as n tends to infinity every subgraph of g with at least 1 1 t 1 e g edges contains a copy of kt this is sharp up to the constant c we also show how to prove sparse analogues of structural results giving two main applications a stability version of the random turan theorem stated above and a sparse hypergraph removal lemma many similar results have recently been obtained independently in a different way by schacht and by friedgut rodl and schacht

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

the cauchy problem for the second member of a rm p_ iv hierarchy a rigorous method the inverse monodromy transform for studying the riemann hilbert rh problem associated with the classical painleve equations is applied to the second member of a fourth painleve hierarchy we show that the cauchy problem for the second member of this hierarchy admits in general a global meromorphic solution in x moreover for a particular choice of the monodromy data the associated rh problem can be reduced to a set of scalar rh problems and a special solution which can be written in terms of the airy function is obtained

second order second degree painleve equations related with painleve i ii iii equations the algorithmic method introduced by fokas and ablowitz to investigate the transformation properties of painleve equations is used to obtain a one to one correspondence between the painleve i ii and iii equations and certain second order second degree equations of painleve type

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

on the initial value problem of the second painleve transcendent the initial value problem associated with the second painleve transcendent is linearized via a matrix discontinuous homogeneous riemann hilbert rh problem defined on a complicated contour six rays intersecting at the origin this problem is mapped through a series of transformations to three different simple riemann hilbert problems each of which can be solved via a system of two fredholm integral equations the connection of these results with the inverse scattering transform in one and two dimensions is also pointed out

barriers of the mckean vlasov energy via a mountain pass theorem in the space of probability measures abstract we show that the empirical process associated with a system of weakly interacting diffusion processes exhibits a form of noise induced metastability the result is based on an analysis of the associated mckean vlasov free energy which for suitable attractive interaction potentials has at least two distinct global minimisers at the critical parameter value c on the torus one of these states is the spatially homogeneous constant state and the other is a clustered state we show that a third critical point exists at this value as a result we obtain that the probability of transition of the empirical process from the constant state scales like exp n with the energy gap at c the proof is based on a version of the mountain pass theorem for lower semicontinuous and geodesically convex functionals on the space of probability measures p 2 m equipped with the 2 wasserstein metric where m is a complete connected and smooth riemannian manifold

schlesinger transformations for painleve vi equation a method to obtain the schlesinger transformations for painleve vi equation is given the procedure involves formulating a riemann hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use