uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

spectral instability of the semiclassical zakharov shabat eigenvalue problem abstract we present some brief numerical results indicating that in the semiclassical limit the zakharov shabat eigenvalue problem exhibits severe spectral instability and that small changes in the potential can result in large changes in the spectrum

the semiclassical limit of the defocusing nls hierarchy we establish the semiclassical limit of the one dimensional defocusing cubic nonlinear schrodinger nls equation complete integrability is exploited to obtain a global characterization of the weak limits of the entire nls hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows consequently this also establishes the zero dispersion limit of the modified korteweg de vries equation that resides in that hierarchy we have adapted and clarified the strategy introduced by lax and levermore to study the zero dispersion limit of the korteweg de vries equation expanding it to treat entire integrable hierarchies and strengthening the limits obtained a crucial role is played by the convexity of the underlying log determinant with respect to the times associated with the commuting flows c 1999 john wiley sons inc

the semiclassical modified nonlinear schrodinger equation i modulation theory and spectral analysis abstract we study an integrable modification of the focusing nonlinear schrodinger equation from the point of view of semiclassical asymptotics in particular i we establish several important consequences of the mixed type limiting quasilinear system including the existence of maps that embed the limiting forms of both the focusing and defocusing nonlinear schrodinger equations into the framework of a single limiting system for the modified equation ii we obtain bounds for the location of the discrete spectrum for the associated spectral problem that are particularly suited to the semiclassical limit and that generalize known results for the spectrum of the nonselfadjoint zakharov shabat spectral problem and iii we present a multiparameter family of initial data for which we solve the associated spectral problem in terms of special functions for all values of the semiclassical scaling parameter we view our results as part of a broader project to analyze the semiclassical limit of the modified nonlinear schrodinger equation via the noncommutative steepest descent procedure of deift and zhou and we also present a selfcontained development of a riemann hilbert problem of inverse scattering that differs from those given in the literature and that is well adapted to semiclassical asymptotics

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

phase transitions and equilibrium measures in random matrix models the paper is devoted to a study of phase transitions in the hermitian random matrix models with a polynomial potential in an alternative equivalent language we study families of equilibrium measures on the real line in a polynomial external field the total mass of the measure is considered as the main parameter which may be interpreted also either as temperature or time our main tools are differentiation formulas with respect to the parameters of the problem and a representation of the equilibrium potential in terms of a hyperelliptic integral using this combination we introduce and investigate a dynamical system system of odes describing the evolution of families of equilibrium measures on this basis we are able to systematically derive a number of new results on phase transitions such as the local behavior of the system at all kinds of phase transitions as well as to review a number of known ones

oscillations of the zero dispersion limit of the korteweg de vries equation we attack the multiphase averaged systems for the zero dispersion limit of the kdv equation attention is paid to the most important case the single phase oscillations a scheme is developed to solve the whitham averaged system single phase averaged system this system under our scheme is transformed to a linear over determined system of euler poisson darboux type whose solution can be written down explicitly we show that for any smooth initial data which has only one hump or is a nontrivial monotone function the weak limit has single phase oscillations within a cusp in the x t plane for a short time after the breaking time for the corresponding burgers equation outside the cusp the limit satisfies the burgers equation more surprisingly we also show that the weak limit has global single phase oscillations within a cusp for any smooth nontrivial monotone initial data with only one inflection point 1993 john wiley sons inc

universality for the focusing nonlinear schrodinger equation at the gradient catastrophe point rational breathers and poles of the tritronquee solution to painleve i the semiclassical zero dispersion limit of solutions to the one dimensional focusing nonlinear schrodinger equation nls is studied in a scaling neighborhood d of a point of gradient catastrophe x 0 t 0 we consider a class of solutions specified in the text that decay as the neighborhood d contains the region of modulated plane wave with rapid phase oscillations as well as the region of fast amplitude oscillations spikes in this paper we establish the following universal behaviors of the nls solutions q near the point of gradient catastrophe i each spike has height and uniform shape of the rational breather solution to the nls scaled to the size ii the location of the spikes is determined by the poles of the tritronquee solution of the painleve i p1 equation through an explicit map between d and a region of the painleve independent variable iii if but lies away from the spikes the asymptotics of the nls solution is given by the plane wave approximation with the correction term being expressed in terms of the tritronquee solution of p1 the relation with the conjecture of dubrovin grava and klein about the behavior of solutions to the focusing nls near a point of gradient catastrophe is discussed we conjecture that the p1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point our technique is based on the nonlinear steepest descent method for matrix riemann hilbert problems and discrete schlesinger isomonodromic transformations 2013 wiley periodicals inc

overdetermined elliptic problems in topological disks abstract we introduce a method based on the poincare hopf index theorem to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk applications to some well known nonlinear elliptic pdes are provided our result can be seen as the analogue of hopf s uniqueness theorem for constant mean curvature spheres but for the general analytic context of overdetermined elliptic problems

numerical inverse scattering for the focusing and defocusing nonlinear schrodinger equations we solve the focusing and defocusing nonlinear schrodinger nls equations numerically by implementing the inverse scattering transform the computation of the scattering data and of the nls solution are both spectrally con vergent initial conditions in a suitable space are treated using the approach of biondini and bui 3 we numerically solve homogeneous robin boundary value problems on the half line finally using recent theoretical developments in the numerical approximation of riemann hilbert problems we prove that under mild assumptions our method of approximating solutions to the nls equations is uniformly accurate in their domain of definition

a hopf differential for constant mean curvature surfaces in s2 r and h2 r a basic tool in the theory of constant mean curvature cmc surfaces in space forms is the holomorphic quadratic differential discovered by h hopf in this paper we generalize this differential to immersed cmc surfaces in the product spaces s r and h r and prove a corresponding result about the geometry of immersed cmc spheres s in these target spaces

the eigenvalue problem for the focusing nonlinear schrodinger equations new solvable cases abstract in this paper we study the semi classical limit of the zakharov shabat eigenvalue problem for the focusing of nls with some specific initial data in all these cases the eigenvalue problem is reduced to connection problems for the hypergeometric equation and for other classical equations the special initial data suppl prog theor phys 55 1974 284 is contained in our family of initial data parameterized by a real parameter as a particular case 0 we find that beyond a certain value of the parameter the pure point spectrum becomes empty and all the scattering information is contained in the reflection coefficient

the zero dispersion limit of the korteweg de vries equation with periodic initial data on etudie le probleme aux valeurs initiales pour l equation de korteweg de vries u t 6uu x e 2 u xxx 0 dans la limite e 0 on calcul la limite faible de la solution pour des donnees initiales periodiques quand e 0

the generation of modulated wavetrains in the solution of the korteweg de vries equation on demontre une relation entre la methode des ondes modulees de whitham et la methode de lax et levermore contenant la limite de petite dispersion

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

nonlinear steepest descent asymptotics for semiclassical limit of integrable systems continuation in the parameter space the initial value problem for an integrable system such as the nonlinear schrodinger equation is solved by subjecting the linear eigenvalue problem arising from its lax pair to inverse scattering and thus transforming it to a matrix riemann hilbert problem rhp in the spectral variable in the semiclassical limit the method of nonlinear steepest descent 4 5 supplemented by the g function mechanism 3 is applied to this rhp to produce explicit asymptotic solution formulae for the integrable system these formule are based on a hyperelliptic riemann surface mathcal r mathcal r x t in the spectral variable where the space time variables x t play the role of external parameters the curves in the x t plane separating regions of different genuses of mathcal r x t are called breaking curves or nonlinear caustics the genus of mathcal r x t is related to the number of oscillatory phases in the asymptotic solution of the integrable system at the point x t the evolution theorem 10 guarantees continuous evolution of the asymptotic solution in the space time away from the breaking curves

semiclassical limit of the focusing nls whitham equations and the riemann hilbert problem approach abstract the main goal of this paper is to put together a the whitham theory applicable to slowly modulated n phase nonlinear wave solutions to the focusing nonlinear schrodinger fnls equation and b the riemann hilbert problem approach to particular solutions of the fnls in the semiclassical small dispersion limit that develop slowly modulated n phase nonlinear wave in the process of evolution both approaches have their own merits and limitations understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fnls

on semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation we calculate the leading order term of the solution of the focusing nonlinear cubic schrodinger equation nls in the semiclassical limit for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 we utilize the riemann hilbert problem formulation of the inverse scattering problem to obtain the leading order term of the solution error estimates are provided 2004 wiley periodicals inc

the zero dispersion limit of the korteweg de vries equation for initial potentials with non trivial reflection coefficient the inverse scattering method is used to determine the distribution limit as 0 of the solution u x t of the initial value problem ut 6uux 2uxxx 0 u x 0 v x where v x is a positive bump which decays sufficiently fast as x x the case v x 0 has been solved by peter d lax and c david levermore 8 9 10 the computation of the distribution limit of u x t as 0 is reduced to a quadratic maximization problem which is then solved

the n soliton of the focusing nonlinear schrodinger equation for n large we present a detailed analysis of the solution of the focusing nonlinear schrodinger equation with initial condition x 0 n sech x in the limit n 1 we begin by presenting new and more accurate numerical reconstructions of the n soliton by inverse scattering numerical linear algebra for n 5 10 20 and 40 we then recast the inverse scattering problem as a riemann hilbert problem and provide a rigorous asymptotic analysis of this problem in the large n limit for those x t where results have been obtained by other authors we improve the error estimates from o n 1 3 to o n 1 we also analyze the fourier power spectrum in this regime and relate the results to the optical phenomenon of supercontinuum generation we then study the n soliton for values of x t where analysis has not been carried out before and we uncover new phenomena the main discovery of this paper is the mathematical mechanism for a secondary caustic phase transition which turns out to differ from the mechanism that generates the primary caustic the mechanism for the generation of the secondary caustic depends essentially on the discrete nature of the spectrum for the n soliton and more significantly cannot be recovered from an analysis of an ostensibly similar riemann hilbert problem in the conditions of which a certain formal continuum limit is taken on an ad hoc basis

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

semiclassical limit of focusing nls for a family of square barrier initial data the small dispersion limit of the focusing nonlinear schrodinger equation nls exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales the non self adjoint scattering problem and ill posed limiting whitham equations associated to focusing nls make rigorous asymptotic results difficult previous studies have focused on special classes of analytic initial data for which the limiting elliptic whitham equations are wellposed in this paper we consider another exactly solvable family of initial data the family of square barriers 0 x q l l for real amplitudes q using riemann hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of focusing nls globally in space and up to an o 1 maximal time in particular we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data to the best of our knowledge this is the first case in which the genus structure of the semiclassical asymptotics for focusing nls have been calculated for nonanalytic initial data 2013 wiley periodicals inc

numerical study of fractional nonlinear schrodinger equations using a fourier spectral method we provide a detailed numerical investigation of dispersive schrodinger type equations involving a fractional laplacian in an one dimensional case by an appropriate choice of the dispersive exponent both mass and energy sub and supercritical regimes can be identified this allows us to study the possibility of finite time blow up versus global existence the nature of the blow up the stability and instability of nonlinear ground states and the long time dynamics of solutions the latter is also studied in a semiclassical setting moreover we numerically construct ground state solutions of the fractional nonlinear schrodinger equation

rotational hypersurfaces of prescribed mean curvature abstract we use a phase space analysis to give some classification results for rotational hypersurfaces in r n 1 whose mean curvature is given as a prescribed function of its gauss map for the case where the prescribed function is an even function in s n we show that a delaunay type classification holds for this class of hypersurfaces we also exhibit examples showing that the behavior of rotational hypersurfaces of prescribed non constant mean curvature is much richer than in the constant mean curvature case

semiclassical eigenvalue distribution of the zakharov shabat eigenvalue problem abstract in this paper we consider the semi classical limit of the non self adjoint zakharov shabat eigenvalue problem we conduct a series of careful numerical experiments which provide strong evidence that the number of eigenvalues scales like 1 just as in the self adjoint case and that the eigenvalues appear to approach a limiting curve one general choice of potential functions produces a y shaped spectrum we give an asymptotic argument which predicts a critical value for the phase for which the straight line spectra bifurcates to produce the y shaped spectra this asymptotic prediction agrees quite well with numerical experiments the asymptotic argument also predicts a symmetry breaking in the eigenfunctions which we are able to observe numerically we also show that the number of eigenvalues living away from the real axis for a restricted class of potentials is bounded by c 1 where c is an explicit constant a complete theory of the shape of the eigenvalue curve and a general bound on the number of eigenvalues is still lacking

on the generation of dispersive shock waves abstract we review various methods for the analysis of initial value problems for integrable dispersive equations in the weak dispersion or semiclassical regime some methods are sufficiently powerful to rigorously explain the generation of modulated wavetrains so called dispersive shock waves as the result of shock formation in a limiting dispersionless system they also provide a detailed description of the solution near caustic curves that delimit dispersive shock waves revealing fascinating universal wave patterns

commuting difference operators spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights abstract the paper has three parts in the first part we apply the theory of commuting pairs of pseudo difference operators to the formal asymptotics of orthogonal polynomials using purely geometrical arguments we show heuristically that the asymptotics for large degrees of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of deift et al on the subject in the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane the strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a riemann hilbert analysis in the third part we use strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves as a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of jordan arcs forming an arbitrary forest of trivalent loop free trees

thin set theorems and cone avoidance the thin set theorem mathsf rt n infty ell asserts the existence for every k coloring of the subsets of natural numbers of size n of an infinite set of natural numbers all of whose subsets of size n use at most ell colors whenever ell 1 the statement corresponds to ramsey s theorem from a computational viewpoint the thin set theorem admits a threshold phenomenon in that whenever the number of colors ell is sufficiently large with respect to the size n of the tuples then the thin set theorem admits strong cone avoidance let d 0 d 1 dots be the sequence of catalan numbers for n geq 1 mathsf rt n infty ell admits strong cone avoidance if and only if ell geq d n and cone avoidance if and only if ell geq d n 1 we say that a set a is mathsf rt n infty ell encodable if there is an instance of mathsf rt n infty ell such that every solution computes a the mathsf rt n infty ell encodable sets are precisely the hyperarithmetic sets if and only if ell 2 n 1 the arithmetic sets if and only if 2 n 1 leq ell d n and the computable sets if and only if d n leq ell

pigeons do not jump high abstract the infinite pigeonhole principle for 2 partitions asserts the existence for every set a of an infinite subset of a or of its complement in this paper we develop a new notion of forcing enabling a fine analysis of the computability theoretic features of the pigeonhole principle we deduce various consequences such as the existence for every set a of an infinite subset of it or its complement of non high degree we also prove that every 0 3 set has an infinite low3 solution and give a simpler proof of liu s theorem that every set has an infinite subset in it or its complement of non pa degree

long time asymptotics of the nonlinear schrodinger equation shock problem the long time asymptotics of two colliding plane waves governed by the focusing nonlinear schrodinger equation are analyzed via the inverse scattering method we find three asymptotic regions in space time a region with the original wave modified by a phase perturbation a residual region with a one phase wave and an intermediate transition region with a modulated two phase wave the leading order terms for the three regions are computed with error estimates using the steepest descent method for riemann hilbert problems the nondecaying initial data requires a new adaptation of this method a new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions also the effect of the collision is felt in the plane wave state well beyond the shock front at large times 2007 wiley periodicals inc

the generation propagation and extinction of multiphases in the kdv zero dispersion limit we study the multiphases in the kdv zero dispersion limit these phases are governed by the whitham equations which are 2g 1 quasi linear hyperbolic equations where g is the number of phases we are interested in both the interaction of two single phases and the breaking of a single phase for general initial data we analyze in detail how a double phase is generated from the interaction or breaking how it propagates in space time and how it collapses to a single phase in a finite time the whitham equations are known to be integrable via a hodograph transform the crucial step in our approach is to formulate the hodograph transform in terms of the euler poisson darboux solutions under our scheme the zeros of the jacobian of the transform are given by the zeros of the euler poisson darboux solution hence the problem of inverting the hodograph transform to give the whitham solution reduces to that of counting the zeros of the euler poisson darboux solution 2002 wiley periodicals inc

the metamathematics of stable ramsey s theorem for pairs we show that over the base theory rca0 stable ramsey s the orem for pairs implies neither ramsey s theorem for pairs nor 0 induction

isoperimetric domains of large volume in homogeneous three manifolds abstract given a non compact simply connected homogeneous three manifold x and a sequence n n of isoperimetric domains in x with volumes tending to infinity we prove that as n 1 the radii of the n tend to infinity 2 the ratios area n vol n converge to the cheeger constant ch x which we also prove to be equal to 2 h x where h x is the critical mean curvature of x 3 the values of the constant mean curvatures h n of the boundary surfaces n converge to 1 2 ch x furthermore when ch x is positive we prove that for n large n is well approximated in a natural sense by the leaves of a certain foliation of x where every leaf of the foliation is a surface of constant mean curvature h x

exact solutions of semiclassical non characteristic cauchy problems for the sine gordon equation abstract the use of the sine gordon equation as a model of magnetic flux propagation in josephson junctions motivates studying the initial value problem for this equation in the semiclassical limit in which the dispersion parameter e tends to zero assuming natural initial data having the profile of a moving 2 kink at time zero we analytically calculate the scattering data of this completely integrable cauchy problem for all e 0 sufficiently small and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small e this sequence of exact solutions is analogous to that of the well known n soliton or higher order soliton solutions of the focusing nonlinear schrodinger equation we then use plots obtained from a careful numerical implementation of the inverse scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit in the limit e 0 one observes the appearance of nonlinear caustics i e curves in space time that are independent of e but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases in the appendices we give a self contained account of the cauchy problem from the perspectives of both inverse scattering and classical analysis picard iteration specifically appendix a contains a complete formulation of the inverse scattering method for generic l 1 sobolev initial data and appendix b establishes the well posedness for l p sobolev initial data which in particular completely justifies the inverse scattering analysis in appendix a

on the long time limit of semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation pure radiation case in a previous paper 13 we calculated the leading order term q0 x t e of the solution ofq x t e the focusing nonlinear cubic schrodinger nls equation in the semiclassical limit e 0 for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 the aim of the present paper is to calculate the long term behavior of the semiclassical solution q x t e in the pure radiation case as before our main tool is the riemann hilbert problem rhp formulation of the inverse scattering problem and the corresponding system of moment and integral conditions known also as a system of modulation equations c 2006 wiley periodicals inc

a hopf theorem for non constant mean curvature and a conjecture of a d alexandrov we prove a uniqueness theorem for immersed spheres of prescribed non constant mean curvature in homogeneous three manifolds in particular this uniqueness theorem proves a conjecture by a d alexandrov about immersed spheres of prescribed weingarten curvature in mathbb r 3 for the special but important case of prescribed mean curvature as a consequence we extend the classical hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in mathbb r 3

on uniform relationships between combinatorial problems the enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic with one of the most common frameworks for doing so being reverse mathematics in this setting one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics since the proofs of such implications take place in classical logic they may in principle involve appeals to multiple applications of a particular theorem or to non uniform decisions about how to proceed in a given construction in practice however if a theorem q implies a theorem p it is usually because there is a direct uniform translation of the problems represented by p into the problems represented by q in a precise sense formalized by weihrauch reducibility we study this notion of uniform reducibility in the context of several natural combinatorial problems and compare and contrast it with the traditional notion of implication in reverse mathematics we show for instance that for all n j k 1 if j k then ramsey s theorem for n tuples and k many colors is not uniformly or weihrauch reducible to ramsey s theorem for n tuples and j many colors the two theorems are classically equivalent so our analysis gives a genuinely finer metric by which to gauge the relative strength of mathematical propositions we also study weak konig s lemma the thin set theorem and the rainbow ramsey s theorem along with a number of their variants investigated in the literature weihrauch reducibility turns out to be connected with sequential forms of mathematical principles where one wishes to solve infinitely many instances of a particular problem simultaneously we exploit this connection to uncover new points of difference between combinatorial problems previously thought to be more closely related

the korteweg de vries equation with small dispersion higher order lax levermore theory we utilize the inverse scattering transformation to study the rapid oscillations which arise in the solution of the initial value problem begin gathered u t 6u u x in 2 u xxx 0 hfill u left x 0 right v left x right hfill end gathered when 0 we refine the lax levermore theory which gives the weak limit of the solution as 0 by introducing a quantum condition this allows us to calculate the waveform of the local oscillations up to phase shifts when is small

some logically weak ramseyan theorems abstract we study four families of consequences of ramsey s theorem from the viewpoint of reverse mathematics the first which we call the achromatic ramsey theorem is from a partition relation introduced by erd s hajnal and rado c d r which asserts that for every f r c there exists an infinite h with f h r d the second and third are the free set theorem and the thin set theorem which were introduced by harvey friedman and the last is the rainbow ramsey theorem we show that most theorems from these families are quite weak i e they are strictly weaker than aca 0 over rca 0 interestingly these families turn out to be closely related we establish the so called strong cone avoidance property of most instances of the achromatic ramsey theorem by an induction of exponents then apply this and a similar induction to obtain the strong cone avoidance property of the free set theorem from the strong cone avoidance property of the achromatic ramsey theorem and the free set theorem we derive the strong cone avoidance property of the thin set theorem and the rainbow ramsey theorem it follows easily that a theorem with the strong cone avoidance property does not imply aca 0 over rca 0