master symmetries for differential difference equations of the volterra type abstract it is demonstrated that in the case of integrable differential difference equations similar to the well known volterra equation unlike the korteweg de vries and nonlinear schrodinger equations there are many instances in which local master symmetries can be found those master symmetries are new interesting examples of local evolution chains explicitly depending on the time and discrete variable and integrable in a special sense the examples are constructed by a direct and elementary approach which enables one to get new integrable equations using miura type transformations

on the general structure of nonlinear evolution equations integrable by the two dimensional matrix spectral problem a generalization of the akns technique to the two dimensional arbitrary order matrix spectral problem is given the general form of the integrable equations and their backlund transformations in 1 2 dimensions are found the reduction problem is discussed

a recursive generation of local higher order sine gordon equations and their backlund transformation a new hierarchy of local nonlinear evolution equations is generated by a recursion operator and its explicit inverse it is shown that this hierarchy satisfies a canonical geometrical scheme and that it contains as special cases the sine gordon and liouville equations in laboratory coordinates a generalization of the well known backlund transformation and nonlinear superposition formula for the sine gordon equation is also obtained

the newton stratification on deformations of local g shtukas bounded local g shtukas are function field analogs for p divisible groups with extra structure we describe their deformations and moduli spaces the latter are analogous to rapoport zink spaces for p divisible groups the underlying schemes of these moduli spaces are a ne deligne lusztig varieties for basic newton polygons the closed newton stratum in the universal deformation of a local g shtuka is isomorphic to the completion of a corresponding a ne deligne lusztig variety in that point this yields bounds on the dimension and proves equidimensionality of the basic a ne deligne lusztig varieties

the concave majorant of brownian motion let s t be a version of the slope at time t of the concave majorant of brownian motion on lbrack 0 infty it is shown that the process s 1 s t t 0 is the inverse of a pure jump process with independent nonstationary increments and that brownian motion can be generated by the latter process and brownian excursions between values of the process at successive jump times as an application the limiting distribution of the l 2 norm of the slope of the concave majorant of the empirical process is derived

evolution equations with high order conservation laws an asymptotic algorithm presented in a previous paper is applied to investigate the possible structures of evolution equations ut um k u um 1 m 3 5 which could be compatible with the existence of a conserved density 0 u depending only on u and with the existence as well of conserved densities with arbitrarily high order derivatives for m 3 it is shown that the calogero degasperis fokas equation is essentially the only nonpolynomial equation of that type for the case ut d u4 q with q u u3 a polynomial we find a very narrow class of admissible structures for q typified by the few particular examples known up to date actually there is in this case an essentially unique structure modulo a miura transformation

backlund transformations soliton solutions and wave functions of kaup newell and wadati konno ichikawa systems using the bargmann darboux method the backlund transformations n soliton solutions and corresponding wave functions of the kaup newell and wadati konno ichikawa systems are obtained these results culminate in an algebraic recursive procedure for the determination of multisoliton solutions and their wave functions of the derivative and mixed derivative nonlinear schrodinger equations iqt qxx i q 2q x q 2q 0 0 0

commutation methods applied to the mkdv equation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided our theory is based on commutation methods i e n 1 supersymmetry underlying miura s transformation that links solutions of the two evolution equations

baklund transformations and singular integral equations abstract a systematic method for deriving backlund transformations for singular linear integral equations is presented the method leads in a natural way to the backlund transformations for the corresponding integrable nonlinear partial differential equations by repeated use of the method a hierarchy of singular integral equations and backlund transformations can be derived corresponding to so called multi modified partial differential equations specific examples include the korteweg de vries equation and the first second and third modified korteweg de vries equation the nonlinear schrodinger equation and the anisotropic heisenberg spin chain the sine gordon equation and the modified sine gordon equation

explicit quasi periodic solutions of the kaup newell hierarchy abstract an algebraic curve of arithmetic genus n is introduced with the aid of the lax matrix from which we construct the meromorphic function on the algebraic curve and investigate its properties further we straighten out all the flows associated with the kaup newell hierarchy under the abel jacobi coordinates finally we achieve the explicit quasi periodic solutions for the whole kaup newell hierarchy including the coupled derivative nonlinear schrodinger equations

codimensions of newton strata for sl3 f in the iwahori case we study the newton stratification on sl 3 f where f is a laurent power series field we provide a formula for the codimensions of the newton strata inside each component of the affine bruhat decomposition on sl 3 f these calculations are related to the study of certain affine deligne lusztig varieties in particular we describe a method for determining which of these varieties is non empty in the case of sl 3 f

mod points on shimura varieties of abelian type we show that the mod p points on a shimura variety of abelian type with hyperspecial level have the form predicted by the conjectures of kottwitz and langlands rapoport along the way we show that the isogeny class of a mod p point contains the reduction of a special point

on miura transformations among nonlinear partial differential equationsa in this paper we study miura transformations u v from partial differential equations uxxx f u ux ut to nonlinear partial differential equations g v vx vt xlv t1v 0 defined using integrable systems on v we classify all such miura transformations under some restrictions and hence generalize the classical miura transformation to a large class of nonlinear partial differential equations for some examples by applying miura transformations found in this paper we derive exact solutions v from known solutions u in particular kink and soliton kink solutions of vt 32vxsin2v 12vx3 vxxx are obtained from constant solutions and soliton solutions of the mkdv equation as another application of miura transformations of this paper we deduce a new backlund transformation for each of vt 32vxsin2v 12vx3 vxxx and vt 32vxsinh2v 12vx3 vxxx from the known backlund transformations for the mkdv equations

on series expansions giving closed form solutions of korteweg de vries like equations korteweg devries kdv like equations with higher degree nonlinearity are solved by a direct algebraic technique due to hereman et al j phys a 19 1986 pp 607 628 for two kdv like equations one with fifth degree nonlinearity the other a combined kdv and mkdv equation for particular choices of the coefficients of the nonlinear terms the kink and antikink solutions found by dey are recovered furthermore soliton solutions of the combined kdv and mkdv equation are found for all values of the coefficients closed form solutions for the calogero degasperis fokas modified mkdv equation are also developed applications of the solutions of these equations in quantum field theory plasma physics and solid state physics are mentioned the hereman et al method is illustrated and slightly extended and this direct series method is briefly compared to hirota s method

gauge transformations and symmetries of integrable systems we analyse several integrable systems in zero curvature form within the framework of invariant gauge theory in the drinfeld sokolov gauge we derive a two parameter family of nonlinear evolution equations which as special cases include the korteweg de vries kdv and harry dym hd equations we find residual gauge transformations which lead to infinitesimal symmetries of this family of equations for kdv and hd equations we find an infinite hierarchy of such symmetry transformations and we investigate their relation with local conservation laws constants of the motion and the bi hamiltonian structure of the equations applying successive gauge transformations of miura type we obtain a sequence of gauge equivalent integrable systems among them the modified kdv and calogero kdv equations

backlund transformations for the schrodinger equation with a spectral dependence in the potential the schrodinger equation on the line with a potential depending linearly on the spectral parameter is considered it is shown that this spectral problem admits one parameter called elementary backlund transformations and two parameter called full backlund transformations the transformations of the spectral data under all these backlund transformations bts are explicitly computed different nonlinear superposition formulae for the different bts considered are given and used to obtain soliton solutions of the evolution equations associated with the spectral problem

the greatest convex minorant of brownian motion meander and bridge this article contains both a point process and a sequential description of the greatest convex minorant of brownian motion on a finite interval we use these descriptions to provide new analysis of various features of the convex minorant such as the set of times where the brownian motion meets its minorant the equivalence of these descriptions is non trivial which leads to many interesting identities between quantities derived from our analysis the sequential description can be viewed as a markov chain for which we derive some fundamental properties

the convex minorant of a levy process we offer a unified approach to the theory of convex minorants of levy processes with continuous distributions new results include simple explicit constructions of the convex minorant of a levy process on both finite and infinite time intervals and of a poisson point process of excursions above the convex minorant up to an independent exponential time the poisson dirichlet distribution of parameter 1 is shown to be the universal law of ranked lengths of excursions of a levy process with continuous distributions above its convex minorant on the interval 0 1

connected components of closed affine deligne lusztig varieties for rese fgln abstract let f be a p adic field and let e f be a finite unramified extension for the weil restriction res e f g l n of the general linear group g l n we determine the connected components of closed affine deligne lusztig varieties of arbitrary parahoric level

hamiltonian structure of the general integrable equations under reductions the general form of the nonlinear evolution equations integrable by the linear matrix spectral problem and their hamiltonian structures are presented for potentials with nonzero constant asymptotic values the reduction problem is discussed within the framework of the akns method a method is proposed for the calculation of the hamiltonian structures for the integrable equations under reductions matrix zn and n reductions and some other reductions are considered as examples general forms of the corresponding reduced equations and their hamiltonian structures are found

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

an algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations an algebraic method is devised to uniformly construct a series of exact solutions for general integrable and nonintegrable nonlinear evolution equations compared with most existing tanh methods the jacobi function expansion method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the solutions according to the values of some parameters the solutions obtained in this paper include a polynomial solutions b exponential solutions c rational solutions d triangular periodic wave solutions e hyperbolic and solitary wave solutions and f jacobi and weierstrass doubly periodic wave solutions the efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper new 2 1 dimensional calogero kdv equation 3 1 dimensional jimbo miwa equation symmetric regular long wave equation drinfel d sokolov wilson equation 2 1 dimensional generalized dispersive long wave equation double sine gordon equation calogero degasperis fokas equation and coupled schrodinger boussinesq equation in addition the links among our proposed method the tanh method the extended method and the jacobi function expansion method are also clarified generally

phase space analysis for the direct algebraic method for nonlinear evolution and wave equations abstract the basis of the direct algebraic method employing a series in real exponential functions for solving nonlinear evolution and wave equations is examined in terms of the general theory of autonomous ordinary differential equations odes and phase space analysis the closed form solutions obtained by the direct algebraic method are identified with special trajectories in phase space that connect a critical point to itself or other critical points the phase portrait is investigated for odes resulting from a combined kdv and mkdv equation a kdv like equation with fifth degree nonlinearity a special case of the calogero degasperis fokas modified mkdv equation and a generalized fisher equation the phase space analysis serves to clarify and partially justify the direct algebraic method

explicit auto backlund relation through gauge transformation application of gauge transformation for generating explicit auto backlund relations is demonstrated for a number of integrable systems e g kdv sg nls dnls mixed dnls modified dnls lle etc this method turns out to be much simpler effective and straightforward using backlund relations all conserved quantities are expressed in a novel derivative free form which simplifies the calculation of important soliton characteristics

hodograph transformations of linearizable partial differential equations in this paper an algorithmic method is developed for transforming quasilinear partial differential equations of the form u t g u u nx f u u x cdots u n 1 x u mx equiv partial m u partial x m where dg du not equiv 0 into semilinear equations i e equations of the above form with g u 1 this crucially involves the use of hodograph transformations i e transformations involving the interchange of dependent and independent variables furthermore the most general quasilinear equation of the above form is found that can be mapped via a hodograph transformation to a semilinear form this algorithm provides a method for establishing whether a given quasilinear equation is linearizable i e is solvable in terms of either a linear partial differential equation or of a linear integral equation in particular this method is used to show how the painleve tests may be applied to quasilinear equations this appears to resolve the problem that solutions of lineariz

exact explicit traveling wave solutions for the cdf equation abstract the traveling wave solutions of the calogero degasperis fokas cdf equation are studied by using the approach of dynamical systems and the theory of bifurcations with the aid of maple all bifurcations and phase portraits in the parametric space are obtained all possible explicit parametric representations of the bounded traveling wave solutions solitary wave solutions kink and anti kink wave solutions periodic wave solutions and breaking wave solutions are given moreover the reason for appearance of breaking waves and for persistence of smoothness of smooth traveling waves is explained

on the hodge newton filtration for p divisible o modules the notions hodge newton decomposition and hodge newton filtration for f crystals are due to katz and generalize messing s result on the existence of the local etale filtration for p divisible groups recently some of katz s classical results have been generalized by kottwitz to the context of f crystals with additional structures and by moonen to ordinary p divisible groups in this paper we discuss further generalizations to the situation of crystals in characteristic p and of p divisible groups with additional structure by endomorphisms

connected components of closed affine deligne lusztig varieties for split reductive algebraic groups we determine the sets of connected components of closed affine deligne lusztig varieties with parahoric level structure

markovian bridges weak continuity and pathwise constructions a markovian bridge is a probability measure taken from a disintegration of the law of an initial part of the path of a markov process given its terminal value as such markovian bridges admit a natural parameterization in terms of the state space of the process in the context of feller processes with continuous transition densities we construct by weak convergence considerations the only versions of markovian bridges which are weakly continuous with respect to their parameter we use this weakly continuous construction to provide an extension of the strong markov property in which the flow of time is reversed in the context of self similar feller process the last result is shown to be useful in the construction of markovian bridges out of the trajectories of the original process

on the calogero degasperis fokas equation in 2 1 dimensions abstract in this paper we study a 2 1 dimensional integrable calogero degasperis fokas equation derivable by using a method proposed by calogero a catalogue of classical symmetry reductions are given these reductions to partial differential equations in 1 1 admit symmetries which lead to further reductions i e to second order ordinary differential equations these odes provide several classes of solutions all of them are expressible in terms of known functions some of them are expressible in terms of the second and third painleve trascendents the corresponding solutions of the 2 1 dimensional equation involve up to three arbitrary smooth functions consequently the solutions exhibit a rich variety of qualitative behaviour indeed by making appropriate choices for the arbitrary functions we exhibit solitary waves coherent structures and bound states

nonlinear evolution equations and nonabelian prolongations a systematic analysis of the class of nonlinear evolution equations ut uxxx u ux 0 is carried out within the estabrook wahlquist prolongation scheme

reduction technique for matrix nonlinear evolution equations solvable by the spectral transform the main purpose of this paper is to describe a technique of reduction whereby from the class of evolution equations for matrices of order n solvable via the spectral transform associated to the matrix linear schrodinger eigenvalue problem one derives subclasses of nonlinear evolution equations involving less than n2 fields to illustrate the method from the equations for matrices of order 2 two subclasses of equations for 2 fields rather than 4 are obtained the first class coincides or rather includes that solvable via the spectral transform associated to the generalized zakharov shabat spectral problem further reduction to nonlinear evolution equations for a single field reproduces a number of well known equations but also yields a novel one highly nonlinear the second class also yields highly nonlinear equations some examples are given including another novel evolution equation for a single field

specializations of integrable systems and affine lie algebras we analyze a number of restrictions on the evolution systems associated with the zero curvature equations corresponding to the extended dynkin diagram a 1 n 1 the resulting specialized evolution systems contain exponential terms like a third order differential equation previously derived in ref 5

from singular manifold equations to integrable evolution equations the singular manifold equations are used to construct nonlinear integrable partial differential equations a number of well known integrable equations hierarchies of korteveg de vries caudrey dodd gibbon kaup kuperschmidt and harry dym are obtained some new integrable equations are presented a possible approach to classification of integrable evolution equations is discussed

bicomplexes and backlund transformations a bicomplex is a simple mathematical structure in particular associated with completely integrable models the conditions defining a bicomplex are a special form of a parameter dependent zero curvature condition we generalize the concept of a darboux matrix to bicomplexes and use it to derive backlund transformations for several models the method also works for moyal deformed equations with a corresponding deformed bicomplex

backlund transformations connecting different isospectral deformation equations we study singular spectral problems on riemann surfaces and isospectral deformation equations connected with them we show that for a given class of reductions all spectral problems on a given riemann surface give rise to the same class of isospectral deformation equations these isospectral deformation equations can be solved using the riemann boundary value problem all possess local conservation laws and backlund transformations

backlund transformations related to the kaup newell spectral problem abstract a hierarchy of isospectral deformation equations related to the kaup newell spectral problem is obtained the generalized gauge transformations of the eigenmatrix of the spectral problem are used to generate backlund and self backlund transformations in a special case the backlund transformation of the null solution for all the equations in the hierarchy has been explicitly solved

connected components of closed affine deligne lusztig varieties we determine the set of connected components of closed affine deligne lusztig varieties for special maximal compact subgroups of split connected reductive groups we show that there is a transitive group action on this set thus such an affine deligne lusztig variety has isolated points if and only if its dimension is 0 we also obtain a description of the set of these varieties that are zero dimensional