hodograph transformations for a camassa holm hierarchy in 2 1 dimensions a generalization of the negative camassa holm hierarchy to 2 1 dimensions is presented under the name chh 2 1 several hodograph transformations are applied in order to transform the hierarchy into a system of coupled cbs calogero bogoyavlenskii schiff equations in 2 1 dimensions that pass the painleve test a non isospectral lax pair for chh 2 1 is obtained through the above mentioned relationship with the cbs spectral problem

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

automorphisms of the fine grading of sl n c associated with the generalized pauli matrices we consider the grading of sl n c by the group n of generalized pauli matrices the grading decomposes the lie algebra into n2 1 one dimensional subspaces in the article we demonstrate that the normalizer of grading decomposition of sl n c in n is the group sl 2 zn where zn is the cyclic group of order n as an example we consider sl 3 c graded by 3 and all contractions preserving that grading we show that the set of 48 quadratic equations for grading parameters splits into just two orbits of the normalizer of the grading in 3

weyl groups of fine gradings on matrix algebras octonions and the albert algebra given a grading a g gag on a nonassociative algebra a by an abelian group g we have two subgroups of aut a the automorphisms that stabilize each component ag as a subspace and the automorphisms that permute the components by the weyl group of we mean the quotient of the latter subgroup by the former in the case of a cartan decomposition of a semisimple complex lie algebra this is the automorphism group of the root system i e the so called extended weyl group a grading is called fine if it cannot be refined we compute the weyl groups of all fine gradings on matrix algebras octonions and the albert algebra over an algebraically closed field of characteristic different from 2 in the case of the albert algebra

grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

nonstandard conserved hamiltonian structures in dissipative damped systems nonlinear generalizations of damped harmonic oscillator in this paper we point out the existence of a remarkable nonlocal transformation between the damped harmonic oscillator and a modified emden type nonlinear oscillator equation with linear forcing x xx x3 x 0 which preserves the form of the time independent integral conservative hamiltonian and the equation of motion generalizing this transformation we prove the existence of nonstandard conservative hamiltonian structure for a general class of damped nonlinear oscillators including lienard type systems further using the above hamiltonian structure for a specific example namely the generalized modified emden equation x xqx x2q 1 0 where and q are arbitrary parameters the general solution is obtained through appropriate canonical transformations we also present the conservative hamiltonian structure of the damped mathews lakshmanan oscillator equation the associated lagrangian description for all the above systems is also briefly discussed

the weyl group of the fine grading of sl n c associated with tensor product of generalized pauli matrices we consider the fine grading of sl n c induced by tensor product of generalized pauli matrices in the paper regard pgl n c as the inner automorphism group of sl n c based on the classification of maximal abelian subgroups of pgl n c consisting of diagonalizable automorphisms by havlicek et al we prove that any such subgroup k of pgl n c is a symplectic abelian group and its weyl group which describes the symmetry of the fine grading induced by the action of k is just the isometry group of the symplectic abelian group k for a finite symplectic abelian group it is also proved that its isometry group is always generated by the transvections contained in it

1 1 spectral problems arising from the manakov santini system this paper deals with the spectral problem of the manakov santini system the point lie symmetries of the lax pair have been identified several similarity reductions arise from these symmetries an important benefit of our procedure is that the study of the lax pair instead of the partial differential equations yields the reductions of the eigenfunctions and also the spectral parameter therefore we have obtained five interesting spectral problems in 1 1 dimensions

korteweg de vries surfaces abstract we consider 2 surfaces arising from the korteweg de vries kdv hierarchy and the kdv equation the surfaces corresponding to the kdv equation are in a three dimensional minkowski m 3 space they contain a family of quadratic weingarten and willmore like surfaces we show that some kdv surfaces can be obtained from a variational principle where the lagrange function is a polynomial function of the gaussian and mean curvatures we also give a method for constructing the surfaces explicitly i e finding their parameterizations or finding their position vectors

slopes of eigencurves over boundary disks let p be a prime number we study the slopes of u p eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra we give a sharp lower bound of the corresponding newton polygon the computation happens over a definite quaternion algebra by jacquet langlands correspondence it generalizes a prior work of jacobs slopes of compact hecke operators thesis university of london imperial college 2004 who treated the case of p 3 with a particular level in case when the modular forms have a finite character of conductor highly divisible by p we improve the lower bound to show that the slopes of u p eigenvalues grow roughly like arithmetic progressions as the weight k increases this is the first very positive evidence for buzzard kilford s conjecture on the behavior of the eigencurve near the boundary of the weight space that is proved for arbitrary p and general level we give the exact formula of a fraction of the slope sequence

modified korteweg de vries surfaces in this work we consider 2 surfaces in r3 arising from the modified korteweg de vries mkdv equation we give a method for constructing the position vector of the mkdv surface explicitly for a given solution of the mkdv equation mkdv surfaces contain willmore like and weingarten surfaces we show that some mkdv surfaces can be obtained from a variational principle where the lagrange function is a polynomial of the gaussian and mean curvatures

on the eigencurve at classical weight 1 points we show that the p adic eigencurve is smooth at classical weight one points which are regular at p and give a precise criterion for etaleness over the weight space at those points our approach uses deformations of galois representations

a nonlocal connection between certain linear and nonlinear ordinary differential equations oscillators we explore a nonlocal connection between certain linear and nonlinear ordinary differential equations odes representing physically important oscillator systems and identify a class of integrable nonlinear odes of any order we also devise a method to derive explicit general solutions of the nonlinear odes interestingly many well known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models

a generalized formula for integrable classes of surfaces in lie algebras we discuss relations between the approach of fokas and gelfand to immersions on lie algebras and the theory of soliton surfaces of sym we show that many results concerning immersions on lie algebras can be reduced to or interpreted within the soliton surfaces approach we present also some new results including a generalization of the fokas gelfand formula for integrable classes of surfaces in lie algebras and in particular in pseudo euclidean n dim spaces the generalized formula is used to formulate a method of constructing integrable classes of surfaces as an example we discuss the class of linear weingarten surfaces defined by the linear relationship between gaussian and mean curvatures we construct explicitly a one parameter family of linear weingarten surfaces parallel equidistant to a given pseudospherical surface

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

integrable theories and generalized graded maillet algebras we present a general formalism to investigate the integrable properties of a large class of non ultralocal models which in principle allows the construction of the corresponding lattice versions our main motivation comes from the subsector of the string theory on ads5 s5 in the uniform gauge where such type of non ultralocality appears in the resulting alday arutyunov frolov aaf model we first show how to account for the second derivative of the delta function in the lax algebra of the aaf model by modifying maillet s r and s matrices formalism and derive a well defined algebra of transition matrices which allows for the lattice formulation of the theory we illustrate our formalism on the examples of the bosonic wadati konno ichikawa shimizu wkis model and the two dimensional free massive dirac fermion model which can be obtained by a consistent reduction of the full aaf model and give the explicit forms of their corresponding r matrices

erratum classification of lie point symmetries for quadratic lienard type equation x f x x2 g x 0 j math phys 54 053506 2013

triple product p adic l functions for balanced weights we construct p adic triple product l functions that interpolate square roots of central critical l values in the balanced region thus our construction complements that of harris and tilouine there are four central critical regions for the triple product l functions and two opposite settings according to the sign of the functional equation in the first case three of these regions are of interpolation having positive sign they are called the unbalanced regions and one gets three p adic l functions one for each region of interpolation this is the harris tilouine setting in the other setting there is only one region of interpolation called the balanced region we produce the corresponding p adic l function our triple product p adic l function arises as p adic period integrals interpolating normalizations of the local archimedean period integrals the latter encode information about classical representation theoretic branching laws the main step in our construction of p adic period integrals is showing that these branching laws vary in a p adic analytic fashion this relies crucially on the ash stevens theory of highest weight representations over affinoid algebras

backlund transformations and explicit solutions of certain inhomogeneous nonlinear schrodinger type equations shows that the darboux bargmann method can be used to derive backlund transformations and construct explicit soliton solutions for certain inhomogeneous nonlinear schrodinger type equations

p adic families of siegel modular cuspforms let p be an odd prime and g 2 an integer we prove that a nite slope siegel cuspidal eigenform of genus g can be p adically deformed over the g dimensional weight space the proof of this theorem relies on the construction of a family of sheaves of locally analytic overconvergent modular forms contents

on the general solution for the modified emden type equation in this paper we demonstrate that the modified equation of emden type mee is integrable either explicitly or by quadrature for any value of and we also prove that the mee possesses appropriate time independent hamiltonian functions for the full range of parameters and in addition we show that the mee is intimately connected with two well known nonlinear models namely the equation of force free duffing type oscillator and the two dimensional lotka volterra equation and thus the complete integrability of the latter two models can also be understood in terms of the mee

hida families and rational points on elliptic curves let e be an elliptic curve over q of conductor n mp having a prime p n of multiplicative reduction because e is modular it corresponds to a normalised weight two eigenform on 0 n whose q expansion is denoted f n anq letx hom z p z p z p 1 z zp which contains z as a dense subset by associating to k z the character x xk 2 denote by a u the ring of cp valued p adic analytic functions on a compact open subset u of x hida s theory associates to f a neighborhood u of 2 x which can be assumed for simplicity to be contained in the residue class of 2 modulo p 1 and a formal q expansion

on the coefficients of the characteristic series of the u operator abstract a conceptual proof is given of the fact that the coefficients of the characteristic series of the u operator acting on families of overconvegent modular forms lie in the iwasawa algebra

harmonic maps backlund darboux transformations and line solution analogues harmonic maps from or one connected domain into and u m are treated the gbdt version of the b cklund darboux transformation is applied to the case of the harmonic maps and a new and simple algebraic procedure to construct new harmonic maps from the initial ones is given using some methods from system theory a new general formula on the gbdt transformations of the sym tafel immersions is derived a new class of the unitary harmonic maps with asymptotics along one line essentially different from the asymptotics in all other directions similar in certain ways to line solutions is obtained explicitly and studied

integrable 1 1 dimensional hierarchies arising from reduction of a non isospectral problem in 2 1 dimensions this work presents a classical lie point symmetry analysis of a two component non isospectral lax pair of a hierarchy of partial differential equations in 2 1 dimensions which can be considered as a modified version of the camassa holm hierarchy in 2 1 dimensions a classification of reductions for this spectral problem is performed non isospectral reductions in 1 1 dimensions are considered of remarkable interest

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

ermakov systems of arbitrary order and dimension structure and linearization ermakov systems of arbitrary order and dimension are constructed these inherit an underlying linear structure based on that recently established for the classical ermakov system as an application alignment of a 2 1 dimensional ermakov and integrable ernst system is shown to produce a novel integrable hybrid of a 2 1 dimensional sinh gordon system and of a conventional ermakov system

backlund transformations of linear weingarten surfaces in minkowski three space the classical backlund transformation which transforms surfaces of constant negative curvature has been extended to surfaces in minkowski three space in this article the tangency requirement of the backlund transformation is relaxed to obtain a transformation of surfaces of minkowski three space with mean and gaussian curvatures satisfying a linear relationship

fine grading of sl p2 c generated by tensor product of generalized pauli matrices and its symmetries study of the normalizer of the mad group corresponding to a fine grading offers the most important tool for describing symmetries in the system of nonlinear equations connected with contraction of a lie algebra one fine grading that is always present in any lie algebra sl n c is the pauli grading the mad group corresponding to it is generated by generalized pauli matrices for such mad group we already know its normalizer its quotient group is isomorphic to the lie group sl 2 zn z2 in this paper we deal with a more complicated situation namely that the fine grading of sl p2 c is given by a tensor product of the pauli matrices of the same order p p being a prime we describe the normalizer of the corresponding mad group and we show that its quotient group is isomorphic to sp 4 fp z2 where fp is the finite field with p elements

universal eigenvarieties trianguline galois representations and p adic langlands functoriality using the overconvergent cohomology modules introduced by ash and stevens we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces building on work of ash stevens urban and others we also formulate a precise modularity conjecture linking trianguline galois representations with overconvergent cohomology classes in the course of giving evidence for this conjecture we establish several new instances of p adic langlands functoriality our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of chenevier s interpolation theorem

a nonlocal connection between certain linear and nonlinear ordinary differential equations extension to coupled equations identifying integrable coupled nonlinear ordinary differential equations odes of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics in this paper we undertake these problems and unearth two classes of integrable coupled nonlinear odes of arbitrary order to achieve these goals we introduce suitable nonlocal transformations in certain linear odes and generate the coupled nonlinear odes in particular we show that the problem of solving these classes of coupled nonlinear odes of any order effectively reduces to solving a single first order nonlinear ode we then describe a procedure to derive explicit general solutions for the identified integrable coupled odes when the above mentioned first order nonlinear ode reduces to a bernoulli equation the equations which we generate and solve include the two coupled versions of modified emden equations in second order coupled versions of chazy equations in third order and their variants higher d

on integrability of the inhomogeneous heisenberg ferromagnet model examination of a new test we present an improved version of the integrability test based on application of lie point transformations acting on the bare linear problem the test is then applied to the inhomogeneous heisenberg ferromagnet model equation the integrable cases selected by this test are identical with those obtained by the standard painleve test

the eigencurve over the boundary of weight space author s liu ruochuan wan daqing xiao liang abstract we prove that the eigencurve associated to a definite quaternion algebra over qq satisfies the following properties as conjectured by coleman mazur and buzzard kilford a over the boundary annuli of weight space the eigencurve is a disjoint union of countably infinitely many connected components each finite and flat over the weight annuli b the u p slopes of points on each fixed connected component are proportional to the p adic valuations of the parameter on weight space and c the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference in particular as a point moves towards the boundary on an irreducible connected component of the eigencurve the slope converges to zero

how can we construct abelian galois extensions of basic number elds recall the almost tautological but very useful way we have of dealing with an irreducible polynomial over a field for example in the case of x5 x 1 irreducible over the rational field q we happily adjoin a root to our base field by merely forming l the quotient of the polynomial ring q x modulo the ideal generated by x5 x 1 then l is a field of algebraic numbers and the image of x is indeed a root of x5 x 1 in l

nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order in this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations odes of arbitrary order related to linear odes through nonlocal relations the procedure makes use of the lie point symmetries of the linear odes and the nonlocal connection to deduce the nonlocal symmetries of the corresponding nonlinear odes using these nonlocal symmetries we obtain reduction transformations and reduced equations to specific examples we find that the reduced equations can be explicitly integrated to deduce the general solutions for these cases we also extend this procedure to coupled higher order nonlinear odes with specific reference to second order nonlinear odes

a new approach to the darboux backlund transformation versus the standard dressing method we present a new approach to the construction of the darboux matrix this is a generalization of a recently formulated method based on the assumption that the square of the darboux matrix vanishes for some values of the spectral parameter we consider the multisoliton case the reduction problem and the discrete case the relationships between our approach the zakharov shabat dressing method and the neugebauer meinel method are discussed in detail

une correspondance de jacquet langlands p adique we extend the jacquet langlands correspondence between the hecke modules of usual and quaternionic modular forms to overconvergent p adic forms of finite slope we show that this correspondence respects p adic families and is induced by an isomorphism between some associated eigencurves

rational solutions for schwarzian integrable hierarchies we give an approach to finding rational solutions of completely intagrable hierarchies which makes use of the relationship between modifications and the schwarzian equations obtained via the singular manifold method this extends the recent work of kudryashov which allowed a simple derivation of the iteration used to construct sequences of such solutions we also give a closed form for the index polynomial of the schwarzian korteweg de vries hierarchy in addition we consider the representation of rational solutions using lower families of the hierarchy we give a simple representation under which the rational solutions remain solutions of every flow of the hierarchy this representation also allows the inclusion of arbitrary data corresponding to negative indices we use our method to derive an alternative form of the backlund transformation for the hierarchy of the second painleve equation as well as new solutions of a hierarchy of breaking soliton equations we also present here for the first time a schwarzian version of this breaking soliton hierarchy

nonlocal symmetries and a working algorithm to isolate integrable geometries a 1 parameter group of nonlocal symmetries of nonhomogeneous nonlinear schrodinger system with the coupling function linear in x is presented on using this symmetry the author is able to insert the spectral parameter into the nonparametric lax pair in all cases which are known to be integrable a working algorithm to isolate integrable geometries proposed by levi sym and tu 1990 is extended

on the determination of non local symmetries the importance of non local symmetries of differential equations lies in their manifestation as lie point symmetries of the equations resulting from reduction of order the reason for the determination of these symmetries in second order equations with only one lie point symmetry is self evident however the disadvantage of non local symmetries is that no systematic approach to their determination exists we present such an approach applicable to differential equations of any order and apply it to some second order ordinary differential equations and show that they have a rich occurrence we also look at possible generalizations of the concept of non local symmetries

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

critical p adic l functions we attach p adic l functions to critical modular forms and study them we prove that those l functions fit in a two variables p adic l function defined locally everywhere on the eigencurve

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

on the properties of a variant of the riccati system of equations a variant of the generalized riccati system of equations is considered it is shown that for 2n 3 the system admits a bilagrangian description and the dynamics has a node at the origin whereas for much smaller than a critical value the dynamics is periodic the origin being a centre it is found that the solution changes from being periodic to aperiodic at a critical point which is independent of the initial conditions this behaviour is explained by finding a scaling argument via which the phase trajectories corresponding to different initial conditions collapse onto a single universal orbit numerical evidence for the transition is shown further using a perturbative renormalization group argument it is conjectured that the oscillator exhibits isochronous oscillations the correctness of the conjecture is established numerically

surfaces immersed in lie algebras associated with elliptic integrals the objective of this work is to adapt the fokas gel fand immersion formula to ordinary differential equations written in the lax representation the formalism of generalized vector fields and their prolongation structure is employed to establish necessary and sufficient conditions for the existence and integration of immersion functions for surfaces in lie algebras as an example a class of second order integrable ordinary differential equations is considered and the most general solutions for the wavefunctions of the linear spectral problem are found several explicit examples of surfaces associated with jacobian and weierstrass elliptic functions are presented

une interpretation modulaire de la variete trianguline using a patching module constructed in recent work of caraiani emerton gee geraghty pask nas and shin we construct some kind of analogue of an eigenvariety we can show that this patched eigenvariety agrees with a union of irreducible components of a space of trianguline galois representations building on this we discuss the relation with the modularity conjectures for the crystalline case a conjecture of breuil on the locally analytic socle of representations occurring in completed cohomology and with a conjecture of bellaiche and chenevier on the complete local ring at certain points of eigenvarieties

nonlocal symmetries and integrable ordinary differential equations x 3xx x3 0 and its generalizations the equation x 3xx x3 0 is well known in many areas of mathematics and physics it possesses the algebra sl 3 r of lie point symmetries hence is equivalent to the equation for a free particle and both left and right painleve series we investigate two higher dimensional analogs in terms of their symmetry and singularity structures we find a drastic reduction in symmetry and a loss of some of the singularity properties from the nonlocal symmetries we are able to determine the complete symmetry group as being represented by a five dimensional abelian algebra

overconvergent modular forms and the fontaine mazur conjecture we prove a conjecture of fontaine and mazur on modularity of representations of g which are potentially semi stable at p for representations coming from finite slope overconvergent eigenforms we also give an application of our technique to a question of gouvea on overconvergence of certain modular forms if is a suitable representation of g s on a two dimensional vector space over a finite field product of m and the s a finite set of primes we construct a rigid analytic subspace of the universal deformation space of which is defined by a purely representation theoretic condition and contains the eigencurve of coleman mazur conjecturally it is equal to this curve

group interpretation of the spectral parameter in the case of nonhomogeneous nonlinear schrodinger system a one parameter group inserting the spectral parameter into the nonparametric lax pair of the nonhomogeneous nonlinear schrodinger system is presented and discussed being a nonlocal extension of a lie point transformation this group is probably not contained in any known class of symmetries

symmetries of finite heisenberg groups for multipartite systems a composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in i 1 k is considered its hilbert space is given by a k fold tensor product of hilbert spaces of dimensions n1 nk the symmetry group of the respective finite heisenberg group is given by the quotient group of certain normalizer this paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type it provides detailed description of the normalizers and the corresponding symmetry groups the new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings as an application a new proof of existence of the maximal set of mutually unbiased bases in hilbert spaces of prime power dimensions is provided

extension of inverse scattering method to nonlinear evolution equation in nonuniform medium by allowing the entire spectrum of certain linear eigenvalue problems to evolve with time a general type of nonlinear evolution equation in nonuniform medium which is exactly integrable by the inverse scattering method has been derived the derivative nonlinear schrodinger equation or the nonlinear schrodinger equation with linear or parabolic density profiles are special cases of this generalized form

p adic banach spaces and families of modular forms let p be a prime cp the completion of an algebraic closure of the p adic numbers qp and k a nite extension of qp contained in cp let v be the valuation on cp such that v p 1 and let be the absolute value on cp such that x p v x for x cp suppose n is a positive integer prime to p let x1 np denote the modular curve over k which represents elliptic curves with 1 np structure and let up be the hecke operator on modular forms on x1 np which takes a form with q expansion n anq n to the modular form with q expansion n anpq n a modular form f is said to have slope q if there is a polynomial r t over cp such that r up f 0 and such that the newton polygon of r t has only one side and its slope is

integrable non isospectral flows associated with the kadomtsev petviashvili equations in 2 1 dimensions the symmetries of the kadomtsev petviashvili kp equations in 2 1 dimensions yield two hierarchies of integrable non linear evolution equations nee one is the familiar family of isospectral flows the kp hierarchy the other is non isospectral and its flows have coefficients which depend linearly on x and y the spectral methods used to solve kp can be used to solve all these nee an underlying infinite dimensional lie algebra is used to determine all the lax pairs for both families and it also determines their symplectic structures constants of the motion are constructed for the non isospectral cases

symmetry reductions of the lax pair for the 2 1 dimensional konopelchenko dubrovsky equation based on the symbolic computational system maple the similarity reduction arising from the classical lie point symmetries of the lax pair for the 2 1 dimensional konopelchenko dubrovsky kd equation is carried out we obtain several interesting reductions by analyzing not only the reduced lax pair but also the kd equation reduced under the same symmetry group we find that the reduced lax pairs do not always lead to the reduced kd equation

completed cohomology of shimura curves and a p adic jacquet langlands correspondence we study indefinite quaternion algebras over totally real fields f and give an example of a cohomological construction of p adic jacquet langlands functoriality using completed cohomology we also study the tame levels of p adic automorphic forms on these quaternion algebras and give an analogue of mazur s level lowering principle

an effective method to compute n fold darboux matrix and n soliton surfaces fairly simple formulas for an n fold darboux matrix and an n soliton addition to an arbitrary soliton surface embedded in e3 are proven they depend directly on the background wave function and are evidently symmetric with respect to n added solitons as an example an explicit expression for the interaction of n solitons on a single vortex filament is presented moreover there is given a simplified version of the n soliton formulas of neugebauer and meinel

weyl groups of some fine gradings on e6 abstract the weyl groups of the fine gradings on the exceptional lie algebra e 6 whose universal grading group is infinite are computed

graded contractions of the pauli graded sl 3 c the lie algebra sl 3 c is considered in the basis of generalized pauli matrices corresponding grading is the pauli grading here it is one of the four gradings of the algebra which cannot be further refined the set s of 48 contraction equations for 24 contraction parameters is solved our main tools are the symmetry group of the pauli grading of sl 3 c which is essentially the finite group sl 2 z3 and the induced symmetry of the system s a list of all equivalence classes of solutions of the contraction equations is provided among the solutions 175 equivalence classes are non parametric and 13 solutions depend on one or two continuous parameters providing a continuum of equivalence classes and subsequently continuum of non isomorphic lie algebras solutions of the contraction equations of pauli graded sl 3 c are identified here as specific solvable lie algebras of dimensions up to 8 earlier algorithms for identification of lie algebras given by their structure constants had to be made more efficient in order to distinguish non isomorphic lie algebras encountered here resulting lie algebras are summarized in tabular form there are 88 indecomposable solvable lie algebras of dimension 8 77 of them being nilpotent there are 11 infinite sets of parametric lie algebra which still deserve further study

hidden symmetries and linearization of the modified painleve ince equation the linearization and hidden symmetries of the modified painleve ince equation y yy y3 0 where and are constants are presented the linearization of this equation by a nonlocal transformation yields a damped stable or growing unstable harmonic oscillator equation for 0 hidden symmetries are analyzed by transforming the modified painleve ince equation to a third order ordinary differential equation ode which in general is invariant under a three parameter group by a riccati transformation a type i hidden symmetry is found of a second order ode found from the third order ode where a symmetry is lost in the reduction of order by the non normal subgroup invariants a type ii hidden symmetry occurs in the third order ode because the symmetries of a second order ode reduced from the third order ode by another set of normal subgroup invariants are increased

on the semi simplicity of the u_p operator on modular forms for and positive integers let denote the vector space of cuspidal modular forms of level and weight this vector space is equipped with the usual hecke operators if we need to consider several levels or weights at the same time we will denote this by or if is a prime number dividing our is also known under the name one of our main results can be stated very easily if and does not divide then the operator is semi simple we can prove the same result for weight under the assumption that certain crystalline frobenius elements are semi simple milne has shown in 11 0 2 that this semi simplicity is implied by tate s conjecture claiming that for 1 projective and smooth over a finite field of characteristic and 23 54 687 9 ab c1d fehgjilknmo equals the order of pq c1r sb at 2 ulmer proved in 17 that is semi simple for 5 t and u not dividing under the assumption of the birch swinnerton dyer conjecture for elliptic curves over function fields in characteristic his method is quite different from ours assuming that is not semi simple he really shows that the birch swinnerton dyer conjecture does not hold for an explicitly given elliptic curve the structure of our proof is as follows using the theory of newforms the problem is shown to be equivalent to the problem of showing that for a normalized newform v of weight prime to level and character w the polynomial x uzy x w8 ba c has no double root this polynomial happens to be the characteristic polynomial of the frobenius element at in the two dimensional galois representations associated to v it is also the characteristic polynomial of the crystalline frobenius asociated to v we show that this crystalline frobenius cannot be a scalar in sections 2 and 3 we prove the results concerning these frobenius elements for d and e f respectively section 2 is quite elementary whereas in section 3 we use a lot of the machinary for comparing adic etale and crystalline cohomology in section 4 we give some applications the ramanujan inequality is a strict inequality in certain cases certain hecke algebras are reduced hence have non zero discriminant section 5 gives some results due to abbes and ullmo concerning the discriminants of certain hecke algebras g partially supported by the institut universitaire de france

hidden symmetries associated with the projective group of nonlinear first order ordinary differential equations hidden symmetries those not found by the classical lie group method for point symmetries are reported for nonlinear first order ordinary differential equations odes which arise frequently in physical problems these are for the special class of the eight nonabelian two parameter subgroups of the eight parameter projective group the first order odes can be transformed by non local transformations to new separable first order odes which then can be reduced to quadratures the first order odes include riccati equations and equations which in particular cases are of the form of abel s equation the procedure demonstrates the feasibility of integrating nonlinear odes that do not show any apparent lie group point symmetry applications to the vlasov characteristic equation and the reaction diffusion equation are given

the symmetries of the fine gradings of sl nk c associated with direct product of pauli groups a grading of a lie algebra is called fine if it could not be further refined for a fine grading of a simple lie algebra we define its weyl group to describe the symmetry of this grading it is already known that the weyl group of the fine grading of sl n c induced by the action of the group n of the generalized pauli matrices of rank n is sl 2 zn where zn is the cyclic group of order n in this paper we consider the fine grading of sl nk c induced by the action of the group of k fold tensor product of the generalized pauli matrices of rank n we prove that its weyl group is sp 2k zn and is generated by transvections therefore this generalizes the previous result

deformations of surfaces associated with integrable gauss mainardi codazzi equations using the formulation of the immersion of a two dimensional surface into the three dimensional euclidean space proposed recently a mapping from each symmetry of integrable equations to surfaces in r3 can be established we show that among these surfaces the sphere plays a unique role indeed under the rigid su 2 rotations all integrable equations are mapped to a sphere furthermore we prove that all compact surfaces generated by the infinitely many generalized symmetries of the sine gordon equation are homeomorphic to a sphere we also find some new weingarten surfaces arising from the deformations of the modified kurteweg de vries and of the nonlinear schrodinger equations surfaces can also be associated with the motion of curves we study curve motions on a sphere and we identify a new integrable equation characterizing such a motion for a particular choice of the curve velocity

the lund regge surface and its motion s evolution equation the system of evolution equations for general motion of surfaces in orthogonal coordinates is analyzed to reduce the number of variables as well as equations the explicit expression of the lund regge surface is obtained when the surface corresponds to the lund regge equation we prove that some components of velocity satisfy the linearizations of the lund regge equation the soliton solution is derived and one special case of the lund regge surface is studied

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

surfaces on lie groups on lie algebras and their integrability it is shown that the problem of the immersion of a 2 dimensional surface into a 3 dimensional euclidean space as well as then dimensional generalization of this problem is related to the problem of studying surfaces in lie groups and surfaces in lie algebras a particular case of the general formalism presented here implies that any surface can be characterized in terms of 2 2 matrices using an arbitrary parametrization it is also shown that this generality of parametrization is useful for studying integrable surfaces i e surfaces described by integrable equations in particular starting from a suitable lax pair i e a suitable integrable equation it is possible to construct explicitly large classes of integrable surfaces

the motion of surfaces in geodesic coordinates and 2 1 dimensional breaking soliton equation the system of evolution equations for general motion of surfaces in geodesic coordinates is analyzed to reduce the number of variables as well as equations a special choice for some variants deduces that the surface and the motion of surface correspond to a nonlinear schrodinger equation and a 2 1 dimensional breaking soliton equation respectively we also study some geometric properties corresponding to the integrals of the motion of surface and the motion of the curve on the surface the spectral parameter is introduced

on the higher order deformed heisenberg spin equation as an exactly solvable dynamical system it is pointed out that the contents of the recent article in this journal by de gang zhang and jie liu 1988 regarding the geometrical equivalence of a deformed continuous spin equation to the hirota equation are merely a special case of a more generalised set up already published in the literature some time

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

on the singularity analysis of ordinary differential equations invariant under time translation and rescaling the pt is applied to the general second and third order ordinary differential equations invariant under the two symmetries associated with time translation and rescaling in order to investigate their solvability and global integrability the effect of the two symmetries on the compatibility conditions is determined and we show that generally these conditions are automatically a consequence of the resonance condition use is made of truncated laurent series both in ascending and descending powers as an example the case of the generalized chazy equation is presented

geometrical and gauge equivalence of the generalized hirota heisenberg and wkis equations with linear inhomogeneities integrable evolution equations can take several equivalent forms in a geometrical sense here we consider the equivalence of generalized versions involving linear inhomogeneities of three important nonlinear evolution equations namely the hirota heisenberg ferromagnetic spin and wadati konno ichikawa shimizu wkis equation through a moving helical space curve formalism and stereographic representation from the geometrical consideration we also construct suitable 2 2 matrix linear eigenvalue equations involving however non isospectral flow the eigenvalues evolve in time however these systems are also gauge equivalent we briefly analyse the scattering problem and show that infinite number of constants of motion can exist for these systems

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve