transformation groups for soliton equations iv a new hierarchy of soliton equations of kp type abstract a new approach to soliton equations based on functions or hirota s dependent variables vertex operators and the clifford algebra of free fermions is applied to study a new hierarchy of kadomtsev petviashvili type equations the bkp hierarchy the infinite dimensional orthogonal group acts on the space of bkp functions the sawada kotera equation is obtained as a reduction of bkp its infinitesimal transformations constitute the euclidean lie algebra a2 2

the adler shiota van moerbeke formula for the bkp hierarchy we study the bkp hierarchy and prove the existence of an adler shiota van moerbeke formula this formula relates the action of the bw1 algebra on tau functions to the action of the additional symmetries on wave functions

on the constrained b type kadomtsev petviashvili hierarchy hirota bilinear equations and virasoro symmetry we derive the bilinear equations of the constrained bkp hierarchy from the calculus of pseudodifferential operators the full hierarchy equations can be expressed in hirota s bilinear form characterized by the functions and besides we also give a modification of the original orlov schulman additional symmetry to preserve the constrained form of the lax operator for this hierarchy the vector fields associated with the modified additional symmetry turn out to satisfy a truncated centerless virasoro algebra

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

nonlinear ordinary differential equations resolvable with respect to an irregular singular point abstract we define a class of systems of nonlinear ordinary differential equations resolvable with respect to an irregular singular point say x it is proved that a resolvable system has a unique formal series solution in some fractional powers of x 1 moreover any system has a formal series solution if and only if it can be reduced to a resolvable one we present a method of constructing a proper solution in a suitable sectorial neighorhood of infinity which has the asymptotics in this region the second painleve equation is considered as an example

exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables abstract we present a numerical study of some billiard tables depending on a perturbative parameter 0 and a hyperbolicity parameter h 0 these tables are ellipses for 0 and circumferences in the limit h 0 elliptic billiard tables are integrable and have four separatrices which break up when 0 we conjecture based on numerical experiments that as h 0 the area of the main lobes of the resulting turnstile which can be interpreted as the difference of the lengths of the symmetric primary homoclinic billiard trajectories behaves like an exponential term e 2 h times an asymptotic series j 0 j h 2 j such that 0 0 this series is gevrey 1 of type 1 2 2 so that its borel transform is convergent on a disk of radius 2 2 in the limit 0 the series j 0 j 0 h 2 j is an analytic function which can be explicitly computed with a discrete melnikov method the asymptotic series j 0 j h 2 j associated to the second exponential term e 2 2 h has the same properties finally we have detected some almost invisible homoclinic bifurcations that take place in an exponentially small region of the parameter space our computations have been performed in multiple precision arithmetic namely with several thousands decimal digits and rely strongly on the expansion of the local invariant curves up to very high orders namely with several hundreds taylor coefficients our programs have been written using the pari system

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable hamiltonian systems abstract in this paper we study a generalization of arnold s original example in which he discussed the existence of a mechanism for instability caused by the splitting of the homoclinic manifolds of the weakly hyperbolic tori that has subsequently been referred to as arnold diffusion in case when the number of degrees of freedom n 3 namely we consider a widely studied model of a pendulum weakly coupled with n 1 rotors with the degeneracies in the unperturbed hamiltonian corresponding to different timescales existing in the problem using an alloy of the iterative and direct methods developed within the last years we give exponentially small upper bounds for the splitting measure of transversality for the case of an even analytic perturbation thus improving the estimate of gallavotti 1994 which he calls quasiflat and generalizing the analogous recent estimate of delshams et al 1996 for the rapidly quasiperiodically forced pendulum to a much larger class of hamiltonian systems in particular the exponentially small upper bound for the transversality measure of the splitting applies when the hamiltonian has extra degeneracies namely when the frequencies on a torus become near resonant in fact we show that in such a case the quantity in question becomes smaller which is the incarnation of the general fact that resonant regions in the action space are more stable in the sense that they have larger nekhoroshev exponent nekhoroshev exponent nevertheless we emphasize that getting uniform estimates for an arbitrary n 3 is very hard unless one makes some additional assumptions on the approximation properties of the frequency vector although recent developments show that the first order of canonical perturbation theory given by melnikov integrals generally cannot be accepted as the leading order answer for the splitting distance for the case of more than two degrees of freedom including the rapidly quasiperiodically forced pendulum problem we suggest an analytic perturbation the majority of whose fourier components are strictly nondashzero for which melnikov integrals can be vindicated as the leading order approximation for the components of the splitting distance in different directions if the frequencies on the invariant tori satisfy certain arithmetic conditions this allows us to bound the splitting distance from below furthermore having such a perturbation for the case of three degrees of freedom we use a simple number theoretical argument to find the asymptotics of the fourier series with exponentially small coefficients involved this enables us to compute the numerous homoclinic orbits for the whiskered tori of asymptotically full measure and by proving the domineering contribution of the first order of perturbation theory for the transversality measure to suggest a leading order answer for this quantity which implies the existence of an infinite number of heteroclinic connections between tori with close diophantine frequencies we elucidate the numerous arithmetic issues that obstruct getting a compact leading order approximation for the splitting size most of which can be overcome in the case of three degrees of freedom as our example shows these obstacles can be also possibly avoided in the same fashion for an arbitrary n 3 if one treats the case when the frequencies of the rotors are near a resonance of multiplicity n 3 or n 2

asymptotics beyond all orders and analytic properties of inverse laplace transforms of solutions a number of modern mathematical and physical problems require the study of delicate asymptotic properties lying beyond the power series asymptotics in this paper we suggest a link between these asymptotic problems and some analytic properties of inverse laplace transforms of the corresponding solutions the main result claims that these inverse transforms are holomorphic in an appropriately cut complex plane a direct consequence of this is the nonexistence of solutions to the class of asymptotics beyond all orders problems such as regular shocks of the kuramoto sivashinsky equaiton gr needle crystal solutions of the simple geometrical model of crystal growth ks solitary wave solutions to a class of the fifth order kortveg de vries equations ko sect 8 gj homoclinic orbits of some singularly perturbed mappings ec hm and others

random matrices virasoro algebras and noncommutative kp department of mathematics brandeis university waltham mass 02254 usa email adler math brandeis edu the support of a national science foundation grant dms 9503246 is gratefully acknowledged department of mathematics kyoto university kyoto 606 japan the hospitality of the university of louvain and brandeis university is gratefully acknowledged department of mathematics universite de louvain 1348 louvain la neuve belgium and brandeis university waltham mass 02254 usa e mail vanmoerbeke geom ucl ac be and math brandeis edu the support of a national science foundation grant dms 9503246 a nato a fnrs and a francqui foundation grant is gratefully acknowledged a corrected version has appeared in the duke mathematical journal 94 379 431 1998

the break up of a heteroclinic connection in a volume preserving mapping abstract we consider a family of three dimensional volume preserving maps depending on a small parameter e as e 0 these maps asymptote to flows which attain a heteroclinic connection we show that for small e the heteroclinic connection breaks up and that the splitting between its components scales with e like e exp e we estimate using the singularities of the e 0 heteroclinic orbit in the complex plane we then estimate using linearization about orbits in the complex plane while these estimates are not proven they are well supported by our numerical calculations the work described here is a special case of the theory derived by amick et al which applies to q dimensional volume preserving mappings

stability of the phase motion in race track microtrons we model the phase oscillations of electrons in race track microtrons by means of an area preserving map with a fixed point at the origin which represents the synchronous trajectory of a reference particle in the beam we study the nonlinear stability of the origin in terms of the synchronous phase the phase of the synchronous particle at the injection we estimate the size and shape of the stability domain around the origin whose main connected component is enclosed by an invariant curve we describe the evolution of the stability domain as the synchronous phase varies we also clarify the role of the stable and unstable invariant curves of some hyperbolic fixed or periodic points

non integrability and continuation of fixed points of 2 n dimensional perturbed twist maps abstract in this paper a simple criterion to prove non integrability of symplectic perturbed twist mappings in 2 n dimensions is developed for sufficiently small perturbations in addition an upper bound for the number of isolating integrals the system can possess is provided a criterion for the analytic continuation of isolated periodic orbits in case of a small nonzero perturbation of twist maps is found the evaluation of their linear stability character by obtaining a simplified expression of the eigenvalues of the jacobian matrix concludes the theoretical part the theory is finally applied to a twist map perturbed by the suris potential and also a four dimensional map with physical interest

a lax representation for the vertex operator and the central extension integrable hierarchies viewed as isospectral deformations of an operatorl may admit symmetries they are time dependent vector fields transversal to and commuting with the hierarchy and forming an algebra in this work the commutation relations for the symmetries are shown to be based on a non commutative lie algebra splitting theorem the symmetries viewed as vector fields onl are expressed in terms of a lax pair

structure of lorentzian tori with a killing vector field all lorentzian tori with a non discrete group of isometries are characterized and explicitly obtained they can lie into three cases a flat b conformally flat but non flat and c geodesically incomplete a detailed study of many of their properties including results on the logical dependence of the three kinds of causal completeness on geodesic connectedness and on prescribed curvature is carried out the incomplete case is specially analyzed and several known examples and results in the literature are generalized from a unified point of view

nonpersistence of breather families for the perturbed sine gordon equation we show that up to one exception and as a consequence of first order perturbation theory only it is impossible that a large portion of the well known family of breather solutions to the sine gordon equation could persist under any nontrivial perturbation of the form u tt u xx sin u varepsilon delta left u right o left varepsilon 2 right where is an analytic function in anarbitrarily small neighbourhood ofu 0 improving known results we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of to be small the single exception is a one dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in e

basic representations of a2l 2 and dl 1 2 and the polynomial solutions to the reduced bkp hierarchies basic representations of a2l 2 and dl 1 2 are studied the weight vectors are represented in terms of schur s q functions the method to get the polynomial solutions to the reduced bkp hierarchies is shown to be equivalent to a certain rule in the maya game

width of stochastic layers in near integrable two dimensional symplectic maps abstract width of stochastic layers around separatrix eight is studied for two dimensional symplectic maps close to integrable we prove that under rather general assumptions the relation w d 1 log holds where w is the width of the stochastic layer d is the width of a lobe domain and is a multiplier at the corresponding hyperbolic point

separatrix splitting for systems with three time scales an exact expression for the determinant of the splitting matrix is derived for three degrees of freedom systems with three time scales it allows us to analyze the asymptotic behaviour needed to amend the large angles theorem proposed in ann inst h poincare b 60 1 1994 the asymptotic validity of mel nikov s integrals is proved for the class of models considered which are polynomial perturbations the technique for exhibiting cancellations is inspired by renormalization theory in quantum electrodynamics and uses an analogue of dyson s equations to prove an infinite family of identities due to symmetries that remind us of ward s identities

a refined formula for the separatrix splitting for the standard map abstract we study the standard map for small values of a parameter e an asymptotic formula is found which describes the size of the splitting of the stable and unstable separatrices of the origin considered as a hyperbolic fixed point the formula contains an exponentially decreasing expression times an asymptotic series on powers of a small parameter the coefficients in the asymptotic expansion are to be found in the process of solving numerically a nonlinear recurrent system of difference equations in the complex domain

on nonlinear boundary value problems ghosts parasites and discretizations ghosts are a new class of approximate solutions to nonlinear boundary value problems bvps they are bounded away from true solutions but solve an initial value problem ivp for the differential equations exactly and approximate boundary and internal matching conditions with exponentially decaying error as a control parameter approaches infinity ghosts are characteristic of singularly perturbed differential equations in which solutions are localized and the ivp depends sensitively on initial data they correspond to shifted localized solutions and they can be observed in imperfect physical experiments and be approximated by convergent discretization schemes the latter property suggests that the concept of the numerically irrelevant solution introduced for discretized second order bvps should be re examined

polynomial tau functions of bkp and dkp hierarchies we construct all polynomial tau functions of the bkp dkp and mdkp hierarches we construct all polynomial tau functions of the bkp dkp and mdkp hierarches

hamiltonian maps in the complex plane abstract the standard map is a nonintegrable discrete time analog of the vertical pendulum detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency the functional dependence of the coordinate q on the canonical angle variable is analtyically continued into the complex plane where natural boundaries are found at constant absolute values of im the boundaries represent the appearance of chaotic motion in the complex plane when the domain of analyticity shrinks to zero the kam invariant curve is destroyed two independent numerical methods with fourier analysis in the angle variable were used one based on a variation annihilation method and the other on a double expansion the results were further checked by direct solution of the complex equations of motion the numerically simpler but intrinsically complex semipendulum and semistandard map are also studied we conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or kam surfaces of general nonintegrable systems with analytic hamiltonians

on the squared eigenfunction symmetry of the toda lattice hierarchy the squared eigenfunction symmetry for the toda lattice hierarchy is explicitly constructed in the form of the kronecker product of the vector eigenfunction and the vector adjoint eigenfunction which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively then after the fay like identities and some important relations about the wave functions are investigated the action of the squared eigenfunction related to the additional symmetry on the tau function is derived which is equivalent to the adler shiota van moerbeke formulas

an analogue of the riemann mapping theorem for lorentz metrics the classical riemann mapping theorem essentially says that a simply connected two dimensional smooth riemannian manifold is conformal to the standard sphere s 2 the euclidean plane e 2 or the unit disk d 2 the analogous problems in the lorentz case are as follows let m be a simply connected two dimensional smooth lorentz manifold note that necessarily m is diffeomorphic to r 2 1 does m admit a conformal embedding in e 1 1 i e r 2 with cartesian coordinates x y and the metric d x d y 2 what are the conformal types of such m s in this paper these questions are partly answered in terms of an ideal boundary 0 m which is a conformal invariant of m a geometric consequence is that among orientable surfaces only r 2 r x s 1 and s 1 x s 1 admit complete lorentz metrics of constant curvature the results of this paper may be regarded as a part of a global study of space times as well as of the second order linear hyperbolic operators in dimension 2

a proof of the exponentially small transversality of the separatrices for the standard map in 1984 v f lazutkin laz84 lst89 obtained an asymptotic formula for the separatrix splitting angle for the standard map the difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter lazutkin s proof was based on two conjectures probably the original form of those conjectures was incorrect but lazutkin s method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices the consequent works laz91 laz92 gls94 and gel96 prepared the base to fill all the gaps of the original proof the present paper contains a complete and self contained proof of a refined version of the original formula formula 1 7 of the present paper in this form the formula was obtained in gls94 the proof is inspired by the ideas of lazutkin s original paper laz84

splitting of separatrices for standard and semistandard mappings abstract we investigate chirikov s standard mapping which is a model for the onset of nonintegrability in nonlinear hamiltonian systems the asymptotic formula when the deviation from an integrable system is small for the angle of intersection of separatrices at a homoclinic point is derived on the basis of approximation by the green percival semistandard mapping in the complex domain computed data on the splitting of separatrices for the semi standard map are presented from which we derive the pre exponential coefficient in the formula for splitting of separatrices in the standard mapping

an asymptotic expression for the splitting of separatrices of the rapidly forced pendulum the measure of the splitting of the separatrices of the rapidly forced pendulum ddot x sin x mu sin frac t varepsilon is considered as a model problem that has been studied by different authors here e are small parameters e 0 but otherwise independent the following formula for the angle between separatrices is established alpha frac pi 2 varepsilon frac mu cosh frac pi 2 varepsilon left 1 o mu varepsilon 2 right

splitting of separatrices and symbolic dynamics for some degenerate standard like maps we consider standard like maps of class c1 with parabolic fixed points and such that the second derivative has a jump discontinuity at the fixed points an asymptotic formula for the splitting of the separatrices is obtained the structure of certain chaotic set in the stochastic layer around the separatrices is described in terms of symbolic dynamics it turns out in particular that swinging and rotating regimes of motion may alternate in time arbitrarily analogous results are stated for the so called separatrix mapping and for the more smooth standard like maps

the ghost symmetry of the bkp hierarchy in this paper we systematically develop the ghost symmetry of the kadomtsev petviashvili sub hierarchy of b type bkp through its actions on the lax operator l the eigenfunctions and the function in this process the spectral representation of the eigenfunctions and a new potential are introduced by using squared eigenfunction potential of the bkp hierarchy moreover the bilinear identity of the constrained bkp hierarchy and adler shiota van moerbeke formula of the bkp hierarchy are rederived compactly by means of the spectral representation and ghost symmetry

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

travelling wave solutions of two nonlinear evolution equations by using the g g expansion method abstract by using the g g expansion method proposed recently we give the exact travelling wave solutions of two different types of nonlinear evolution equations in this paper the travelling wave solutions are expressed by three types of functions the computation for the method appears to be easier and faster by general mathematical software

splitting of a small separatrix loop near the saddle center bifurcation in area preserving maps abstract when the saddle center bifurcation occurs in an analytic family of area preserving maps a parabolic fixed point first appears at the origin and then this point bifurcates creating an elliptic fixed point and a hyperbolic one separatrices of the hyperbolic fixed point form a small loop around the elliptic point in general the separatrices intersect transversely and the splitting is exponentially small with respect to the perturbation parameter we derive an asymptotic formula which describes the splitting and study the properties of the pre exponential factor

from additional symmetries to linearization of virasoro symmetries abstract we construct the additional symmetries and derive the adler shiota van moerbeke formula for the two component bkp hierarchy considered as certain reductions of the two component bkp hierarchy the drinfeld sokolov hierarchies of type d are proved to possess symmetries written as the linear action of a series of virasoro operators on the tau function it results in that the drinfeld sokolov hierarchies of type d coincide with dubrovin and zhang s hierarchies associated to the frobenius manifolds for coxeter groups of type d and that every solution of such a hierarchy together with the string equation is annihilated by certain combinations of the virasoro operators and the time derivations of the hierarchy

study of the double mathematical pendulum ii investigation of exponentially small homoclinic intersections we consider the double mathematical pendulum in the limit when the ratio of pendulum masses is close to zero and the ratio of pendulum lengths is close to infinity we found that the limit system has a hyperbolic periodic trajectory whose invariant manifolds intersect transversally and the intersections are exponentially small in this case we obtain an asymptotic formula of the homoclinic invariant for the limit system

weakly nonlocal solitary waves in a singularly perturbed korteweg de vries equation a fifth order korteweg de vries equation is considered where the fifth order derivative term is multiplied by a small parameter it is known that solitary wave solutions of this model equation are nonlocal in that the central core of the wave is accompanied by copropagating trailing oscillations here using the techniques of exponential asymptotics these solutions are reexamined and it is established that they form a one parameter family characterized by the phase shift of the trailing oscillations explicit asymptotic formula relating the oscillation amplitude to the phase shift are obtained

asymptotic expansion of homoclinic structures in a symplectic mapping the phenomenon of separatrix splitting and complicated homoclinic structures constitute a symptom of chaos as pointed out by poincare more than a century ago taking a time discrete dynamical system with a double well potential as an example this interesting feature is demonstrated analytically on the basis of asymptotic expansions beyond all orders violent undulations of the unstable manifold in the extreme vicinity of hyperbolic fixed points are recovered excellently comparison with results for standard and henon maps is also made

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

quasifinite highest weight modules over the lie algebra of differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw 1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t t m 1 the unitary ones are classified as well similar results are obtained for the sin algebras

continuum limit of lattice approximation schemes boundary layer perturbation theory problems are inherently singular however it is known that discretizing the problem by introducing a lattice may convert such problems into regular perturbation problems the singular nature of boundary layer problems is then relegated to and hidden in the continuum limit the subtle limit in which the lattice spacing tends to zero if the lattice is introduced cavalierly then extrapolating to zero lattice spacing gives a sequence of extrapolants that at first approaches the correct limit and then veers off thereby revealing the asymptotic nature of such problems however discretizing the problem following the procedures described here yields lattice approximations that have a smooth and regular continuum limit these ideas are illustrated by three nonlinear ordinary differential equations the cubic equation that describes instantons an oscillator equation having a quadratic nonlinearity and the blasius equation

dynamics in chaotic zones of area preserving maps close to separatrix and global instability zones abstract the purpose of this paper is to study phenomena in chaotic zones of area preserving maps using simpler models which are easier to analyse theoretically and numerically first of all the study of the dynamics in a neighbourhood of the separatrices of a resonant zone is carried out the well known separatrix map defined on a figure eight when needed is used to determine the location of rotational invariant curves r i c inside and outside the resonance the interest in this part is on a quantitative description of the dynamics in a neighbourhood of the separatrices to produce theoretical estimates of the width of the stochastic zone distance to the r i c existence of tiny islands close to the separatrices in every one of the studied items one has tried to complement the limit analytic study with realistic numerical simulations describing the analogy when possible after this study we focus on the formation of larger domains without r i c e g birkhoff domains to this end we introduce the biseparatrix map model although this is a qualitative model the mechanism of destruction of the last r i c and hence the process of creation of zones without r i c is clarified by means of this simple model several numerical examples illustrate the results obtained and are used as a test of the theoretical quantitative predictions

new exact complex traveling wave solutions for 2 1 dimensional bkp equation by using some exact solutions of the auxiliary ordinary differential equation a direct algebraic method is obtained to construct the exact complex solutions for non linear partial differential equations the method is implemented for the 2 1 dimensional bkp equation new exact complex solutions are obtained

sato s backlund transformations additional symmetries and asvm formula for the discrete kp hierarchy two kinds of symmetries sato s backlund transformations and additional symmetries for the discrete kp dkp hierarchy are introduced and the asvm formula which demonstrates the equivalence of these two kinds of symmetries is obtained in this process the fay identity and the difference fay identity of the dkp hierarchy are introduced and the asvm formula in the form of tau function is calculated

on formal solutions of linear matrix differential difference equations abstract in this paper we construct a formal solution to the matrix differential difference equation dde y t a t y t 1 where a t is a matrix power series in t 1 in many cases solutions to the latter equation and to the matrix differential equation y t a t y t have the same form however these solutions may have different forms when the spectrum of a 0 the leading term of a t contains e 1

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given

the additional symmetries for the btl and ctl hierarchies in this paper we construct the additional symmetries for the toda lattice hierarchies of b type and c type the btl and ctl hierarchies and show their algebraic structures are w b w b and w c w c respectively and also we discuss the generating functions of the additional symmetries