on the toda and kac van moerbeke systems given a solution of the toda lattice we explicitly construct a solution of the kac van moerbeke system related to each other by a miura type transformation as an illustration of our method we derive the n soliton solutions of the kac van moerbeke lattice

on the toda and kac van moerbeke hierarchies we provide a comprehensive treatment of the single and double commutation method as a tool for constructing soliton solutions of the toda and kac van moerbeke hierarchy on arbitrary background in addition we present a novel construction based on the single commutation method as an illustration we compute the n soliton solution of the toda and kac van moerbeke hierarchy

characterizations of δ volterra lattice a symmetric orthogonal polynomials interpretation abstract in this paper we introduce the volterra lattice which is interpreted in terms of symmetric orthogonal polynomials it is shown that the measure of orthogonality associated with these systems of orthogonal polynomials evolves in t like 1 x 2 1 t x where is a given positive borel measure moreover the volterra lattice is related to the toda lattice from miura or backlund transformations the main ingredients are orthogonal polynomials which satisfy an appell condition with respect to the forward difference operator and the characterization of the point spectrum of a jacobian operator that satisfies a volterra equation lax type theorem we also provide an explicit example of solutions of volterra and toda lattices and connect this example with the results presented in the paper

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

new classes of toda soliton solutions we provide a detailed investigation of limits of n soliton solutions of the toda lattice as n tends to infinity our principal results yield new classes of toda solutions including in particular new kinds of soliton like i e reflectionless solutions as a byproduct we solve an inverse spectral problem for one dimensional jacobi operators and explicitly construct tri diagonal matrices that yield a purely absolutely continuous spectrum in 1 1 and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of mr bs 1 1

explicit construction of solutions of the modified kadomtsev petviashvili equation abstract given a solution of the kadomtsev petviashvili equation we explicitly construct a solution of the modified kadomtsev petviashvili equation related to one another by a generalized miura transformation the construction is modeled after a previous treatment of the modified korteweg de vries case as an illustration of our method we derive the soliton solutions of the modified kadomtsev petviashvili equation

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

constructing solutions of the mkdv equation abstract using commutation methods i e n 1 supersymmetry underlying miura s transformation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided

spectral deformations of jacobi operators we extend recent work concerning isospectral deformations for one dimensional schrodinger operators to the case of jacobi operators we provide a complete spectral characterization of a new method that constructs isospectral deformations of a given jacobi operator hu n a n u n 1 a n 1 u n 1 b n u n our technique is connected to dirichlet data that is the spectrum of the operator h n0 on 2 n0 n0 with a dirichlet boundary condition at n0 the transformation moves a single eigenvalue of h n0 and perhaps flips which side of n0 the eigenvalue lives on the remainder of the spectrum the transformation is realized by a unitary operator

on the backlund transformation for the gel fand dickey equations we study the backlund transformation of the gel fand dickey equations and in particular how the factorization ofnth order differential operators leads to lax type equations for first order operators generalizing work of adler and moser 1 in a similar fashion we study the toda equations

curvatures and inhomogeneities an improved common reflection surface approach abstractmultiparameter stacking is an important tool to obtain a first reliable time image of the subsurface in addition it provides wavefield attributes which form the basis for many important applications the quality of the image and the attribute estimates relies heavily on the accuracy of the traveltime moveout description the commonly used hyperbolic common reflection surface crs operator reduces to the nmo hyperbola in the common midpoint gather its accuracy however depends on the curvature of the reflector under consideration the conventional multifocusing mf operator a time shifted double square root expression leads to good results for high reflector curvatures and moderate inhomogeneities of the overburden we used a new implicit crs formulation that combines the robustness of crs regarding heterogeneities with the high sensitivity to curvature of the mf approach it assumes reflectors to be locally circular and can be applied in an iterative fashion for simple but common acquisi

algebro geometric solution of the 2 1 dimensional burgers equation with a discrete variable the quasiperiodic solution of the 2 1 dimensional burgers equation with a discrete variable is obtained through three steps a decomposition into a symplectic map plus two finite dimensional hamiltonian systems b straightening out of both the discrete and the continuous flows in the jacobian variety c inversion into the original variables inner relation with the modified kadomtsev petviashvili equation is presented the explicit theta function solutions for these two 2 1 integrable models are given

on weyl titchmarsh theory for singular finite difference hamiltonian systems we develop the basic theory of matrix valued weyl titchmarsh m functions and the associated green s matrices for whole line and half line self adjoint hamiltonian finite difference systems with separated boundary conditions

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

matrix toda and volterra lattices abstract we consider matrix toda and volterra lattice equations and their relation with matrix biorthogonal polynomials from that relation we give a method for constructing a new solution of these systems from another given one an illustrative example is presented

toda flows with infinitely many variables abstract the toda flow and related flows extend naturally to operators in hilbert space and the purpose of this paper is to describe these flows and to analyse some of their special properties

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

soliton solutions of the korteweg de vries and the kadomtsev petviashvili equations the wronskian technique the well known soliton solutions of the kadomtsev petviashvili equations are written in terms of determinants of wronskian form by using this compact representation together with the hirota bilinear form of the equations it is demonstrated by elementary algebraic methods that the n soliton solution satisfies the evolution equation and the n and n 1 soliton solutions satisfy the associated backlund transformation the relation of these results to the eigensolutions of the inverse scattering method and to the more usual representation of the n soliton solution is also given

hessenberg varieties slodowy slices and integrable systems this work is intended to contextualize and enhance certain well studied relationships between hessenberg varieties and the toda lattice thereby building on the results of kostant peterson and others one such relationship is the fact that every lagrangian leaf in the toda lattice is compactified by a suitable choice of hessenberg variety it is then natural to imagine the toda lattice as extending to an appropriate union of hessenberg varieties we fix a simply connected complex semisimple linear algebraic group g and restrict our attention to a particular family of hessenberg varieties a family that includes the peterson variety and all toda leaf compactifications the total space of this family x h 0 is shown to be a poisson variety with a completely integrable system defined in terms of mishchenko fomenko polynomials this leads to a natural embedding of completely integrable systems from the toda lattice to x h 0 we also show x h 0 to have an open dense symplectic leaf isomorphic to g z times s text reg where z is the centre of g and s text reg is a regular slodowy slice in the lie algebra of g this allows us to invoke results about integrable systems on g times s text reg as developed by rayan and the second author lastly we witness some implications of our work for the geometry of regular hessenberg varieties

the pfaff lattice on symplectic matrices the pfaff lattice is an integrable system arising from the sr group factorization in an analogous way to how the toda lattice arises from the qr group factorization in our earlier paper kodama and pierce 2007 int math res not arxiv 0705 0510 we studied the pfaff lattice hierarchy for the case where the lax matrix is defined to be a lower hessenberg matrix in this paper we deal with the case of a symplectic lower hessenberg lax matrix this forces the lax matrix to take a 2 2 block tridiagonal shape we then show that the odd members of the pfaff lattice hierarchy are trivial while the even members are equivalent to the indefinite toda lattice hierarchy defined in kodama and ye 1996 physica d 91 321 39 this is analogous to the case of the toda lattice hierarchy in relation to the kac van moerbeke system in the case with the initial matrix having only real or imaginary eigenvalues the fixed points of the even flows are given by 2 2 block diagonal matrices with zero diagonals we also consider a family of skew orthogonal polynomials with a symplectic recursion relation related to the pfaff lattice and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite toda lattice

on critical and subcritical sturm liouville operators we study critical and subcritical sturm liouville operators k 1 ddx p ddx v in l2 a b k dx a b r defined as friedrichs extensions under most general conditions on k p and v in particular we prove the equivalence of the notions of criticality as introduced by simon and murata for an important class of potentials v