the q onsager algebra and the positive part of uq slˆ2 the positive part uq of uq sl 2 has a presentation by two generators x yx y that satisfy the q serre relations the q onsager algebra oqoq has a presentation by two generators a ba b that satisfy the q dolan grady relations we give two results that describe how uq and oqoq are related first we consider the filtration of oqoq whose nth component is spanned by the products of at most n generators we show that the associated graded algebra is isomorphic to uq second we introduce an algebra q q and show how it is related to both uq and oqoq the algebra q q is defined by generators and relations the generators are xi i z4 xi i z4 where z4z4 is the cyclic group of order 4 for i z4i z4 the generators xi xi 1xi xi 1 satisfy a q weyl relation and xi xi 2xi xi 2 satisfy the q serre relations we show that q q is related to uq in the following way let qeven resp qodd denote the subalgebra of q q generated by x0 x2x0 x2 resp x1 x3x1 x3 we show that i there exists an algebra isomorphism uq qeven that sends x x0x x0 and y x2y x2 ii there exists an algebra isomorphism uq qodd that sends x x1x x1 and y x3y x3 iii the multiplication map qeven qodd q u v uvu v uv is an isomorphism of vector spaces we show that q q is related to oqoq in the following way for nonzero scalars a ba b there exists an injective algebra homomorphism oq qoq q that sends a ax0 a 1x1a ax0 a 1x1 and b bx2 b 1x3b bx2 b 1x3

direct minimization of the least squares spectral element functional part i direct solver this paper describes an equivalent but improved least squares formulation for the numerical approximation of the solution of partial differential equations instead of using variational analysis to impose the conditions for minimizing the residual the residuals are minimized directly thus leading to a method we will denote by direct minimization dm dm circumvents setting up the normal equations which consists of matrix matrix multiplications matrix matrix multiplications are expensive may lead to loss of accuracy and destroy the sparsity pattern present in the original system the condition number of the dm formulation is the square root of the condition number which would be obtained if variational analysis was employed an element by element procedure will be presented which allows for parallelization of dm a computational comparison between dm and the conventional least squares formulation based on variational analysis will be presented

linear transformations that are tridiagonal with respect to both eigenbases of a leonard pair let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v let x denote the set of linear transformations x v v such that the matrix representing x with respect to the basis i is tridiagonal and the matrix representing x with respect to the basis ii is tridiagonal we show that x is spanned by i a a aa a a and these elements form a basis for x provided the dimension of v is at least 3

birkhoff coordinates for the focusing nls equation in this paper we construct birkhoff coordinates for the focusing nonlinear schrodinger equation near the zero solution

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

tridiagonal pairs of q racah type the double lowering operator ψ and the quantum algebra uq sl2 let kk denote an algebraically closed field and let v denote a vector space over kk with finite positive dimension we consider an ordered pair of linear transformations a v va v v and a v va v v that satisfy the following four conditions i each of a a a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1a vi vi 1 vi vi 1 for 0 i d0 i d where v 1 0v 1 0 and vd 1 0vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a a such that avi vi 1 vi vi 1 for 0 i 0 i where v 1 0 and v 1 0 iv there does not exist a subspace w of v such that aw waw w a w wa w w w 0w 0 w vw v we call such a pair a tridiagonal pair on v it is known that d d to avoid trivialities assume d 1d 1 we assume that a a a a belongs to a family of tridiagonal pairs said to have q racah type this is the most general type of tridiagonal pair let ui i 0d and ui i 0d denote the first and second split decompositions of v in an earlier paper we introduced the double lowering operator v v v v one feature of is that both ui ui 1 ui ui 1 and ui ui 1 for 0 i d0 i d where u 1 0u 1 0 and u 1 0 define linear transformations k v vk v v and b v vb v v such that k qd 2ii ui 0 k qd 2ii ui 0 and b qd 2ii ui 0 for 0 i d0 i d our results are summarized as follows using k b we obtain two actions of uq sl2 uq sl2 on v for each of these uq sl2 uq sl2 module structures the chevalley generator e acts as a scalar multiple of for each of the uq sl2 uq sl2 module structures we compute the action of the casimir element on v we show that these two actions agree using this fact we express as a rational function of k 1 b 1k 1 b 1 in several ways eliminating from these equations we find that k and b are related by a quadratic equation

the split decomposition of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i iv below i each of a a is diagonalizable ii there exists an ordering v0 v1 vd of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 vd 1 0 iii there exists an ordering v 0 v 1 v of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 v 1 0 iv there is no subspace w of v such that both aw w a w w other than w 0 and w v we call such a pair a tridiagonal pair on v in this note we obtain two results first we show that each of a a is determined up to affine transformation by the vi and v i secondly we characterize the case in which the vi and v i all have dimension one we prove both results using a certain decomposition of v called the split decomposition

sobolev preconditioning for the poisson boltzmann equation abstract this paper presents an overview of least squares steepest descent using sobolev gradients for several prototype differential equations in the linear case the method is viewed as a very effective preconditioning strategy for the basic iterative method which arises from steepest descent in particular it acts to smooth the euclidean gradient results are given for the one dimensional possion boltzmann equation from semiconductor device modeling

leonard pairs in classical mechanics leonard pairs lp are matrices with the property of mutual tri diagonality we introduce and study a classical analogue of lp we show that corresponding classical leonard dynamical variables satisfy non linear relations of the aw type with respect to poisson brackets

homoclinic orbits in the near integrable double discrete sine gordon equation we establish the existence of homoclinic orbits for the near integrable double discrete sine gordon ddsg equation under periodic boundary conditions the hyperbolic structure and homoclinic orbits are constructed through the backlund transformation and lax pair a geometric perturbation method based on mel nikov analysis is used to establish necessary criteria for the persistent of temporally homoclinic orbits for the class of ddsg equations with dissipative perturbations

matrix units associated with the split basis of a leonard pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v it is known that there exists a basis for v with respect to which the matrix representing a is lower bidiagonal and the matrix representing a is upper bidiagonal in this paper we give some formulae involving the matrix units associated with this basis

on the shape of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of v i v i v d i v d i coincide we denote this common dimension by i in this paper we prove that i 0 d i for 0 i d it is already known that 0 1 if k is algebraically closed

the q version of a theorem of bochner askey and wilson 1985 found a family of orthogonal polynomials in the variable s k 1 2 k 1 k that satisfy a q difference equation of the form a k p n s qk p n s k b k p n s k q p n s k theta n p n s k n 0 1 we show here that this property characterizes the askey wilson polynomials the proof is based on an operator identity of independent interest this identity can be adapted to prove other characterization results indeed it was used in grunbaum and haine 1996 to give a new derivation of the result of bochner alluded to in the title of this paper we give the appropriate identity for the case of difference equations leading to the wilson polynomials but pursue the consequences only in the case of q difference equations leading to the askey wilson and big q jacobi polynomials this approach also works in the discrete case and should yield the results in leonard 1982

sharp tridiagonal pairs let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of k linear transformations a v v and a v v that satisfies the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide we say the pair a a is sharp whenever dimv0 1 a conjecture of tatsuro ito and the second author states that if k is algebraically closed then a a is sharp in order to better understand and eventually prove the conjecture in this paper we begin a systematic study of the sharp tridiagonal pairs our results are summarized as follows assuming a a is sharp and using the data a vi i 0d a vi i 0d we define a finite sequence of scalars called the parameter array we display some equations that show the geometric significance of the parameter array we show how the parameter array is affected if is replaced by a vi i 0d a vi i 0d or a vd i i 0d a vi i 0d or a vi i 0d a vd i i 0d we prove that if the isomorphism class of is determined by the parameter array then there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for all u v v

a classification of sharp tridiagonal pairs let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide the pair a a is called sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp in this paper we classify up to isomorphism the sharp tridiagonal pairs as a corollary we classify up to isomorphism the tridiagonal pairs over an algebraically closed field we obtain these classifications by proving the conjecture

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

the structure of quotients of the onsager algebra by closed ideals we study the onsager algebra from the ideal theoretic point of view a complete classification of closed ideals and the structure of quotient algebras are obtained we also discuss the solvable algebra aspect of the onsager algebra through the use of formal lie algebras

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

balanced leonard pairs let k denote a field and let v denote a vector space over k with finite positive dimension by a leonard pair on v we mean an ordered pair of linear transformations a v v and a v v that satisfy the following two conditions i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal let v0 v1 vd respectively v0 v1 vd denote a basis for v that satisfies i respectively ii for 0 i d let ai denote the coefficient of vi when we write avi as a linear combination of v0 v1 vd and let ai denote the coefficient of vi when we write a vi as a linear combination of v0 v1 vd in this paper we show a0 ad if and only if a0 ad moreover we show that for d 1 the following are equivalent i a0 ad and a1 ad 1 ii a0 ad and a1 ad 1 iii ai ad i and ai ad i for 0 i d these give a proof of a conjecture by the second author we say a a is balanced whenever ai ad i and ai ad i for 0 i d we say a a is essentially bipartite respectively essentially dual bipartite whenever ai respectively ai is independent of i for 0 i d observe that if a a is essentially bipartite or dual bipartite then a a is balanced for d 2 we show that if a a is balanced then a a is essentially bipartite or dual bipartite

energy minimization related to the nonlinear schrodinger equation in this the window of the sobolev gradient technique to the problem of minimizing a schrodinger functional associated with a nonlinear schrodinger equation we show that gradients act in a suitably chosen sobolev space sobolev gradients can be used in finite difference and finite element settings in a computationally efficient way to find minimum energy states of schrodinger functionals

a lax pair for the two dimensional euler equation a lax pair for the 2d euler equation is found a lax pair for the 2d euler equation is found

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

energy minimization using sobolev gradients application to phase separation and ordering a common problem in physics and engineering is the calculation of the minima of energy functionals the theory of sobolev gradients provides an efficient method for seeking the critical points of such a functional we apply the method to functionals describing coarse grained ginzburg landau models commonly used in pattern formation and ordering processes

discretizations integrable systems and computation discretizations and associated numerical computation of solutions of certain integrable systems such as the nonlinear schrodinger equation nls and the sine gordon sg equations with periodic boundary values can lead to instabilities chaotic and spurious results the chaos can be due to truncation errors or even roundoff errors and can be traced to the fact that these integrable systems are strongly unstable when the initial values are in the neighbourhood of homoclinic manifolds by using the associated nonlinear spectral transform of the nls equation and tracking the evolution of relevant eigenvalues one can observe and relate crossing of homoclinic manifolds to the temporal chaos in the waveforms when the initial data is even for general initial values even though there is no crossing of the unperturbed homoclinic manifolds the waveforms still exhibit chaotic phenomena which can be related to the evolution of the spectrum this paper reviews the current understanding of this intriguing phenomena and also compares the implementation of certain symplectic integrators and runge kutta algorithms for the nls and sg equations in regions of phase space proximate to the homoclinic manifolds

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine gordon equation abstract in this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two degree of freedom hamiltonian systems our methods are fundamentally different than other global perturbation methods e g standard melnikov theory in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation our methods combine the higher dimensional melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds we apply our methods to a modified model of the forced and damped sine gordon equation developed by bishop et al we give explicit conditions in terms of the system parameters for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle focus type chaotic dynamics follow from a theorem of silnikov this provides a mechanism for chaotic dynamics geometrically similar to that observed by bishop et al namely a random jumping between two spatially dependent states with an intermediate passage through a spatially independent state however in order for this type of silnikov dynamics to exist we require a different and unphysical type of damping compared to that used by bishop et al

a refinement of the split decomposition of a tridiagonal pair let v denote a nonzero finite dimensional vector space over a field k and let a a denote a tridiagonal pair on v of diameter d let v u0 ud denote the split decomposition and let i denote the dimension of ui in this paper at first we show there exists a unique integer h 0 h d 2 such that i 1 i for d h i d we call h the height of the tridiagonal pair for 0 r h we define subspaces ui r r i d r by ui r ri r ur kerrd 2r 1 where r denotes the rasing map we show v is decomposed as a direct sum v r 0h i rd rui r this gives a refinement of the split decomposition define u r i rd rui r and observe v r 0hu r we show lu r u r 1 u r u r 1 for 0 r h where we set u 1 u h 1 0 let f r v u r denote the projection we show the lowering map l is decomposed as l l l 0 l where l r 1hf r 1 lf r l 0 r 0hf r lf r and l r 0h 1f r 1 lf r these maps satisfy l u r u r 1 l 0 u r u r and l u r u r 1 for 0 r h the main results of this paper are the following i for 0 r h 1 and r 2 i d r 1 rl l r holds on ui r for some scalar ii for 1 r h and r i d r 1 rl l r holds on ui r for some scalar iii for 0 r h and r 1 i d r 1 rl 0 l 0 r i holds on ui r for some scalars moreover we give explicit expressions of

towards a classification of the tridiagonal pairs let k denote a field and let v denote a vector space over k with finite positive dimension let end v denote the k algebra consisting of all k linear transformations from v to v we consider a pair a a end v that satisfy i iv below i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v let e0 denote the element of end v such that e0 i v0 0 and e0 vi 0 for 1 i d let d resp d denote the k subalgebra of end v generated by a resp a in this paper we prove that the span of e0 dd de0 equals the span of e0 de0 de0 and that the elements of e0 de0 mutually commute we relate these results to some conjectures of tatsuro ito and the second author that are expected to play a role in the classification of tridiagonal pairs

multi dimensional homoclinic jumping and the discretized nls equation we consider a class of dynamical systems that arise frequently in multi mode truncations and discretizations of partial differential equations including the perturbed nls we develop a general method to detect the existence of multi pulse solutions that are doubly asymptotic to an invariant manifold with two different time scales we use our method together with some recent results of li and mclaughlin to show the existence of several families of multi pulse orbits for the ablowitz ladik discretization of the perturbed nls these orbits include n pulse heteroclinic orbits and n pulse silnikov type orbits for arbitrarily large n

steepest descent using smoothed gradients this paper reviews solving differential equations using the least squares steepest descent method with sobolev gradients the method s superiority over standard steepest descent with a euclidean gradient is explained in terms of stability and the classical courant freiderichs lewy condition for the path of steepest descent the spectra for several of the operators arising from a cononical example are also computed kantorovich s inequality then gives explicit estimates on the rate of convergence for the two processes in this way use of the sobolev gradient is viewed as a very effective preconditioning strategy for the linear part of the differential equation

tridiagonal pairs and the q tetrahedron algebra abstract the q tetrahedron algebra q was recently introduced and has been studied in connection with tridiagonal pairs in this paper we further develop this connection let k denote an algebraically closed field and let q denote a nonzero scalar in k that is not a root of unity let v denote a vector space over k with finite positive dimension and let a a denote a tridiagonal pair on v let i i 0 d resp i i 0 d denote a standard ordering of the eigenvalues of a resp a ito and terwilliger have shown that when i q 2 i d and i q d 2 i 0 i d there exists an irreducible q module structure on v such that the q generators x 01 x 23 act as a a respectively in this paper we examine the case in which there exists a nonzero scalar c in k such that i q 2 i d and i q 2 i d cq d 2 i for 0 i d in this case we associate to a a a polynomial p in one variable and prove the following theorem as our main result theorem the following are equivalent 1 i there exists a q module structure on v such that x 01 acts as a and x 30 cx 23 acts as a where x 01 x 30 x 23 are standard generators for q 2 ii p q 2 d 2 q q 1 2 0 suppose i and ii hold then the q module structure on v is unique and irreducible

onsager s algebra and the dolan grady condition in the non self dual case in this paper it is shown that a pair of operators that satisfy the dolan grady conditions generate an onsager algebra even in the case that they are not self dual briefly considered will be the implications for the spectra of superintegrable spin chains in various sectors

q krawtchouk polynomials as spherical functions on the hecke algebra of type b the hecke algebra for the hyperoctahedral group contains the hecke algebra for the symmetric group as a subalgebra inducing the index representation of the subalgebra gives a hecke algebra module which splits multiplicity free the corresponding zonal spherical functions are calculated in terms of q krawtchouk polynomials using the quantised enveloping algebra for sl 2 c the result covers a number of previously established interpretations of q krawtchouk polynomials on the hyperoctahedral group finite groups of lie type hypergroups and the quantum su 2 group

onsager s algebra and superintegrability the author considers the irreducible representations of the onsager algebra and shows that for a finite system possessing such an algebra leads to an ising like structure in the spectra of associated hamiltonians and transfer matrices the chiral potts model is considered as an example for transfer in the diagonal direction it is known to be superintegrable for transfer in the principal direction a new superintegrable solution manifold is found

high order sobolev preconditioning abstract this paper compares the use of first and second order sobolev gradients to solve differential equations using the method of least squares steepest descent the use of high order sobolev gradients offers a very effective preconditioning strategy for the linear part of a nonlinear differential equation

generic non selfadjoint zakharov shabat operators in this paper we develop tools to study within a family of non selfadjoint operators l varphi depending on a parameter varphi in a real hilbert space those with partially simple spectrum as a case study we consider the zakharov shabat operators l varphi appearing in the lax pair of the focusing nls on the circle in particular the main result implies that the set of potentials varphi of sobolev class h n n ge 0 so that all non real eigenvalues of l varphi are simple is path connected and dense

multi pulse jumping orbits and homoclinic trees in a modal truncation of the damped forced nonlinear schro dinger equation abstract in this paper we prove the existence of multi pulse orbits homoclinic to a slow manifold in a two mode truncation of the damped forced nonlinear schrodinger equation first suggested by bishop et al these orbits are jumping i e the corresponding solutions keep switching in time between neighborhoods of the two charateristic breathers of the integrable limit in the case of no damping we find multi pulse smale horseshoes in the two mode model while in the dissipative case we establish the existence of structurally stable multi pulse heterolinic connections between two unstable equilibria the orbits we construct are not amenable to melnikov type perturbation methods in both the hamiltonian and the dissipative case we find homoclinic trees which describe the repeated bifurcations of multi pulse solutions to illustrate the theoretical predictions we also present visualizations of these complicated structures

the structure of a tridiagonal pair let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of k linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide in this paper we show that the following i iv hold provided that k is algebraically closed i each of v0 v0 vd vd has dimension 1 ii there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for all u v v iii there exists a unique anti automorphism of end v that fixes each of a a iv the pair a a is determined up to isomorphism by the data i i 0d i i 0d i i 0d where i resp i is the eigenvalue of a resp a on vi resp vi and i i 0d is the split sequence of a a corresponding to i i 0d and i i 0d

tridiagonal pairs of krawtchouk type let f denote an algebraically closed field with characteristic 0 and let v denote a vector space over f with finite positive dimension let a a denote a tridiagonal pair on v with diameter d we say that a a has krawtchouk type whenever the sequence d 2i i 0d is a standard ordering of the eigenvalues of a and a standard ordering of the eigenvalues of a assume a a has krawtchouk type we show that there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for u v v we show that the following tridiagonal pairs are isomorphic i a a ii a a iii a a iv a a we give a number of related results and conjectures

a family of tridiagonal pairs and related symmetric functions a family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in detail the corresponding eigenvalue sequences eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described the overlap functions between the two dual bases are shown to satisfy a coupled system of recurrence relations and a set of discrete second order q difference equations which generalize those associated with the askey wilson orthogonal polynomials with a discrete argument normalizing the fundamental solution to unity the hierarchies of solutions are rational functions of one discrete argument explicitly derived in some simplest examples the weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given

the switching element for a leonard pair abstract let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v let v i i 0 d resp w i i 0 d denote a basis for v referred to in i resp ii we show that there exists a unique linear transformation s v v that sends v 0 to a scalar multiple of v d fixes w 0 and sends w i to a scalar multiple of w i for 1 i d we call s the switching element we describe s from many points of view

the geometry of real sine gordon wavetrains the characterization ofreal n phase quasiperiodic solutions of the sine gordon equation has been an open problem in this paper we achieve this result employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature a significant by product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds and an understanding of how the real isospectral manifolds are embedded by placing the problem in this general context these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions

a mean field statistical theory for the nonlinear schrodinger equation abstract a statistical model of self organization in a generic class of one dimensional nonlinear schrodinger nls equations on a bounded interval is developed the main prediction of this model is that the statistically preferred state for such equations consists of a deterministic coherent structure coupled with fine scale random fluctuations or radiation the model is derived from equilibrium statistical mechanics by using a mean field approximation of the conserved hamiltonian and particle number l2 norm squared for finite dimensional spectral truncations of the nls dynamics the continuum limits of these approximated statistical equilibrium ensembles on finite dimensional phase spaces are analyzed holding the energy and particle number at fixed finite values the analysis shows that the coherent structure minimizes total energy for a given value of particle number and hence is a solution to the nls ground state equation and that the remaining energy resides in gaussian fluctuations equipartitioned over wave numbers some results of direct numerical integration of the nls equation are included to validate empirically these properties of the most probable states for the statistical model moreover a theoretical justification of the mean field approximation is given in which the approximate ensembles are shown to concentrate on the associated microcanonical ensemble in the continuum limit

tridiagonal pairs of shape 1 2 1 abstract let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfies the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and that for 0 i d the dimensions of v i v d i v i v d i coincide we denote this common value by i the sequence i i 0 d is called the shape of the pair in this paper we assume the shape is 1 2 1 and obtain the following results we describe six bases for v one diagonalizes a another diagonalizes a and the other four underlie the split decompositions for a a we give the action of a and a on each basis for each ordered pair of bases among the six we give the transition matrix at the end we classify the tridiagonal pairs of shape 1 2 1 in terms of a sequence of scalars called the parameter array

mel nikov analysis of numerically induced chaos in the nonlinear schro dinger equation abstract certain hamiltonian discretizations of the periodic focusing nonlinear schrodinger equation nls have been shown to be responsible for the generation of numerical instabilities and chaos in this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the nls using heuristic mel nikov methods the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system the structure of the persistent homoclinic orbits that are predicted by the mel nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos the mel nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments in a neighborhood of the persistent homoclinic orbits the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study

tridiagonal pairs and the askey wilson relations abstract the notion of a tridiagonal pair was introduced by ito tanabe and terwilliger let v denote a nonzero finite dimensional vector space over a field f a tridiagonal pair on v is a pair a a where a v v and a v v are linear transformations that satisfy some conditions assume a a is a tridiagonal pair on v recently terwilliger and vidunas showed that if a is multiplicity free on v then a a satisfy the following askey wilson relation for some scalars a 2 a aa a a a 2 aa a a a a 2 a i a 2 a a aa aa 2 a a aa a a 2 a i in the present paper we show that if a tridiagonal pair a a satisfy the askey wilson relations then the eigenspaces of a and the eigenspaces of a have one common dimension and moreover if f is algebraically closed then that common dimension is 1

instability zones of a periodic 1d dirac operator and smoothness of its potential let l be the differential operator where p x q x are 1 periodic functions such that the operator l considered on 0 1 with periodic y 0 y 1 or antiperiodic y 0 y 1 boundary conditions is self adjoint and moreover for large n it has close to n a pair of periodic if n is even or antiperiodic if n is odd eigenvalues n n we study the relationship between the decay rate of the instability zone sequence n n n n and the smoothness of the potential function p x

mock tridiagonal systems we introduce the notion of a mock tridiagonal system this is a generalization of a tridiagonal system in which the irreducibility assumption is replaced by a certain nonvanishing condition we show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications this paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism

transition maps between the 24 bases for a leonard pair abstract let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below 1 i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal 2 ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v in an earlier paper we described 24 special bases for v one feature of these bases is that with respect to each of them the matrices that represent a and a are i diagonal and irreducible tridiagonal or ii irreducible tridiagonal and diagonal or iii lower bidiagonal and upper bidiagonal or iv upper bidiagonal and lower bidiagonal for each ordered pair of bases among the 24 there exists a unique linear transformation from v to v that sends the first basis to the second basis we call this the transition map in this paper we find each transition map explicitly as a polynomial in a a

the tetrahedron algebra and its finite dimensional irreducible modules abstract recently terwilliger and the present author found a presentation for the three point sl 2 loop algebra via generators and relations to obtain this presentation we defined a lie algebra by generators and relations and displayed an isomorphism from to the three point sl 2 loop algebra in this paper we classify the finite dimensional irreducible modules

two commuting operators associated with a tridiagonal pair let k denote a field and let v denote a vector space over k with finite positive dimension we consider an ordered pair of linear transformations a v v and a v v that satisfy the following four conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there does not exist a subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d to avoid trivialities assume d 1 we show that there exists a unique linear transformation v v such that i vi v0 v1 vi 1 and vi vi 1 vd v0 v1 vd i for 0 i d we show that there exists a unique linear transformation v v such that vi vi 1 vi vi 1 and vi v0 v1 vi 2 for0 i d where i 0 d 1 and 0 resp d denotes the eigenvalue of a associated with v0 resp vd we characterize in several ways there are two well known decompositions of v called the first and second split decomposition we discuss how act on these decompositions we also show how relate to each other along this line we have two main results our first main result is that commute in the literature on td pairs there is a scalar used to describe the eigenvalues our second main result is that each of 1 is a polynomial of degree d in under a minor assumption on

poisson algebras for some generalized eigenvalue problems we introduce particular poisson algebras designed to describe after quantization generalized eigenvalue problems for two tridiagonal matrices these algebras are naturally implemented upon various elliptic curves

linear covariance algebra for slq 2 the askey wilson algebra aw 3 is shown to serve as a hidden covariance algebra for quantum algebra slq 2 the generators of aw 3 are chosen to be linear combinations of slq 2 generators with operator valued coefficients

homoclinic orbits for a perturbed nonlinear schrodinger equation in recent years there have been extensive studies on the existence of homoclinic orbits for nearly integrable hamiltonian pdes which are closely related to chaos in this work we consider a perturbed nonlinear schrodinger equation for u even and periodic in x the diffusion ieuxx is an unbounded perturbation term when the diffusion is replaced by its bounded fourier truncation li mclaughlin shatah and wiggins 26 proved the existence of homoclinic orbits for the perturbed equation the method was based on invariant manifolds foliations and melnikov analysis the unboundedness of the diffusion prevents the equation from being solved for t 0 for general initial values and destroys some geometric structures however we overcome these difficulties and prove the existence of homoclinic orbits for the diffusively perturbed nls 2000 john wiley sons inc

tridiagonal pairs and the μ conjecture let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide we say the pair a a is sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp a conjectured classification of the sharp tridiagonal pairs was recently introduced by t ito and the second author we present a result which supports the conjecture given scalars i i 0d i i 0d in f that satisfy the known constraints on the eigenvalues of a tridiagonal pair we define an f algebra t by generators and relations we consider the f algebra e0 te0 for a certain idempotent e0 t let f x1 xd denote the polynomial algebra over f involving d mutually commuting indeterminates we display a surjective f algebra homomorphism f x1 xd e0 te0 we conjecture that is an isomorphism we show that this conjecture implies the classification conjecture and that the conjecture holds for d 5

a bispectral q hypergeometric basis for a class of quantum integrable models for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain class of parameters and if the n variables belong to a discrete support in this case the bispectral property finds a natural interpretation within the framework of tridiagonal pairs in the third part eigenfunctions of the q dolan grady hierarchy are considered in the polynomial basis in particular invariant subspaces are identified for certain conditions generalizing nepomechie s relations in the fourth part the analysis is extended to the special case q 1 this framework provides a q hypergeometric formulation of quantum integrable models such as the open xxz spin chain with generic integrable boundary conditions q 1 for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain

persistent homoclinic orbits for a perturbed nonlinear schrodinger equation the persistence of homoclinic orbits for certain perturbations of the integrable nonlinear schrodinger equation under even periodic boundary conditions is established more specifically the existence of a symmetric pair of homoclinic orbits is established for the perturbed nls equation through an argument that combines melnikov analysis with a geometric singular perturbation theory for the pde 1996 john wiley sons inc

gauge theory of ba cklund transformations ii abstract a class of gauge transformations is constructed for hamiltonian hierarchies of completely integrable systems on semi simple lie algebras these transformations create or annihilate poles in the scattering data hence create or annihilate solitons in the potential q in that sense they generalize backlund transformations to which they reduce precisely for the 2 2 akns systems to each backlund gauge transformation is associated an automorphism in the weyl group of the lie algebra two proofs are given to show that these backlund gauge transformations preserve the nonlinear evolution equation one proof uses the transformation of the scattering data under the backlund transformation the second proof is purely gauge theoretic

long time behavior for the focusing nonlinear schroedinger equation with real spectral singularities we consider the effect of real spectral singularities on the long time behavior of the solutions of the focusing nonlinear schroedinger equation we find that for each spectral singularity such an effect is limited to the region of the x t plane in which is close to the point of stationary phase lambda 0 tfrac x 4t the phase here being defined in a standard way by say the evolution of the jost functions in that region the solution performs decaying oscillations of the same form as in the other regions but with different parameters the order of decay is o left left tfrac log t t right 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right we prove our result by using the riemann hilbert factorization formulation of the inverse scattering problem we recover our asymptotics by transforming our problem to one which is equivalent for large time and which can be interpreted as the one corresponding to the genus 0 algebro geometric solution of the equation

multiplicities of the eigenvalues of periodic dirac operators abstract let us consider the dirac operator l ij d dx u j 1 0 0 1 u 0 a cos 2 x a cos 2 x 0 where a 0 is real on i 0 1 with boundary conditions bc per i e f 1 f 0 and bc per i e f 1 f 0 f f 1 f 2 h 1 i then l bc l bc and all per l u are of multiplicity 2 while per l u are simple theorem 15 this is an analogue of ince s statement for mathieu hill operator links between the spectra of dirac and hill operators lead to detailed information about the spectra of hill operators with potentials of the ricatti form v p p 2 section 3 it helps to get analogues of grigis results ann sci ecole norm sup 4 20 1987 641 on the zones of instability of hill operators with polynomial potentials and their asymptotics for the case of dirac operators as well section 4 2

tridiagonal pairs of q racah type abstract let f denote an algebraically closed field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that a w w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d for 0 i d let i resp i denote the eigenvalue of a resp a associated with v i resp v i the pair a a is said to have q racah type whenever i a b q 2 i d c q d 2 i and i a b q 2 i d c q d 2 i for 0 i d where q a b c a b c are scalars in f with q b c b c nonzero and q 2 1 1 this type is the most general one we classify up to isomorphism the tridiagonal pairs over f that have q racah type our proof involves the representation theory of the quantum affine algebra u q sl 2

a characterization of leonard pairs using the notion of a tail abstract let v denote a vector space with finite positive dimension we consider an ordered pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v in this paper we characterize the leonard pairs using the notion of a tail this notion is borrowed from algebraic graph theory

morse and melnikov functions for nls pde s the theory of the focusing nls equation under periodic boundary conditions together with the floquet spectral theory of its associated zakharov shabat linear operator hat l is developed in sufficient detail for later use in studies of perturbations of the nls equation counting lemmas for the non selfadjoint operator hat l are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discd in the complex eigenvalue plane the radius of the discd is controlled by theh 1 norm of the potential overrightarrow q for this integrable nls hamiltonian system unstable tori are identified and backlund transformations are then used to construct global representations of their stable and unstable manifolds whiskered tori for the nls pde the floquet discriminant delta lambda overrightarrow q is used to introduce a natural sequence of nls constants of motion f j overrightarrow q equiv delta lambda lambda j c overrightarrow q overrightarrow q where denotes thej th critical point of the floquet discriminant a taylor series expansion of the constants f j overrightarrow q with explicit representations of the first and second variations is then used to study neighborhoods of the whiskered tori in particular critical tori with hyperbolic structure are identified through the first and second variations of f j overrightarrow q which themselves are expressed in terms of quadratic products of eigenfunctions of hat l the second variation permits identification within the disc d of important bifurcations in the spectral configurations of the operator hat l the constant f j overrightarrow q as the height of the floquet discriminant over the critical point admits a natural interpretation as a morse function for nls isospectral level sets this morse interpretation is studied in some detail it is valid globally for the infinite tail f j overrightarrow q j n which is associated with critical points outside the discd within this disc the interpretation is only valid locally with the same obstruction to its global validity as to a global ordering of the spectrum nevertheless this local morse theory together with the backlund representations of the whiskered tori produces extremely clear pictures of the stratification of nls invariant sets near these whiskered tori pictures which are useful in the study of perturbations of nls finally a natural connection is noted between the constants f j overrightarrow q of the integrable theory and melnikov functions for the theory of perturbations of the nls equation this connection generates a simple but general representations of the melnikov functions

the stationary equations of a coupled nonlinear schrodinger system abstract the integrable coupled nonlinear schrodinger cnls equations under periodic boundary conditions are known to possess linearized instabilities in both the focussing and defocussing cases m g forest d w mclaughlin d muraki o c wright non focussing instabilities in coupled integrable nonlinear schrodinger pdes in preparation d j muraki o c wright d w mclaughlin birefringent optical fibers modulational instability in a near integrable system nonlinear processes in physics proceedings of iii postdam v kiev workshop 1991 pp 242 245 o c wright modulational stability in a defocussing coupled nonlinear schrodinger system physica d 82 1995 1 10 whereas the scalar nls equation is linearly unstable only in the focussing case m g forest j e lee geometry and modulation theory for the periodic schrodinger equation in dafermas et al eds oscillation theory computation and methods of compensated compactness i m a math appl 2 1986 35 70 these instabilities indicate the presence of crossed homoclinic orbits similar to those in the phase plane of the unforced duffing oscillator y li d w mclaughlin morse and melnikov functions for nls pde s commun math phys 162 1994 175 214 d w mclaughlin e a overman whiskered tori for integrable pde s chaotic behaviour in near integrable pde s in keller et al eds surveys in applied mathematics vol 1 chapter 2 plenum press new york 1995 the homoclinic orbits and the near homoclinic tori that are connected to the unstable wave trains of the nls and the cnls reside in the finite dimensional phase space of certain stationary equations s p novikov funct anal prilozen 8 3 1974 54 66 of the infinite hierarchy of integrable commuting flows the correct stationary equations must be matched to the unstable torus through the analytic structure of the spectral curves o c wright near homoclinic orbits of the focussing nonlinear schrodinger equation preprint thus in this paper the stationary equations of the cnls are derived and the analytic structure of the trigonal spectral curve is examined providing a basis for further study of the near homoclinic orbits of the cnls system

leonard pairs and the q racah polynomials abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following two conditions i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal wecall such a pair a leonard pair on v in the appendix to linear algebra appl 330 2001 p 149 we outlined a correspondence between leonard pairs and a class of orthogonal polynomials consisting of the q racah polynomials and some related polynomials of the askey scheme we also outlined how for the polynomials in this class the 3 term recurrence difference equation askey wilson duality and orthogonality can be obtained in a uniform manner from the corresponding leonard pair the purpose of this paper is to provide proofs for the assertions which we made in that appendix

the algebra uq sl2 in disguise abstract we discuss a connection between the algebra u q sl 2 and the tridiagonal pairs of q racah type to describe the connection let x y 1 z denote the equitable generators for u q sl 2 let u q denote the subalgebra of u q sl 2 generated by x y 1 z using a tridiagonal pair of q racah type we construct two finite dimensional u q modules the constructions yield two nonstandard presentations of u q by generators and relations these presentations are investigated in detail

introduction to leonard pairs in this survey paper we give an elementary introduction to the theory of leonard pairs a leonard pair is defined as follows let k denote a field and let v denote a vector space over k with finite positive dimension by a leonard pair on v we mean an ordered pair of linear transformations a v v and b v v that satisfy conditions i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we give several examples of leonard pairs using these we illustrate how leonard pairs arise in representation theory combinatorics and the theory of orthogonal polynomials

some trace formulae involving the split sequences of a leonard pair let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v let diag 0 1 d denote the diagonal matrix referred to in ii above and let diag 0 1 d denote the diagonal matrix referred to in i above it is known that there exists a basis u0 u1 ud for v and there exist scalars 1 2 d in k such that aui iui ui 1 0 i d 1 aud dud a ui iui 1 i ui 1 i d a u0 0 u0 the sequence 1 2 d is called the first split sequence of the leonard pair it is known that there exists a basis v0 v1 vd for v and there exist scalars 1 2 d in k such that avi d ivi vi 1 0 i d 1 avd 0vd a vi ivi 1 i vi 1 i d a v0 0 v0 the sequence 1 2 d is called the second split sequence of the leonard pair we display some attractive formulae for the first and second split sequence that involve the trace function

spectral stability analysis for periodic traveling wave solutions of nls and cgl perturbations abstract we consider cnoidal traveling wave solutions to the focusing nonlinear schrodinger equation nls that have been shown to persist when the nls is perturbed to the complex ginzburg landau equation cgl we show that while these periodic traveling waves are spectrally stable solutions of nls with respect to periodic perturbations they are unstable with respect to bounded perturbations furthermore we use an argument based on the fredholm alternative to find an instability criterion for the persisting solutions to cgl

the determinant of aa a a for a leonard pair a a abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v in this paper we investigate the commutator aa a a our results are as follows abbreviate d dim v 1 and first assume d is odd we show aa a a is invertible and display several attractive formulae for the determinant next assume d is even we show that the null space of aa a a has dimension 1 we display a nonzero vector in this null space we express this vector as a sum of eigenvectors for a and as a sum of eigenvectors for a

numerical inverse spectral transform for the periodic sine gordon equation theta function solutions and their linearized stability abstract the inverse spectral transform for integrable nonlinear ordinary and partial differential equations such as the toda lattice korteweg de vries sine gordon and nonlinear schrodinger equations provides explicit algorithms to generate exact solutions under periodic or quasiperiodic boundary conditions these oscillatory wavetrains may be prescribed a priori to consist of a nonlinear superposition of n phases j x t j x j t 0 j j 1 n where the wave is 2 periodic independently in each phase this paper exhibits the numerical implementation of the inverse spectral solution of the sine gordon equation the general construction is outlined and then implemented for n 1 2 and 3 we compute 1 the exact theta function solutions 2 the floquet spectrum of x periodic solutions 3 the labelling of linearized instabilities of n phase solutions in terms of spectral data and 4 the linearized growth rate in each unstable mode the associated surfaces q n x t are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system