some remarks on a wkb method for the nonselfadjoint zakharov shabat eigenvalue problem with analytic potentials and fast phase abstract a formal method for approximating eigenvalues of the nonselfadjoint zakharov shabat eigenvalue problem in the semiclassical scaling is described analyticity of the potential is assumed and appears to be crucial the method involves finding appropriate paths between pairs of complex turning points and reproduces the y shaped spectra observed by bronski physica d 97 1996 376 an application to all optical ultrashort pulse generation is briefly described and the kind of tools that are required to make the results rigorous are indicated

conjugacy problem for braid groups and garside groups abstract we present a new algorithm to solve the conjugacy problem in artin braid groups which is faster than the one presented by birman ko and lee adv math 139 1998 322 353 this algorithm can be applied not only to braid groups but to all garside groups which include finite type artin groups and torus knot groups among others

on orthogonal polynomials with respect to varying measures on the unit circle let io d i be a system of orthonormal polynomials on the unit circle with respect to d l and yi m dli be a system of orthonormal polynomials on the unit circle with respect to the varying measures d u l iw z 2 z eio where wn z is a sequence of polynomials deg w n whose zeros wn i n lie in izi 1 the asymptotic behavior of the ratio of the two systems on and outside the unit circle is obtained

global solvability of the derivative nonlinear schrodinger equation the derivative nonlinear schrodinger equation dnls iqt qxx q q2 x i q x t i a77xr z jffjt was first derived by plasma physicists 9 10 this equation was used to in terpret the propagation of circular polarized nonlinear alfven waves in plasma kaup and newell obtained the soliton solutions of dnls in 1978 5 the au thor obtained the local solvability of dnls in his dissertation 6 in this paper we obtain global existence in time of schwartz class solutions of dnls if the z 2 norm of the generic initial data q x 0 is bounded

a density problem for orthogonal rational functions abstract let n n 1 be a sequence of points in the open unit disk in the complex plane and let b 0 1 and b n z k 0 n k k k z 1 k z n 1 2 k k 1 when k 0 we put l span b n n 0 1 2 and we consider the following moment problem given a positive definite hermitian inner product on l l find a nondecreasing function on or a positive borel measure on such that f g f e i g e i d for f g l we give a necessary and sufficient condition called n extremality on a solution of the moment problem in order that l is dense in l2

the eigenvalue problem for the focusing nonlinear schrodinger equations new solvable cases abstract in this paper we study the semi classical limit of the zakharov shabat eigenvalue problem for the focusing of nls with some specific initial data in all these cases the eigenvalue problem is reduced to connection problems for the hypergeometric equation and for other classical equations the special initial data suppl prog theor phys 55 1974 284 is contained in our family of initial data parameterized by a real parameter as a particular case 0 we find that beyond a certain value of the parameter the pure point spectrum becomes empty and all the scattering information is contained in the reflection coefficient

on the semiclassical limit of the general modified nls equation we study the semiclassical limit of the so called general modified nonlinear schrodinger equation for initial data with sobolev regularity before shocks appear in the limit system the strict hyperbolicity and genuine nonlinearity are proved for the dispersion limit of the cubic nonlinear case the limiting transition from the mnls equation to the nls equation is also discussed

quadrature and orthogonal rational functions abstract classical interpolatory or gaussian quadrature formulas are exact on sets of polynomials the szeg quadrature formulas are the analogs for quadrature on the complex unit circle here the formulas are exact on sets of laurent polynomials in this paper we consider generalizations of these ideas where the laurent polynomials are replaced by rational functions that have prescribed poles these quadrature formulas are closely related to certain multipoint rational approximants of cauchy or riesz herglotz transforms of a positive or general complex measure we consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence

on the derivative nonlinear schro dinger equation abstract in this paper we discuss the cauchy problem for the derivative nonlinear schrodinger equation i t 2i x 2 0 0 x x where 0 under an explicit smallness condition of the initial data we prove the unique global existence of solutions to this problem in the usual sobolev spaces in the weighted sobolev spaces and in the schwartz class we describe the smoothing effect in detail furthermore for the data decaying exponentially at infinity we prove that the above equation has unique local solutions which are analytic in the space direction

convergence analysis of the extended krylov subspace method for the lyapunov equation the extended krylov subspace method has recently arisen as a competitive method for solving large scale lyapunov equations using the theoretical framework of orthogonal rational functions in this paper we provide a general a priori error estimate when the known term has rank one special cases such as symmetric coefficient matrix are also treated numerical experiments confirm the proved theoretical assertions

the semiclassical modified nonlinear schrodinger equation i modulation theory and spectral analysis abstract we study an integrable modification of the focusing nonlinear schrodinger equation from the point of view of semiclassical asymptotics in particular i we establish several important consequences of the mixed type limiting quasilinear system including the existence of maps that embed the limiting forms of both the focusing and defocusing nonlinear schrodinger equations into the framework of a single limiting system for the modified equation ii we obtain bounds for the location of the discrete spectrum for the associated spectral problem that are particularly suited to the semiclassical limit and that generalize known results for the spectrum of the nonselfadjoint zakharov shabat spectral problem and iii we present a multiparameter family of initial data for which we solve the associated spectral problem in terms of special functions for all values of the semiclassical scaling parameter we view our results as part of a broader project to analyze the semiclassical limit of the modified nonlinear schrodinger equation via the noncommutative steepest descent procedure of deift and zhou and we also present a selfcontained development of a riemann hilbert problem of inverse scattering that differs from those given in the literature and that is well adapted to semiclassical asymptotics

foreword to the proceedings of the orthoquad 2014 conference orthoquad 2014 was an international symposium on orthogonality quadrature and related topics held on january 20 24 2014 in puerto de la cruz tenerife spain it was held in memory of professor pablo gonzalez vera 1955 2012

semiclassical eigenvalue distribution of the zakharov shabat eigenvalue problem abstract in this paper we consider the semi classical limit of the non self adjoint zakharov shabat eigenvalue problem we conduct a series of careful numerical experiments which provide strong evidence that the number of eigenvalues scales like 1 just as in the self adjoint case and that the eigenvalues appear to approach a limiting curve one general choice of potential functions produces a y shaped spectrum we give an asymptotic argument which predicts a critical value for the phase for which the straight line spectra bifurcates to produce the y shaped spectra this asymptotic prediction agrees quite well with numerical experiments the asymptotic argument also predicts a symmetry breaking in the eigenfunctions which we are able to observe numerically we also show that the number of eigenvalues living away from the real axis for a restricted class of potentials is bounded by c 1 where c is an explicit constant a complete theory of the shape of the eigenvalue curve and a general bound on the number of eigenvalues is still lacking

on semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation we calculate the leading order term of the solution of the focusing nonlinear cubic schrodinger equation nls in the semiclassical limit for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 we utilize the riemann hilbert problem formulation of the inverse scattering problem to obtain the leading order term of the solution error estimates are provided 2004 wiley periodicals inc

macroscopic dynamics in quadratic nonlinear lattices fully nonlinear modulation equations are obtained for plane waves in a discrete system with quadratic nonlinearity in the limit when the modulational scales are long compared to the wavelength and period of the modulated wave the discrete system we study is a model for second harmonic generation in nonlinear optical waveguide arrays and also for exciton waves at the interface between two crystals near fermi resonance the modulation equations predict their own breakdown by changing type from hyperbolic to elliptic modulational stability hyperbolicity of the modulation equations is explicitly shown to be implied by linear stability but not vice versa when the plane wave parameters vary slowly in regions of linear stability the modulation equations are hyperbolic and accurately describe the macroscopic behavior of the system whose microscopic dynamics is locally given by plane waves we show how the existence of riemann invariants allows one to test modulated wave initial data to see whether the modulating wave will avoid all linear instabilities and ultimately resolve into simple disturbances that satisfy the hopf or inviscid burgers equation we apply our general results to several important limiting cases of the microscopic model in question

an indeterminate rational moment problem and caratheodory functions let a n n 1 be a sequence of points in the open unit disk in the complex plane and letb 0 1andb n z email protected email protected email protected a k a email protected email protected z n 1 2 a email protected a k 1 when a k 0 we put l span b n n 0 1 2 and we consider the following moment problem given a positive definite hermitian inner product in l find all positive borel measures n on email protected p such that email protected p pf e i q g e i q email protected q forf email protected l we assume that this moment problem is indeterminate under some additional condition on the a n we will describe a one to one correspondence between the collection of all solutions to this moment problem and the collection of all caratheodory functions augmented by the constant

purely imaginary eigenvalues of zakharov shabat systems zakharov shabat systems with single hump and real but not necessarily symmetric potentials are shown to have purely imaginary eigenvalues only coupled with examples of double hump potentials with nonimaginary eigenvalues this establishes that confinement of zakharov shabat eigenvalues to the imaginary axis is a characteristic of potentials whose energy is concentrated in a single region of the time axis