long time behavior of the solution to the mkdv equation with step like initial data we consider the modified korteweg de vries equation on the line the initial data is the pure step function i e q x 0 cr for x 0 and q x 0 cl for x cr 0 are arbitrary real numbers long time behavior of the solution to the mkdv equation in the case cl cr 0 and t was studied recently in kotlyarov and minakov 2010 j math phys 51 093506 where the structure of the compression wave was obtained in the form of a modulated elliptic wave the goal of this paper is to study the asymptotic behavior of the solution of the initial value problem as t i e we study the long time dynamics of the rarefaction wave using the steepest descent method and the so called g function mechanism we deform the original oscillatory matrix riemann hilbert problem to the explicitly solvable model forms and show that the solution of the initial value problem has a different asymptotic behavior in different regions of the xt plane in the regions x 6c2rt the main term of asymptotics of the solution is equal to cl and cr respectively in the region the asymptotics of the solution tends to an influence of the dispersion is also studied the second term of the asymptotics is obtained in the region x 6c2rt where the background constant cr is perturbed by the self similar vanishing wave

riemann hilbert problems and the mkdv equation with step initial data short time behavior of solutions and the nonlinear gibbs type phenomenon we produce a family of matrix riemann hilbert rh problems parametrized by a given scalar function r k the jump matrix of the rh problem depends on r k and external real parameters x and t we prove the unique solvability of such a problem a solution m x t k of this matrix rh problem gives a smooth solution q x t of the modified korteweg de vries mkdv equation qt 6q2qx qxxx 0 for a special choice of the function r k we show that the corresponding initial data q x 0 is a heaviside type function we consider the small time limit of the solution q x t the leading order term is explicitly calculated as an integral of the airy function the first summand is an exact solution to the linear kdv equation pt pxxx 0 with initial data thus this specific solution q x t to the mkdv equation being smooth with respect to and t 0 takes pointwise its limit values and exhibits a gibbs type phenomenon with rapid oscillations near x 0 as t 0

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

periodic boundary data for an integrable model of stimulated raman scattering long time asymptotic behavior we study the long time asymptotic behavior of the solution to the initial boundary value ibv problem in the quarter plane x 0 t 0 for nonlinear integrable equations of stimulated raman scattering we consider the case of zero initial condition and periodic boundary data p e i t using the steepest descent method for oscillatory matrix riemann hilbert problems we show that the solution of the ibv problem has different asymptotic behavior in different regions the solution takes the form of a plane wave of finite amplitude when 0 2 t for the ibv problem with nonzero initial condition and the same periodic boundary data the solution to this problem is qualitatively similar to that of this study with the only difference that the solitons of finite amplitude can appear in the region x 2 t

modulated elliptic wave and asymptotic solitons in a shock problem to the modified korteweg de vries equation we study the long time asymptotic behavior of the cauchy problem for the modified korteweg de vries equation with an initial function of the step type this function rapidly tends to zero as and to some positive constant c as in 1989 khruslov and kotlyarov have found khruslov and kotlyarov 1989 inverse problems 5 1075 88 that for a large time the solution breaks up into a train of asymptotic solitons located in the domain is a constant the number n of these solitons grows unboundedly as in 2010 kotlyarov and minakov have studied temporary asymptotics of the solution of the cauchy problem on the whole line kotlyarov and minakov 2010 j math phys 51 093506 and have found that in the domain this solution is described by a modulated elliptic wave we consider here the modulated elliptic wave in the domain our main result shows that the modulated elliptic wave also breaks up into solitons which are similar to the asymptotic solitons in khruslov and kotlyarov 1989 inverse problems 5 1075 88 but differ from them in phase it means that the modulated elliptic wave does not represent the asymptotics of the solution in the domain the correct asymptotic behavior of the solution is given by the train of asymptotic solitons given in khruslov and kotlyarov 1989 inverse problems 5 1075 88 however in the asymptotic regime as in the region we can watch precisely a pair of solitons with numbers n one of them is the asymptotic soliton while the other soliton is generated from the elliptic wave their phases become closer to each other for a large n i e these solitons are also close to each other this result gives the answer on a very important question about matching of the asymptotic formulas in the mentioned region where the both formulas are well defined thus we have here a new and previously unknown mechanism 5 35 of matching of the asymptotics of the solution in the adjacent regions

finite dimensional irreducible representations of the quantum algebra uq c2 in this article the communtation relations of the generators of quantum algebra uq c2 are analyzed and irreducible q tensor operators of rank 1 2 of quantum algebra suq 2 are constructed by means of the property of a q tensor operator finite irreducible representations of uq c2 can be easily obtained some low dimensional representations are tabulated

comment on the q analogues of the harmonic oscillator the heisenberg relation for the q analogues of the quantum harmonic oscillator introduced independently by macfarlane and biedenharn 1989 for the quantum group su 2 q is derived by a method analogous to that used by schwinger 1968 for the su 2 case the author also speculates on the possible use of the quantum group as a generalisation of quantum mechanics

simpson s rule to approximate hilbert integral and its application abstract in this paper the computation of hilbert singular integral with generalized composite simpson s rule for is discussed when singular points coincide with some a priori known point the convergence rate of simpson s rule higher than global one we obtain the pointwise superconvergence phenomenon which means the especial function equal zero the superconvergence points are got then choosing the superconvergence point as the collocation points we get a collocation scheme for solving the relevant hilbert integral equation at last some numerical examples are presented to validate the theoretical analysis

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

the superconvergence of the newton cotes rule for cauchy principal value integrals we consider the general composite newton cotes method for the computation of cauchy principal value integrals and focus on its pointwise superconvergence phenomenon which means that the rate of convergence of the newton cotes quadrature rule is higher than what is globally possible when the singular point coincides with some a priori known point the necessary and sufficient conditions satisfied by the superconvergence point are given moreover the superconvergence estimate is obtained and the properties of the superconvergence points are investigated finally some numerical examples are provided to validate the theoretical results

irreducible highest weight representations of quantum groupsu q gl n c explicit recurrence formulas of canonical realization boson representation for quantum enveloping algebrasu q gl n c are given using them irreducible highest weight representations ofu q gl n c are obtained as restriction of representation of fock space to invariant subspace generated by vacuum as a cyclic vector

the multicomponent q coherent states of the quantum algebra slq 3 the multicomponent q coherent states associated with the quantum algebra slq 3 are presented the expansions are discussed for arbitrary operators in terms of the multicomponent q coherent states of slq 3 an inhomogeneous differential realization of slq 3 in this multicomponent q coherent state space is obtained

tensor like technique and quantum algebra suq 3 a new method is presented for constructing the irreducible representation and for calculating the clebsch gordan coefficient of the quantum algebra suq 3 the method is different from others and does not depend on any concrete realization of suq 3 this means that the result holds in general a set of recurrence formula has been obtained for calculating the clebsch gordan coefficient in fact the recurrence formula is aimed at the so called scalar factor of suq 3 and means that the racah factorization lemma also holds for the quantum algebra suq 3

a comparison of some quadrature methods for approximating cauchy principal value integrals cauchy principal value integrals are evaluated by the imt quadrature scheme which like the tanh quadrature scheme is essentially a trapezoidal scheme after making a transformation of the variable of integration numerical results for some test problems demonstrate that the imt scheme is superior to the tanh scheme while both these methods are comparable to or even better than the standard methods like the gaussian or the chebyshev schemes in terms of accuracy and simplicity

reduced coefficients of quantum algebra suq 4 irreducible representation and a recurrent formula to calculate reduced coefficients of quantum algebra suq 4 suq 2 suq 2 has been obtained through a tensorlike method

asymptotics of step like solutions for the camassa holm equation abstract we study the long time asymptotics of solution of the cauchy problem for the camassa holm equation with a step like initial datum by using the nonlinear steepest descent method and the so called g function approach we show that the camassa holm equation exhibits a rich structure of sharply separated regions in the x t half plane with qualitatively different asymptotics which can be described in terms of a modulated finite gap elliptic function and a finite number of solitons

the construction of the q analogues of the harmonic oscillator operators from ordinary oscillator operators the q analogues of boson operators are constructed from ordinary boson operators by an embedding approach accordingly the q deformed quantum algebras su n q can be obtained straightforwardly from corresponding universal enveloping algebras of lie algebra su n

a new riemann hilbert problem in a model of stimulated raman scattering the riemann hilbert problem proposed in 2 for the integrable stimulated raman scattering srs model was shown to be solvable under an additional condition the boundary data have to be chosen in such a way that a corresponding spectral problem has no spectral singularities in the general case it can be shown that a spectral singularity occurs at k 0 on the other hand the initial boundary value ibv problem for the srs equations is known to be well posed using pde techniques this has been established in 3 therefore it seems natural to try to find a new rh problem that is solvable in the presence of arbitrary spectral singularities the formulation of such a rh problem is the main aim of the paper then the solution of the nonlinear initial boundary value problem for the srs equations is expressed in terms of the solution of a linear problem which is the riemann hilbert problem for a sectionally analytic matrix function

an error analysis of the cauchy principal value integral evaluation by the imt scheme we provide an error analysis for the cauchy principal value integral evaluation by the imt method for analytic functions we get the discretization error e d in terms of a contour integral and we derive an asymptotic estimate for this error integral which shows that e d o e n 2 n 3 4 we give two numerical examples where the asymptotically estimated errors are compared with known actual values

a new method of determining an irreducible representation of quantum slq 3 algebra i the explicit forms of the irreducible representation matrices for the quantum slq 3 enveloping algebra are computed by a new technique

long time asymptotics of the nonlinear schrodinger equation shock problem the long time asymptotics of two colliding plane waves governed by the focusing nonlinear schrodinger equation are analyzed via the inverse scattering method we find three asymptotic regions in space time a region with the original wave modified by a phase perturbation a residual region with a one phase wave and an intermediate transition region with a modulated two phase wave the leading order terms for the three regions are computed with error estimates using the steepest descent method for riemann hilbert problems the nondecaying initial data requires a new adaptation of this method a new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions also the effect of the collision is felt in the plane wave state well beyond the shock front at large times 2007 wiley periodicals inc

exponentially small asymptotics of solutions to the defocusing nonlinear schrodinger equation ii the riemann hilbert problem approach is used to derive a special set of leading order exponentially small asymptotics as t such that x t 0 of solutions to the cauchy problem for the defocusing nonlinear schrodinger equation i t u x2 u 2 u 2 1 u 0 with finite density initial data u x 0 x exp i 11 1 o 1 0 2

generation of asymptotic solitons of the nonlinear schrodinger equation by boundary data this article is about the focusing nonlinear schrodinger equation on the half line the initial function vanishes at infinity while boundary data are local perturbations of periodic or quasi periodic finite gap functions we study the corresponding scattering problem for the zakharov shabat compatible differential equations the representation of the solution of the nonlinear schrodinger equation in the quarter of the x t plane through functions which satisfy marchenko integral equations we use this formalism to investigate the asymptotic behavior of the solution for large time we prove that under certain conditions a periodic quasi periodic behavior at infinity of boundary data generates an unbounded train of asymptotic solitons running away from the boundary the asymptotics of the solution shows that boundary data with periodic behavior as time tends to infinity generates a train of such asymptotic solitons even in the case when the initial function is identically zero

on a certain quadrature formula traduction anglaise d un article en japonais publie en aout 1970 dans la serie kokyuroku vol 91 pp 82 118