Point vortices and classical orthogonal polynomials

Stationary equilibria of point vortices in the plane and on the cylinder in the presence of a background flow are studied. Vortex systems with an arbitrary choice of circulations are considered. Differential equations satisfied by generating polynomials of vortex configurations are derived. It is shown that these equations can be reduced to a single one. It is found that polynomials that are Wronskians of classical orthogonal polynomials solve the latter equation. As a consequence vortex equilibria at a certain choice of background flows can be described with the help of Wronskians of classical orthogonal polynomials.     

New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations

In this paper we consider a special fifth-order KdV equation with constant coefficients and we obtain traveling wave solutions for it, using the projective Riccati equation method. By mean of a scaling, exact solutions to general Kaup-Kupershmidt (KK) equation are obtained. As a particular case, exact solutions to standard KK equation can be derived. Using the same method, we obtain exact solutions to standard Ito equation. By mean of scaling, new exact solutions to general Ito equation are formally derived.     

Max-min of polynomials and exponential diophantine equations

In this paper we study the maximum-minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|?B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.     

Characteristic functional equations of polynomials and the morera-carleman theorem

Several characteristic functional equations satisfied by classes of polynomials of bounded degree are examined in connection with certain generalizations of the Morera-Carleman Theorem. Certain functional equations which have nonanalytic polynomial solutions are also considered.     

Dual and triple equations and q-orthogonal polynomials

In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also introduce and solve a system of dual series equations when the kernel is the q-Laguerre polynomials. Examples are included.     

Binomial Thue equations,ternary equations and power values of polynomials

We explicitly solve the equation Ax ⁿ   − By ⁿ  = ±1 and, along the way, we obtain new results for a collection of equations Ax ⁿ   − By ⁿ  = z ^m with m ∈ {3, n}, where x, y, z, A, B, and n are unknown nonzero integers such that n ≥ 3, AB = p ^α q ^β with nonnegative integers α and β and with primes 2 ≤ p < q < 30. The proofs depend on a combination of several powerful methods, including the modular approach, recent lower bounds for linear forms in logarithms, somewhat involved local considerations, and computational techniques for solving Thue equations of low degree.     

Dual exponential polynomials and linear differential equations

We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.     

Differential equations for deformed Laguerre polynomials

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painlevé transcendent. The generating function of a certain discontinuous linear statistic of the Laguerre unitary ensemble can similarly be expressed in terms of a solution of the fifth Painlevé equation. The methodology used to derive these results rely on two theories regarding differential equations for orthogonal polynomial systems, one involving isomonodromic deformations and the other ladder operators. We compare the two theories by showing how either can be used to obtain a characterization of a more general Laguerre unitary ensemble average in terms of the Hamiltonian system for Painlevé V.     

耦合Kaup-Kupershmidt方程显式行波解

主要利用tanh函数方法,对耦合Kaup-Kupershmidt方程进行了讨论,通过行波约化及Riccati方程,将两个五阶的非线性演化方程转化为两个包含若干参变量的代数系统,借助于Mathematica软件符号运算功能,最终得到了上述耦合方程的显式行波解,包括类孤子解,三角函数周期解以及有理解.     

Construction of soliton equations using special polynomials

A simple, algorithmic approach is proposed for the construction of the most general family of equations of a given scaling weight, possessing, at least, the same single-soliton solution as a given, lower scaling weight equation. The construction exploits special polynomials–differential polynomials in the solution, u, of an evolution equation, which vanish identically when u is a single-soliton solution. Applying the approach to different types of evolution equations yields new results concerning the most general families of evolution equations in a given scaling weight, which possess solitary wave solutions. The same method can be applied in the identification of families of evolution equations of mixed scaling weight (and, in general, of any structure), which admit single-soliton solutions of a desired form.     

Dual series equations involving Jacobi and Laguerre polynomials

An exact solution for some dual series equations involving Jacobi polynomials is obtained. Results for similar dual series equations involving Laguerre polynomials are also deduced by applying a limit process.     

Point vortex equilibria related to Bessel polynomials

The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way.     

Ill-posedness of Kawahara equation and Kaup-Kupershmidt equation

In this paper we consider the Cauchy problems for the Kawahara equation and the Kaup-Kupershmidt equation. By using the general well-posedness principle introduced by I. Bejenaru and T. Tao (2006) [1], we prove that the Kawahara equation is ill-posed for the initial data in Hs(R) with and the Kaup-Kupershmidt equation is ill-posed for the initial data in Hs(R) with .     

Discrete orthogonal polynomials and difference equations of several variables

The goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions. Under some mild assumptions, we give a complete solution of the problem.     

Complete decomposition of Dickson-type polynomials and related Diophantine equations

We characterize decomposition over C of polynomials defined by the generalized Dickson-type recursive relation (n?1)

     

Integral equations and potential-theoretic type integrals of orthogonal polynomials

Abel's theorem is used to solve the Fredholm integral equations of the first kind with Gauss's hypergeometric kernel. The case that the kernel takes general form is discussed in detail and the solution is given. Here, the works of some authors are considered as special cases. The discussion focuses on the three dimensional contact problems in the theory of elasticity with general kernel.     

Asymptotic solutions of the Navier-Stokes equations describing periodic systems of localized vortices

We construct asymptotic solutions of the Navier-Stokes equations describing the two-phase Taylor-scale structures consisting of periodic systems of localized vortex filaments. Such solutions are related to certain topological invariants of divergence-free vector fields on the two-dimensional cylinder or the torus. The equations describing the evolution of the vortex system are defined on a graph which is the set of trajectories of a divergence-free field.