On the Lax Pairs of the Symmetric Painlevé Equations

The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N + 1 variables that admit the action of an extended affine Weyl group of type     , as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of ( N + 1) × ( N + 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type     on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group.     

组合KdV方程带修正函数的格子Boltzmann模型

采用一种带修正函数的新格子Boltzmann模型模拟了组合Kdv方程,分析了由此得出的迭代格式的单调性和稳定性,并且得到了格式的单调性条件.在文中的单调性条件下,迭代格式是L∞稳定的.文中的数值模拟结果表明该格式是可行的.     

The Discrete KP and KdV Equations over Finite Fields

We propose an algebro-geometric method for constructing solutions of the discrete KP equation over a finite field. We also perform the corresponding reduction to the finite-field version of the discrete KdV equation. We write formulas that allow constructing multisoliton solutions of the equations starting from vacuum wave functions on an arbitrary nonsingular curve.     

Paley-Wiener spaces with vanishing conditions and Painlevé VI transcendents

We modify the classical Paley-Wiener spaces PWx of entire functions of finite exponential type at most x>0, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points z₁,…,zn. We compute the reproducing kernels and relate their variations with respect to x to a Krein differential system, whose coefficient (which we call the μ-function) and solutions have determinantal expressions. Arguments specific to the case where the “trivial zeros” z₁,…,zn are in arithmetic progression on the imaginary axis allow us to establish for expressions arising in the theory a system of two non-linear first order differential equations. A computation, having this non-linear system at his start, obtains quasi-algebraic and among them rational Painlevé transcendents of the sixth kind as certain quotients of such μ-functions.     

Justification of Asymptotic Formulas for the Fourth Painlevé Equation

A special case of the fourth Painlevé equation is studied. The existence theorem is proved, and asymptotic formulas for the two parametric family of solutions near negative infinity are obtained.     

与KdV及BO方程相关的一类方程之全局解<英>

本文中,对下述方程 u_t (u~h/h)_x _x(—_x~2)~qu=0的柯西问题解建立了全局存在及唯一性,其中h≥2为自然数并且q≥1/2为实数,另外,讨论了当|t|→ ∞时问题解的渐近性。     

Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation

In this article our concern is with the third Painlevé equation
d² y /d x ²= (1/ y )(d y /d x )²− (1/ x )(d y /d x ) + ( αy ²+ β )/ x + γy ³+ δ / y
where α, β, γ, and δ are arbitrary constants. It is well known that this equation admits a variety of types of solution and here we classify and characterize many of these. Depending on the values of the parameters the third Painlevé equation can admit solutions that may be either expressed as the ratio of two polynomials in either x or x ^1/3 or related to certain Bessel functions. It is thought that all exact solutions of (1) can be categorized into one or other of these hierarchies. We show how, given a few initial solutions, it is possible to use the underlying structures of these hierarchies to obtain many other solutions. In addition, we show how this knowledge concerning the continuous third Painlevé equation (1) can be adapted and used to derive exact solutions of a suitable discretized counterpart of (1). Both the continuous and discrete solutions we find are of potential importance as it is known that the third Painlevé equation has a large number of physically significant applications.     

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

In this paper two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painlevé hierarchy are studied. The structure of the roots of these polynomials is shown to be highly regular in the complex plane. Further representations are given of the associated special polynomials in terms of Schur functions. The properties of these polynomials are compared and contrasted with the special polynomials associated with rational solutions of the fourth Painlevé equation.     

On an Inverse Problem for the KdV Equation with Variable Coefficient

Mathematical Notes -     

Semibounded Nonlinear Evolution Equations with Application to the KdV Equations

Existence of solutions for semibounded nonlinear evolution equations is established. This gives more accurate estimate of solutions and conditions of existence axe more easily validated. Our results are successfully applied to prove existence and uniqueness of solutions for some KdV type equations.     

Integrability of a coupled KdV system: Painlevé property, Lax pair and Bäcklund transformation

The integrability of a coupled KdV system is studied by prolongation technique and singularity analysis. As a result, Bäcklund transformation and linear spectral problem associated with this system are obtained. Some special solutions of the system are also proposed.     

On One- and Two-Periodic Wave Solutions of the Ninth-Order KdV Equation

Mathematical Notes - In the text of our article, the following correction must be made: the name of the first author should be changed from S. A. Jumabaev to S. A. Jumabayev.     

On the Kadomtsev-Petviashvili Equation and Associated Constraints

The initial-boundary-value problem for the Kadomtsev-Petviashvili equation in infinite space is considered. When formulated as an evolution equation, found that a symmetric integral is the appropriate choice in the nonlocal term; namely, . If one simply chooses , then an infinite number of constraints on the initial data in physical space are required, the first being . The conserved quantities are calculated, and it is shown that they must be suitably regularized from those that have been used when the constraints are imposed.     

Linear Volterra Integral Equations as the Limit of Discrete Systems

We consider the multidimensional abstract linear integral equation of Volterra type x（t） （*）∫Rtα（s)x(s)ds=f(t),t∈R,（1）as the limit of discrete Stieltjes-type systems and we prove results on the existence of continuous solutions. The functions x, α and f are Bauach space-valued defined on a compact interval R of R^n Rt is a subinterval of R depending on t ∈ R and (*) f denotes either the Bochner-Lebesgue integral or the Henstock integral. The results presented here generalize those in [1] and are in the spirit of [3]. As a consequence of our approach, it is possible to study the properties of (1) by transferring the properties of the discrete systems, The Henstock integral setting enables us to consider highly oscillating functions.     

Painlevé Classification of Binomial Type Ordinary Differential Equations of the Arbitrary Order

The complete Painlevé classification of the binomial ordinary differential equations of the arbitrary order n ≥ 4 is built. Six classes of equations with Painlevé property are obtained. All of these equations are solved in terms of elementary functions and known Painlevé transcendents.     

一个离散晶格方程的N波Darboux变换和无穷守恒律

根据已知离散晶格方程的Lax对,构建了该方程的Ⅳ波Darboux变换和无穷守恒律,通过应用Darboux变换,得到离散晶格方程的范德蒙行列式形式的精确解,通过画图给出了该方程一类特殊的单孤子结构.     

关于离散HJB方程的下解与上解

本文讨论HJB方程上、下解的几个性质,并讨论了从上解出发的一个迭代法的收敛性.