A Riemann-Hilbert Approach to the Harry-Dym Equation on the Line

In this paper, the authors consider the Harry-Dym equation on the line with decaying initial value. They construct the solution of the Harry-Dym equation via the solution of a 2 × 2 matrix Riemann-Hilbert problem in the complex plane. Further, onecusp soliton solution is expressed in terms of the Riemann-Hilbert problem.     

基于Riemann-Hilbert问题建模求解孤立子解

基于矩阵谱问题构造了一种实用的方法来对一类实轴上的可积方程的Riemann-Hilbert问题进行建模。当跳跃矩阵是单位矩阵时,孤立子解通过特殊约化的Riemann-Hilbert问题显性表示。作为一个范例,对于具有任意阶矩阵谱问题的多分量非线性薛定谔方程,给出了该方法的具体应用。     

TD孤子方程族的可积耦合及其哈密顿结构

用拓展谱问题方法构造TD族的可积耦合,并应用二次型恒等式寻求拓展的TD族哈密顿结构.     

Cauchy-Riemann方程的Riemann-Hilbert边值问题

本文考虑多柱域上非齐次的Cauchy-Riemann方程的Riemann-Hilbert边值问题.讨论了上述边值问题可解的充分必要条件,并给出了边值问题解的积分表达式.     

一类Riemann-Hilbert边值逆问题

给出解析函数的一类R iem ann-H ilbert边值逆问题的数学提法,依据解析函数R iem ann-H ilbert边值问题的经典理论,讨论了此边值问题的可解性,给出了该边值问题的可解条件和解的表示式.     

A free boundary problem of diffusion equation with integral condition

We consider a free boundary problem of heat equation with integral condition on the unknown free boundary. Results of solution regularity and problem well-posedness are presented.     

A new approach to factorization of a class of almost-periodic triangular symbols and related Riemann-Hilbert problems

The factorization of almost-periodic triangular symbols, G, associated to finite-interval convolution operators is studied for two classes of operators whose Fourier symbols are almost periodic polynomials with spectrum in the group αZ+βZ+Z (α,β∈]0,1[, α+β>1, α/β∉Q). The factorization problem is solved by a method that is based on the calculation of one solution of the Riemann-Hilbert problem GΦ₊=Φ₋ in L_∞(R) and does not require solving the associated corona problems since a second linearly independent solution is obtained by means of an appropriate transformation on the space of solutions to the Riemann-Hilbert problem. Some unexpected, but interesting, results are obtained concerning the Fourier spectrum of the solutions of GΦ₊=Φ₋. In particular it is shown that a solution exists with Fourier spectrum in the additive group αZ+βZ whether this group contains Z or not. Possible application of the method to more general classes of symbols is considered in the last section of the paper.     

Uniform asymptotics for Jacobi polynomials with varying large negative parameters-- a Riemann-Hilbert approach

An asymptotic expansion is derived for the Jacobi polynomials with varying parameters and , where and are constants. Our expansion is uniformly valid in the upper half-plane . A corresponding expansion is also given for the lower half-plane . Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993). The two asymptotic expansions hold, in particular, in regions containing the curve , which is the support of the equilibrium measure associated with these polynomials. Furthermore, it is shown that the zeros of these polynomials all lie on one side of , and tend to as .

     

可换四元数空间中某些双曲型方程的Riemann-Hilbert边值问题

考察了在可交换四元数空间(基元为1,i,j,k满足条件i~2=j~2=-1,ij=ji= k)中的某些双曲型方程的Riemann-Hilbert边值问题,分别在不同的情况下获得了问题的可解条件和通解.     

Riemann-Hilbert theory for problems with vanishing coefficients that arise in nonlinear hydrodynamics

Motivated by a question from mathematical hydrodynamics, this paper studies the solution set of Riemann-Hilbert problems on the unit disc D in of the form

     

Special Monodromy Groups and the Riemann-Hilbert Problem for the Riemann Equation

In this paper, we solve the Riemann-Hilbert problem for the Riemann equation and for the hypergeometric equation. We describe all possible representations of the monodromy of the Riemann equation. We show that if the monodromy of the Riemann equation belongs to SL(2, ℂ), then it can be realized not only by the Riemann equation, but also by the more special Riemann-Sturm-Liouville equation. For the hypergeometric equation, we construct a criterion for its monodromy group to belong to SL(2, ℤ).__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 753–767.Original Russian Text Copyright ©2005 by V. A. Poberezhnyi.     

三类多双曲复方程组的Riemann-Hilbert边值问题

用函数论的方法研究了三类超定双曲型方程组的解,即三类多双曲复方程组在一般柱形域上的Riemann-Hilbert边值问题的提法,可解条件,解的表示,唯一性和存在性.     

Determination of the source parameter in a heat equation with a non-local boundary condition

In this article we consider the inverse problem of identifying a time dependent unknown coefficient in a parabolic problem subject to initial and non-local boundary conditions along with an overspecified condition defined at a specific point in the spatial domain. Due to the non-local boundary condition, the system of linear equations resulting from the backward Euler approximation have a coefficient matrix that is a quasi-tridiagonal matrix. We consider an efficient method for solving the linear system and the predictor–corrector method for calculating the solution and updating the estimate of the unknown coefficient. Two model problems are solved to demonstrate the performance of the methods.     

A numerical approach for a semilinear parabolic equation with a nonlocal boundary condition

A semilinear reaction-diffusion problem with a nonlocal boundary condition is studied. This paper presents a new and very easy implementable numerical algorithm for computations. This is based on a suitable linearization in time and on the principle of linear superposition. Any method for the space discretization (FEM was taken in this analysis) can be chosen. The derived algorithm is implicit and it does not need any iteration scheme to get a solution with the nonlocal boundary condition. Stability analysis has been performed and the optimal error estimates have been derived. Numerical results have been compared with other known techniques.     

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.     

Matrix Riemann-Hilbert problems and factorization on Riemann surfaces

The Wiener-Hopf factorization of 2×2 matrix functions and its close relation to scalar Riemann-Hilbert problems on Riemann surfaces is investigated. A family of function classes denoted C(Q₁,Q₂) is defined. To each class C(Q₁,Q₂) a Riemann surface Σ is associated, so that the factorization of the elements of C(Q₁,Q₂) is reduced to solving a scalar Riemann-Hilbert problem on Σ. For the solution of this problem, a notion of Σ-factorization is introduced and a factorization theorem is presented. An example of the factorization of a function belonging to the group of exponentials of rational functions is studied. This example may be seen as typical of applications of the results of this paper to finite-dimensional integrable systems.     

Oscillatory Riemann-Hilbert problems and the corona theorem

The paper is devoted to the Riemann-Hilbert problem with matrix coefficient having in Hardy spaces [H_p^±]²,1<p?∞, on half-planes . Under the assumption of existence of a non-trivial solution of corresponding homogeneous Riemann-Hilbert problem in [H_∞^±]² we study the solvability of the non-homogeneous Riemann-Hilbert problem in [H_p^±]²,1<p<∞, and get criteria for the existence of a generalized canonical factorization and bounded canonical factorization for G as well as explicit formulas for its factors in terms of solutions of two associated corona problems (in and ). A separation principle for constructing corona solutions from simpler ones is developed and corona solutions for a number of corona problems in H_∞⁺ are obtained. Making use of these results we construct explicitly canonical factorizations for triangular bounded measurable or almost periodic 2×2 matrix functions whose diagonal entries do not possess factorizations. Such matrices arise, e.g., in the theory of convolution type equations on finite intervals.     

Existence of nonlinear boundary layer to the Boltzmann equation with reverse reflection boundary condition

Many physical models have boundaries. When the Boltzmann equation is used to study a physical problem with boundary, there usually exists a layer of width of the order of the Knudsen number along the boundary. Hence, the research on the boundary layer problem is important both in mathematics and physics. Based on the previous work, in this paper, we consider the existence of boundary layer solution to the Boltzmann equation for hard sphere model with positive Mach number. The boundary condition is imposed on incoming particles of reverse reflection type, and the solution is assumed to approach to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 3 (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian. Moreover, there is an implicit solvability condition on the boundary data. According to the solvability condition, the co-dimension of the boundary data related to the number of the positive characteristic speeds is obtained.