A Class of Quasi-Linear Riemann-Hilbert Problems for General Holomorphic Functions in the Unit Disk

In this paper, a class of quasi linear Riemann Hilbert problems for general holomorphic functions in the unit disk was studied. Under suitable hypotheses, the existence of solutions of the Hardy class H 2 to this problem was proved by means of Tikhonov's fixed point theorem and corresponding theories for general holomorphic functions.

The Riemann Hilbert Problem for General Elliptic Complex Equations of Fourth Order

This paper deals with the existence theorem and Riemann Hilbert boundary value problem for general nonlinear elliptic complex equations of fourth order. Firstly we give the representation and existence theorem of solutions for the complex equations. Moreover,we propose the Riemann Hilbert problem and its well posedness,and then we give the representation of solutions for the modified boundary value problem and prove its solvsbility,and finally derive solvability conditions of the original Riemann Hilbert problem.

A RIEMANN-HILBERT PROBLEM IN A RIEMANN SURFACE

One of the inspirations behind Peter Lax＇s interest in dispersive integrable systems, as the small dispersion parameter goes to zero, comes from systems of ODEs discretizing 1-dimensional compressible gas dynamics [17]. For example, an understanding of the asymptotic behavior of the Toda lattice in different regimes has been able to shed light on some of von Neumann＇s conjectures concerning the validity of the approximation of PDEs by dispersive systems of ODEs. Back in the 1990s several authors have worked on the long time asymptotics of the Toda lattice [2, 7, 8, 19]. Initially the method used was the method of Lax and Levermore [16], reducing the asymptotic problem to the solution of a minimization problem with constraints （an ＂equilibrium measure＂ problem）. Later, it was found that the asyraptotic method of Deift and Zhou （analysis of the associated Riemann-Hilbert factorization problem in the complex plane） could apply to previously intractable problems and also produce more detailed information. Recently, together with Gerald Teschl, we have revisited the Toda lattice; instead of solu- tions in a constant or steplike constant background that were considered in the 1990s we have been able to study solutions in a periodic background. Two features are worth noting here. First, the associated Riemann-Hilbert factorization problem naturally lies in a hyperelliptic Riemann surface. We thus generalize the Deift- Zhou ＂nonlinear stationary phase method＂ to surfaces of nonzero genus. Second, we illustrate the important fact that very often even when applying the powerful Riemann-Hilbert method, a Lax-Levermore problem is still underlying and understanding it is crucial in the analysis and the proofs of the Deift-Zhou method!

Riemann-Hilbert problem for first order complex equations of mixed type with degenerate curve

This paper considers the Riemann-Hilbert problem for linear mixed（elliptichyperbolic） complex equations of first order with degenerate curve in a simply connected domain. We first give the representation theorem and uniqueness of solutions for such boundary value problem. Then by using the methods of successive iteration and parameter extension, the existence of solutions for this problem is proved.

Equivalence of three kinds of well-posed-ness of discontinuous Riemann-Hilbert problem for elliptic complex equation in multiply connected domains

In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-ness for the complex equation. Then we verify the equivalence of three kinds of well-posed-ness. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.

Explicit Solutions of the Coupled mKdV Equation by the Dressing Method via Local Riemann-Hilbert Problem

We study the coupled mKdV equation by the dressing method via local Riemann-Hilbert problem. With the help of the Lax pairs, we obtain the matrix Riemann-Hilbert problem with zeros. The explicit solutions for the coupled mKdV equation are derived with the aid of the regularization of the Riemann-Hilbert problem.

RIEMANN-HILBERT PROBLEMS OF A SIX-COMPONENT MKDV SYSTEM AND ITS SOLITON SOLUTIONS

Based on a 4 x 4 matrix spectral problem, an AKNS soliton hierarchy with six potentials is generated. Associated with this spectral problem, a kind of Riemann-Hilbert problems is formulated for a six-component system of mKdV equations in the resulting AKNS hierarchy. Soliton solutions to the considered system of coupled mKdV equations are computed, through a reduced Riemann-Hilbert problem where an identity jump matrix is taken.

Approximate Solutions to the Discontinuous Riemann-Hilbert Problem of Elliptic Systems of First Order Complex Equations

Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this article, we discuss approximate solutions to discontinuous Riemann-Hilbert boundary value problems, which have various applications in mechanics and physics. We first formulate the discontinuous Riemann-Hilbert problem for elliptic systems of first order complex equations in multiply connected domains and its modified well-posedness, then use the parameter extensional method to find approximate solutions to the modified boundary value problem for elliptic complex systems of first order equations, and then provide the error estimate of approximate solutions for the discontinuous boundary value problem.

Approximate Method of Riemann-Hilbert Problem for Elliptic Complex Equations of First Order in Multiply Connected Unbounded Domains

In this article, we discuss the approximate method of solving the Riemann-Hilbert boundary value problem for nonlinear uniformly elliptic complex equation of first order (0.1) with the boundary conditions (0.2) in a multiply connected unbounded domain D, the above boundary value problem will be called Problem A. If the complex Equation (0.1) satisfies the conditions similar to Condition C of (1.1), and the boundary condition (0.2) satisfies the conditions similar to (1.5), then we can obtain approximate solutions of the boundary value problems (0.1) and (0.2). Moreover the error estimates of approximate solutions for the boundary value problem is also given. The boundary value problem possesses many applications in mechanics and physics etc., for instance from (5.114) and (5.115), Chapter VI, [1], we see that Problem A of (0.1) possesses the important application to the shell and elasticity.

Multi-Cuspon Solutions of the Wadati-Konno-Ichikawa Equation by Riemann-Hilbert Problem Method

In this paper, we consider the initial value problem for a complete integrable equation introduced by Wadati-Konno-Ichikawa (WKI). The solution ?is reconstructed in terms of the solution of a ?matrix Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at one non-singularity point, i.e., . Then, the one-cuspon solution, two-cuspon solutions and three-cuspon solution are discussed in detail. Further, the numerical simulations are given to show the dynamic behaviors of these soliton solutions.

A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRDINGER EQUATION

We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann- Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit （x, t） dependence, and it has jumps across {ξ∈C｜Imξ^4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a（ξ）, b（ξ）}, {A（ξ）, B（ξ）}, and {A（ξ）, B（ξ）}, which in turn are defined in terms of the initial data q0（x） = q（x, 0）, the bound- ary data g0（t）= q（0, t）, g1（t） = qx（0, t）, and another boundary values f0（t） = q（L, t）, f1（t） = qx（L, t）. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

The Riemann-Hilbert Problem for Mixed Complex Equations of First Order with Degenerate Rank 0

This article deals with the Riemann-Hilbert boundary value problem for quasilinear mixed （elliptic- hyperbolic） complex equations of first order with degenerate rank 0. Firstly, we give the representation theorem and prove the uniqueness of solutions for the boundary value problem. Afterwards, by using the method of successive iteration, the existence and estimates of solutions for the boundary value problem are verified. The above problem possesses the important applications to the Tricomi problem of mixed type equations of second order. In this article, the proof of HSlder continuity of a singular double integer is very difficult and interesting as in this Section 4 below.

The Discontinuous Riemann–Hilbert Problem for Elliptic Complex Equations of First Order

In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.

Hilbert Problem 15 and Ritt-Wu Method(Ⅱ)

This paper proves three statements of Schubert about cuspal cubic curves in a plane by using the concept of generic point of Van der Waerden and Weil and Ritt-Wu methods.They are relations of some special lines:1)For a given point,all the curves containing this point are considered.For any such curve,there are five lines.Two of them are the tangent lines of the curve passing through the given point.The other three are the lines connecting the given point with the cusp,the inflexion point and the intersection point of the tangent line at the cusp and the inflexion line.2)For a given point,the curves whose tangent line at the cusp passes through this point are considered.For any such curve,there are four lines.Three of them are the tangent lines passing through this point and the other is the line connect the given point and the inflexion point.3)For a given point,the curves whose cusp,inflexion point and the given point are collinear are considered.For any such curve,there are five lines.Three of them are tangent lines passing through the given point.The other two are the lines connecting the given point with the cusp and the intersection point of the tangent line at the cusp and the inflexion line.

RIEMANN-HILBERT PROBLEMS OF DEGENERATE HYPERBOLIC SYSTEM

This paper is concerned with the Riemann-Hilbert problems of degenerate hyperbolic system in two general domains, where the boundary curves are given by the parameter equations of the arc length s. We prove the existence and uniqueness of solutions to the Riemann-Hilbert problems by conformal deformations. The corre-sponding representations of solutions to the problems are also presented.

ON HILBERT BOUNDARY VALUE PROBLEMS WITH RADICAL

The Hilbert boundary value problem Re{λ（t） p√ψ＋（t）} = c（t）, t∈L of normal type with Holder continuous coefficients is discussed, where L is the unit circle ｜t｜ = 1,p ≥2 is any definite integer,ψ^＋（t） is the boundary value of the unknown function ψ（z） holomorphic in ｜z｜ 〈 1 with single-valued continuous p√ψ＋（t） on L.

THE HILBERT BOUNDARY VALUE PROBLEM FOR GENERALIZED ANALYTIC FUNCTIONS IN CLIFFORD ANALYSIS

Let R0,n be the real Clifford algebra generated by e1, e2,... , en satisfying eiej＋ejei=-2δij,i,j=1,2…,ne0 is the unit element.Let Ω be an open set. A function f is called left generalized analytic in ft if f satisfies the equation Lf=0,where ……qi 〉0, i =-, 1, - ……, n. In this article, we first give the kernel function for the generalized analytic function. Further, the Hilbert boundary value problem for generalized analytic functions in Rn＋1 will be investigated.

Hilbert Problem 15 and Ritt-Wu Method(Ⅰ)

Hilbert problem 15 requires to understand Schubert's book. In this book, there is a theorem in §23, about the relation of the tangent lines from a point and the singular points of cubed curves with cusp near a 3-multiple straight line, which was obtained by the so called main trunk numbers, while for these numbers, Schubert said that he obtained them by experiences. So essentially Schubert even did not give any hint for the proof of this theorem. In this paper, by using the concept of generic point in the framework of Van der Waerden and Weil on algebraic geometry, and realizing Ritt-Wu method on computer, the authors prove that this theorem of Schubert is completely right.

THE RIEMANN-HILBERT BOUNDARY VALUE PROBLEM FOR THE MOISIL-THEODORSCO SYSTEM

This article studies the inhomogeneous Moisil-Theodorsco system in the space R3, gives the integral expression of its solution, proves the Holder continuity of the solution. Moreover the author studies the Riemann-Hilbert boundary value problem for the Moisil-Theodorsco system in a cylindrical domain of R3, and gives the solvability conditions and the integral expressions of solutions. The Holder continuity of the solutions is proved.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part III. An Original Solution of Hilbert’s Fourth Problem

This article refers to the “Mathematics of Harmony” by Alexey Stakhov [1], a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries—New Geometric Theory of Phyl-lotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci -Goniometry ( is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scien-tific ideas—The “golden mean,” which had been introduced by Euclid in his Elements, and its generalization—The “metallic means,” which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.