Schwarzian derivative and general Besov-type domains

We study univalent functions f for which belongs to a class of general Besov-type spaces. We give a characterization of the general Besov-type domains in terms of Carleson measure conditions on the Schwarzian derivative Sf, and mention some further connection to Teichmüller theory.     

Solution to n-dimensional Sturm-Liouville-like equations using multi-dimensional Schwarzian

In the present work we produce the solution to the n-dimensional Sturm-Liouville-like equations in Rn. To make it, we define the multi-dimensional Schwarzian derivative of a real-valued function of n variables and show that its basic properties related to its invariance under the action of a group of multi-dimensional Möbius transformations defined in Rn correspond to a straightforward generalization of those of the one-dimensional Schwarzian.     

Some properties of meromorphic solutions of q-difference equations

We investigate the growth of transcendental meromorphic solutions of some complex q-difference equations and find lower bounds for Nevanlinna lower order for meromorphic solutions of such equations. We also obtain a q-difference version of Tumura-Clunie theorem.     

多复变数的Schwarz导数(Ⅴ)

本文从Thurston的观点出发,用二阶逼近来定义与讨论矩阵空间C~(m×n)（m≤n）中的域上全纯映照的Schwarz导数及高阶Schwarz导数,证明：如果它们存在的话,那么它们是在R_I（m,n）的紧对偶空间CG（m,n）的全纯自同构群下的相似不变量．并证明：这样得到的Schwarz导数与前几文［1－4］中由Ahlfors的观点得到的Schwarz导数是相一致的．此外,还应用这种观点定义与讨论了C~N中的域上全纯映照的Schwarz导数．     

Growth and poles of meromorphic solutions of some difference equations

This paper is devoted to studying the growth property and the pole distribution of meromorphic solutions f of some complex difference equations with all coefficients being rational functions or of growth S(r,f). We find the lower bound of the lower order, or the relation between lower order and the convergence exponent of poles of meromorphic solutions of such equations.     

Dark solitons for a combined potential KdV and Schwarzian KdV equations with t-dependent coefficients and forcing term

In this work we formally derive the dark soliton solutions for the combined potential KdV and Schwarzian KdV equations. The combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term are then investigated to obtain dark soliton solutions. The solitary wave ansatz is used to carry out the analysis for both models.     

The Schwarzian derivative in several complex variables (III)

In the point view of Lie group, the cross ratio and Schwarzian derivative in Cⁿ are defined and discussed, especially the Schwarzian derivative of holomorphic mappings on the domains in matrix space C^{m x n} is defined and discussed. It is proved that it is invariant up to similarity under the group of holomorphic automorphism of the Grassmann manifold CG(m, n). And it is also proved that the Schwarzian derivative equals zero if and only if the mapping is liaearly fractional. Project supported by the National Natural Science Foundation of China.     

The representation of meromorphic solutions for a class of odd order algebraic differential equations and its applications

In this paper, we employ the Nevanlinna's value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations. We obtain the representations of all meromorphic solutions for a class of odd order algebraic differential equations with the weak ?p,q?and dominant conditions. Moreover, we give the complex method to find all traveling wave exact solutions of corresponding partial differential equations. As an example, we obtain all meromorphic solutions of the Kuramoto–Sivashinsky equation by using our complex method. Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd.     

Some results related to complex linear and nonlinear difference equations

In this paper, we investigate the complex oscillation problems of meromorphic solutions to some linear difference equations with meromorphic coefficients, and obtain some results about the relationships between the exponent of convergence of zeros, poles and the order of growth of meromorphic solutions to complex linear difference equations. We also study the existence of solution of certain types of nonlinear differential-difference equations, and partially answer a conjecture concerning the above problem posed by Yang and Laine (C.C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A Math. Sci. 86(1) (2010), pp. 10–14).     

On the sequences of zeros of holomorphic solutions of linear second-order differential equations

We find necessary and sufficient conditions under which a finite or infinite sequence of complex numbers is the sequence of zeros of a holomorphic solution of the linear differential equation f″ + a ₀ f = 0 with a meromorphic coefficient a ₀ that has second-order poles. In addition, we present a criterion for all solutions of second-order linear equations to be meromorphic.     

某些常微分方程的亚纯解表示与应用

本文运用Nevanlinna值分布理论研究了某些常微分方程亚纯解的存在性. 对于某些具有控制项的常系数常微分方程, 本文得到了亚纯解的表示, 并且给出了求相应偏微分方程精确解的一种方法.作为例子, 本文运用此方法得到了著名的KdV方程的所有亚纯行波精确解. 结果显示该方法比其他方法简单.     

Meromorphic solutions of some algebraic differential equations related Painlevé equation IV and its applications

In this paper, the complex method is used to derive meromorphic solutions to some algebraic differential equations related Painlevé equation IV, and then we illustrate our main result by some computer simulations. By the application of our result, we obtain meromorphic solutions of a nonlinear evolution equation. We can apply the idea of this study for other nonlinear evolution equations in mathematical physics.     

The Schwarzian derivative in several complex variables IV

Using two different Lie groups, two different Schwarzian derivatives of holomorphic mappings on domains in Cⁿ are defined and discussed. The necessary and sufficient conditions for annihilation of these two different Schwarzian derivatives are given. Project supported by the National Natural Science Foundation of China.     

On the hyperbolic order

We define the hyperbolic order of any locally injective holomorphic function between arbitrary hyperbolic domains of the complex plane and study the relation between the hyperbolic order and the Schwarzian derivative for locally injective holomorphic functions from the unit disk into itself.     

关于某类Briot-Bouquet微分方程若干结果的推广

本文研究了某类形式的Briot-Bouquet微分方程的亚纯解结构问题.利用Nevanlinna值分布理论的有关知识,获得了该类微分方程的所有可能形式,以及当方程的阶为偶数时,给出了每一个形式的方程的亚纯解结构,推广了某些特定的Briot-Bouquet微分方程亚纯解结构的一些结果.     

On the action of Virasoro algebra on the space of univalent functions

We obtain explicit expressions for differential operators defining the action of the Virasoro algebra on the space of univalent functions. We also obtain an explicit Taylor decomposition for Schwarzian derivative and a formula for the Grunsky coefficients.     

Linear differential equations and multiple zeta values. II. A generalization of the WKB method

For linear differential equations x(n)+a₁x(n−1)+?+anx=0 (and corresponding linear differential systems) with large complex parameter λ and meromorphic coefficients aj=aj(t;λ) we prove existence of analogues of Stokes matrices for the asymptotic WKB solutions. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We also discuss a generalization of the Hukuhara-Levelt-Turritin theorem about formal reduction of a linear differential system near an irregular singular point t=0 to a normal form with ramified change of time to the case of systems with large parameter. These results are applied to some hypergeometric equations related with generating functions for multiple zeta values.     

Injectivity Conditions for Some Classes of Domains. II

New sufficient conditions are found for continuously differentiable mappings in some plane domains to be injective. This result is obtained by using the locally homeomorphic extension method. New univalence criteria for meromorphic functions are given by means of the quasiconformal extension method. These criteria are stated in terms of coefficients of a differential equation related to the Schwarzian derivative of the function under consideration. Bibliography: 13 titles.     

On meromorphic solutions of nonlinear partial differential equations of first order

We prove a uniqueness theorem in terms of value distribution for meromorphic solutions of a class of nonlinear partial differential equations of first order, which shows that such solutions f are uniquely determined by the zeros and poles of f−cj (counting multiplicities) for two distinct complex numbers c₁ and c₂.