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.     

A Sufficient Condition for Rigidity in Extremality of Teichmuller Equivalence Classes by Schwarzian Derivative

The Strebel point is a TeichmOller equivalence class in the Teichmuller space that has a certain rigidity in the extremality of the maximal dilatation. In this paper, we give a sufficient condition in terms of the Schwarzian derivative for a Teichmuller equivalence class of the universal Teichmuller space under which the class is a Strebel point. As an application, we construct a Teichmuller equivalence class that is a Strebel point and that is not an asymptotically conformal class.     

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.     

Dirichlet and VMOA domains via Schwarzian derivative

Short proofs of the following results concerning a bounded conformal map g of the unit disc D are presented: (1) logg^′ belongs to the Dirichlet space if and only if the Schwarzian derivative Sg of g satisfies Sg(z)(1−²|z|)∈L²(D); (2) logg^′∈VMOA if and only if ²|Sg(z)|³(1−²|z|) is a vanishing Carleson measure on D. Analogous results for Besov and Qp,0 spaces are also given.     

Comments on “New exact kink solutions, solitons and periodic form solutions for a combined KdV and Schwarzian KdV equation”

In this paper, we demonstrate that 14 solutions from 34 of the combined KdV and Schwarzian KdV equation obtained by Li [Z.T. Li, Appl. Math. Comput. 215 (2009) 2886-2890] are wrong and do not satisfy the equation. The other a number of exact solutions are equivalent each other.     

区域的对数导数单叶性内径

在万有Teichmller空间的对数导数嵌入模型T1(△)中,我们证明了存在无穷多个点[h]∈LT1(△),h(△)相互不Mbius等价,它们到边界的距离均为1,而在万有Teichmller空间的Schwarz导数嵌入模型T(△)中,只有一个点Sid具有类似性质.论文还得到了万有Teichmller空间两类嵌入模型的测地线的一些新的性质.     

On Schottky groups arising from the hypergeometric equation with imaginary exponents

In an article by Sasaki and Yoshida (2000), we encountered Schottky groups of genus 2 as monodromy groups of the hypergeometric equation with purely imaginary exponents. In this paper we study automorphic functions for these Schottky groups, and give a conjectural infinite product formula for the elliptic modular function .

     

Distortion theorems for higher order Schwarzian derivatives of univalent functions

Let denote the class of functions which are univalent and holomorphic on the unit disc. We derive a simple differential equation for the Loewner flow of the Schwarzian derivative of a given . This is used to prove bounds on higher order Schwarzian derivatives which are sharp for the Koebe function. As well we prove some two-point distortion theorems for the higher order Schwarzians in terms of the hyperbolic metric.

     

Boundary behavior of SLE

We show that the normalized (pre-)Schwarzian derivative of SLE, after we subtract a negligible term, is a complex BMO martingale. Its BMO norm gives strong evidence for Duplantier's duality conjecture. We also show that it has correlations that decay exponentially in the hyperbolic distance.

We reexamine S. Rohde and O. Schramm's derivative expectation to derive the conjectured sharp estimate for the Hölder exponent unless the parameter of SLE is 4.

     

Extinction, persistence and global stability in models of population growth

First, we systematize earlier results on the global stability of discrete model An+1=λAn+F(An−m) of population growth. Second, we invent the effect of delay m when F is unimodal. New, deep and strong results are discussed in Section 4, although Theorems 3-5 (Section 3) are still freshly new. This paper may be considered as a discrete version of our earlier work on the model [D.V. Giang, Y. Lenbury, Nonlinear delay differential equations involving population growth, Math. Comput. Modelling 40 (2004) 583-590]. We are mainly using ω-limit set of persistent solution, which is discussed in more general by P. Walters [An Introduction to Ergodic Theory, Springer-Verlag, Berlin, 1982].