arXiv subjects

Chengfa Wu

Publications and source records attributed to Chengfa Wu.

13 recordsLinked to original sources

arXiv2026

Zeros of the generalized Wronskian-Hermite polynomials

In this paper, we study generalized Wronskian-Hermite (WH) polynomials associated with arithmetic-progression index sets. We verify the conjecture that all nonzero roots are simple for three subclasses of these polynomials, which, in a natural sense, cover more than half of the relevant parameter range. We also provide an interpretation of the root multiplicity at $z=0$ in terms of Young diagrams. In addition, we show that certain members of generalized WH polynomials provide representations of rational solutions of the Noumi-Yamada systems, a family of higher-order Painlev\'e equations. Finally, we apply our results to the large-parameter asymptotic analysis of rogue wave patterns for the multi-component Hirota equation.

math.CV↗

arXiv2026

Soliton solutions to the coupled Sasa-Satsuma equation under mixed boundary conditions

In this paper, we derive general bright-dark soliton solutions to the coupled Sasa-Satsuma (CSS) equation using the Kadomtsev-Petviashvili (KP) reduction method. Since the CSS equation is a special case of the four-component Hirota equation, our approach begins with the construction of two-bright-two-dark soliton solutions for the four-component Hirota equation. By imposing specific parameter constraints, these solutions are subsequently reduced to the bright-dark soliton solutions of the CSS equation. Finally, the dynamical behaviors of the one- and two-bright-dark soliton solutions are thoroughly analyzed and illustrated.

nlin.SI↗

arXiv2025

Methods for exact solutions of nonlinear ordinary differential equations\

In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations, numerous tricks have been proposed. The goal of this short review is to recall classical, 19th-century results, completed in 2006 by Eremenko, which can be turned into algorithms, thus avoiding \textit{ad hoc} assumptions, able to provide \textit{all} (as opposed to some) solutions in a precise class. To illustrate these methods, we present some new such exact solutions, physically relevent.

nlin.SI↗

arXiv2023

Universal convexity and range problems of shifted hypergeometric functions

In the present paper, we study the shifted hypergeometric function $f(z)=z\Gauss(a,b;c;z)$ for real parameters with $0<a\le b\le c$ and its variant $g(z)=z\Gauss(a,b;c;z^2).$ Our first purpose is to solve the range problems for $f$ and $g$ posed by Ponnusamy and Vuorinen in their 2001 paper. Ruscheweyh, Salinas and Sugawa developed in their 2009 paper the theory of universal prestarlike functions on the slit domain $\C\setminus[1,+\infty)$ and showed universal starlikeness of $f$ under some assumptions on the parameters. However, there has been no systematic study of universal convexity of the shifted hypergeometric functions except for the case $b=1.$ Our second purpose is to show universal convexity of $f$ under certain conditions on the parameters.

math.CV↗

arXiv2022

Connectivity of Fatou Components of Meromorphic Functions

In this paper, we show that there exist transcendental meromorphic functions with a cycle of 2-periodic Fatou components, where one is simply connected while the other is doubly connected. In particular, the doubly connected Fatou component can be an attracting, parabolic, or Baker domain. Thus, this settles the problem of whether a doubly connected periodic Fatou component must be a Herman ring. We also prove that there exists a transcendental meromorphic function with a wandering domain that has no eventual connectivity. In addition, we show that the connectivity sequence of this wandering domain is periodic with period two. This solves a problem about the nonexistence of the eventual connectivity of wandering domains and gives a different example constructed by Ferreira [J. London Math. Soc. (2022), DOI:10.1112/jlms.12613].

math.CV↗

arXiv2022

Rogue waves and their patterns in the vector nonlinear Schr\"{o}dinger equation

In this paper, we study the general rogue wave solutions and their patterns in the vector (or $M$-component) nonlinear Schr\"{o}dinger (NLS) equation. By applying the Kadomtsev-Petviashvili hierarchy reduction method, we derived an explicit solution for the rogue wave expressed by $\tau$ functions that are determinants of $K\times K$ block matrices ($K=1,2,\cdots, M$) with an index jump of $M+1$. Patterns of the rogue waves for $M=3,4$ and $K=1$ are thoroughly investigated. We find that when a specific internal parameter is large enough, the wave patterns are linked to the root structures of generalized Wronskian-Hermite polynomial hierarchy in contrast with rogue wave patterns of the scalar NLS equation, the Manakov system and many others. Moreover, the generalized Wronskian-Hermite polynomial hierarchy includes the Yablonskii-Vorob'ev polynomial hierarchy and Okamoto polynomial hierarchies as special cases, which have been used to describe the rogue wave patterns of the scalar NLS equation and the Manakov system, respectively. As a result, we extend the most recent results by Yang {\it et al.} for the scalar NLS equation and the Manakov system. It is noted that the case $M=3$ displays a new feature different from the previous results. The predicted rogue wave patterns are compared with the ones of the true solutions for both cases of $M=3,4$. An excellent agreement is achieved.

nlin.SI↗

arXiv2022

Equivalence of two canonical lists of elliptic functions

We establish a one-to-one correspondence between,on one hand the four types of transcendental meromorphic solutions of the autonomous Schwarzian differential equations which are elliptic,on the other hand the four binomial equations of Briot and Bouquet possessing an elliptic solution.

math.CV↗

arXiv2022

General rogue wave solutions to the Sasa-Satsuma equation

General rogue wave solutions to the Sasa-Satsuma equation are constructed by the Kadomtsev-Petviashvili (KP) hierarchy reduction method. These solutions are presented in three different forms. The first form is expressed in terms of recursively defined differential operators while the second form shares a similar solution structure except that the differential operators are no longer recursively defined. Instead of using differential operators, the third form is expressed by Schur polynomials.

nlin.SI↗

arXiv2021

Closed-form meromorphic solutions of some third order boundary layer ordinary differential equations

We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic methods recently developed, for the generic cases, it is shown that all their meromorphic solutions must be rational, or rational in one exponential, and then we find all of them explicitly. For a few non-generic cases, some solutions, which are meromorphic or singlevalued, are also obtained. Our results also explain why it is so difficult to obtain new closed-form solutions of the Falkner-Skan equation.

math.CV↗

arXiv2021

Multi-breather solutions to the Sasa-Satsuma equation

General breather solution to the Sasa-Satsuma (SS) equation is systematically investigated in this paper. We firstly transform the SS equation into a set of three Hirota bilinear equations under proper plane wave background. Starting from a specially arranged tau-function of the Kadomtsev-Petviashvili hierarchy and a set of eleven bilinear equations satisfied, we implement a series steps of reduction procedure, i.e., C-type reduction, dimension reduction and complex conjugate reduction, and reduce these eleven equations to three bilinear equations for the SS equation. Meanwhile, general breather solution to the SS equation is found in determinant of even order. The one- and two-breather solutions are calculated and analyzed in details.

nlin.SI↗

arXiv2021

Exact meromorphic solutions of Schwarzian differential equations

This paper studies exact meromorphic solutions of the autonomous Schwarzian differential equations. All transcendental meromorphic solutions of five canonical types (among six) of the autonomous Schwarzian differential equations are constructed explicitly. In particular, the solutions of four types are shown to be elliptic functions. Also, all transcendental meromorphic solutions that are locally injective or possess a Picard exceptional value are characterized for the remaining canonical type.

math.CV↗

arXiv2021

Iterates of Meromorphic Functions on Escaping Fatou Components

In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component may be bounded even if the orbit of the component contains an infinite modulus annulus sequence and this case cannot happen when the maximal modulus of the meromorphic function is large enough. Therefore, we extend the related results for entire functions to ones for meromorphic functions with infinitely many poles. And we investigate the fast escaping Fatou components of meromorphic functions defined in [44] in terms of the Nevanlinna characteristic instead of the maximal modulus in [12] and show that the multiply-connected wandering domain is a part of the fast escaping set under a growth condition of the maximal modulus. Finally we give examples of wandering domains escaping at arbitrary fast rate and slow rate.

math.CV↗

arXiv2018

Singularity methods for meromorphic solutions of differential equations

This chapter is mainly a tutorial introduction to methods recently developed in order to find all (as opposed to some) meromorphic particular solutions of given nonintegrable, autonomous, algebraic ordinary differential equations of any order. The examples are taken from physics and include Kuramoto-Sivashinsky and complex Ginzburg-Landau equations.

nlin.SI↗