arXiv subjects

Shaoxuan Huang

Publications and source records attributed to Shaoxuan Huang.

3 recordsLinked to original sources

arXiv2026

Existence Conditions for Darboux Curves and Analytic First Integrals of a Liénard-Type Quadratic Vector Field

We study the rational quadratic differential equation \[ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{ay^2+by+cx}{y^2}, \qquad a,b,c\in\C, \] under the non-degeneracy assumptions \[ c\neq 0,\qquad 2ay+b\not\equiv 0. \] Equivalently, after clearing the denominator, we consider the polynomial vector field \[ \dot{x}=y^2,\qquad \dot{y}=ay^2+by+cx. \] We give a complete, directly checkable classification of its Darboux curves. If $a=0$, no non-constant Darboux polynomial exists. If $a\neq 0$, a non-constant Darboux polynomial exists if and only if \[ c=-ab\qquad\text{or}\qquad c=-2ab. \] In these two cases the unique irreducible Darboux polynomials, up to non-zero constant multiples, are respectively \[ y-ax,\qquad y^2-2bx. \] Consequently every non-constant Darboux polynomial is a non-zero constant multiple of a positive integral power of the corresponding irreducible factor. We then place this classification in the framework of Riccati (R-)integrability and rational potentials, and formulate the analytic integrability theorem asserting that a global Riccati-type analytic first integral occurs precisely on the branch $c=-ab$. Finally, we relate this branch to the Quartic Inverse Riccati (QIR) class and discuss an invariant-based classification problem for quartic Abel equations.

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arXiv2025

Generalized ODE reduction algorithm with bounded degree transformation

As a generalization of our previous result\cite{huang2025algorithm}, this paper aims to answer the following question: Given a 2-dimensional polynomial vector field $y^{\prime}=\frac{M(x,y)}{N(x,y)}$, how to find a rational transformation $y \to \frac{A(x,y)}{B(x,y)}$ with bounded degree numerator, the inverse of which transforms this vector field into a simpler form $y^{\prime}=\sum_{i=0}^nf_i(x)y^i$. Such a structure, often known as the generalized Abel equation and has been studied in various areas, provides a deeper insight into the property of the original vector field. We have implemented an algorithm with considerable performance to tackle this problem and the code is available in \href{https://www.researchgate.net/publication/393362858_Generalized_ODE_reduction_algorithm}{Researchgate}.

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arXiv2025

Generalized ODE reduction algorithm for bounded degree transformation

The integrability problem of rational first-order ODEs $y^{\prime}=\frac{M(x,y)}{N(x,y)}$, where $M,N \in \mathbb{R}[x,y]$ is a long-term research focus in the area of dynamical systems, physics, etc. Although the computer algebra system such as Mathematica, Maple has developed standard algorithms to tackle its first integral expressed by Liouvillian or special function, this problem is quite difficult and the general method requires specifying a tight degree bound for the Darboux polynomial. Computing the bounded degree first integral, in general, is very expensive for a computer algebra system\cite{duarte2021efficient}\cite{cheze2020symbolic} and becomes impractical for ODE of large size. In \cite{huang2025algorithm}, we have proposed an algorithm to find the inverse of a local rational transformation $y \to \frac{A(x,y)}{B(x,y)}$ that transforms a rational ODE to a simpler and more tractable structure $y^{\prime}=\sum_{i=0}^nf_i(x)y^i$, whose integrability under linear transformation $\left\{x \to F(x),y \to P(x)y+Q(x)\right\}$ can be detected by Maple efficiently \cite{CHEBTERRAB2000204}\cite{cheb2000first}. In that paper we have also mentioned when $M(x,y),N(x,y)$ of the reducible structure are not coprime, canceling the common factors in $y$ will alter the structure which makes that algorithm fail. In this paper, we consider this issue. We conclude that with the exact tight degree bound for the polynomial $A(x,y)$ given, we have an efficient algorithm to compute such transformation and the reduced ODE for "quite a lot of" cases where $M,N$ are not coprime. We have also implemented this algorithm in Maple and the code is available in researchgate.

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