Yongyu Qiang | Searcharxiv

arXiv2026

On the final-state problem for the 1D cubic NLS

We consider the one-dimensional cubic nonlinear Schrödinger equation $$ \ii\partial_tu+\frac12\partial_{xx}u=\la|u|^2u,\,λ=\pm 1 $$ and solve the final-state (modified wave operator) problem for small asymptotic data. More precisely, given a small $W(ξ)$, we construct a solution $u$ such that \begin{equation*} u\rightarrow (2π)^{-1/2}(\ii t)^{-1/2}e^{\ii x^2/(2t)}\, W\!\Big(\frac{x}{t}\Big)\exp(-\ii\la|W(x/t)|^2\log t). \end{equation*} Crucially, we design a contraction map, so that we can run the analysis in the spirit of Kato--Pusateri \cite{KP} for $w$ with a forcing term depending {\it only} on the final data $W$. This scheme is easy to adapt to solving final state problems with a complete theory for the forward problems.

math.AP↗

arXiv2026

Classification of Rational Functions of Degree Three over Finite Fields

We study rational functions over finite fields under PGL-equivalence. We say that $f, g \in \Bbb F_q(X)$ are \emph{equivalent} if there exist $ψ, ϕ\in \Bbb F_q(X)$ of degree one such that $g = ψ\circ f \circ ϕ$. Most properties of rational functions over finite fields as they appear in theory and applications are preserved under this equivalence. In a recent work, Mattarei and Pizzato classified rational functions of degree three over finite fields in even characteristic. In the present paper, we classify all rational functions of degree three over finite fields in odd characteristic. Our approach is based on careful analyses of the value frequencies and the ramification points of the degree three rational functions. The completion of our classification also relies on an explicit formula for the number of equivalence classes of degree three rational functions over finite fields recently obtained by the first author.

math.NT↗