Long-time asymptotics of the Newell equation on the line
Abstract
In 1978, A. C. Newell [SIAM J. Appl. Math. 35(4) (1978) 650-664] proposed an exactly solvable model called Newell equation, which simulates the investigation of significant interaction mechanism between long and short waves. Nearly fifty years have passed, yet the long-time asymptotics of the Newell equation remains an open problem to date, with no results reported. In this work, the long-time asymptotic behaviors of the solutions to this model under Schwartz class initial conditions are studied by using the Riemann-Hilbert formulation. Through direct and inverse scattering analysis, the corresponding Riemann-Hilbert problem is formulated, and its relationship with the solution to the initial-value problem of the Newell equation is established. The existence and uniqueness of the solution to the Riemann-Hilbert problem is proved by vanishing lemma. Subsequently, the asymptotic expressions of the solution to the initial-value problem in the dispersive wave region are obtained by using the Deift-Zhou nonlinear steepest descent method. This work extends Newell's original results, providing a rigorous proof for the findings presented in Section 4 of his paper, along with explicit expressions. Furthermore, the comparison between direct numerical simulations and the theoretical results obtained in this paper demonstrates the reliability of the asymptotic expressions.
Keywords
Cite
@article{arxiv.2604.25703,
title = {Long-time asymptotics of the Newell equation on the line},
author = {Deng-Shan Wang and Yingmin Yang},
journal= {arXiv preprint arXiv:2604.25703},
year = {2026}
}