Long-time asymptotics of the Newell equation on the line

TL;DR

This paper rigorously analyzes the long-time behavior of solutions to the Newell equation using Riemann-Hilbert techniques, extending previous results and confirming findings through numerical simulations.

Contribution

It provides the first rigorous proof of the long-time asymptotics of the Newell equation and derives explicit asymptotic expressions using inverse scattering methods.

Findings

Derived explicit asymptotic formulas for solutions in the dispersive wave region.

Proved existence and uniqueness of the Riemann-Hilbert problem solution.

Validated asymptotic results with numerical simulations.

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.