arXiv subjects

Yingmin Yang

Publications and source records attributed to Yingmin Yang.

4 recordsLinked to original sources

arXiv2026

Asymptotic stability of solutions to the good Boussinesq equation in dispersive wave region

This work studies the asymptotic stability of solutions to the good Boussinesq equation in dispersive wave region when the reflection coefficients associated with the initial data belong to weighted Sobolev space. The Dbar-steepest descent method is applied to the Riemann-Hilbert problem and the long-time asymptotic expansion of the solution are obtained up to an optimal error of order $\mathcal{O}(t^{-3/4})$. Compared with previous results, we extend the initial data from the rapidly decaying Schwartz space to a weighted Sobolev space, and prove the asymptotic stability of the solution in dispersive wave region.

math-ph↗

arXiv2026

Long-time asymptotics of the Newell equation on the line

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.

math-ph↗

arXiv2025

Long-time behaviors of the two-component nonlinear Klein-Gordon equation: higher-order asymptotics

This work investigates the long-time asymptotic behaviors of solutions to the initial value problem of the two-component nonlinear Klein-Gordon equation by inverse scattering transform and Riemann-Hilbert formulism. Two reflection coefficients are defined and their properties are analyzed in detail. The Riemann-Hilbert problem associated with the initial value problem is constructed in term of the two reflection coefficients. The Deift-Zhou nonlinear steepest descent method is then employed to analyze the Riemann-Hilbert problem, yielding the long-time asymptotics of the solution in different regions. Specifically, a higher-order asymptotic expansion of the solution inside the light cone is provided, and the leading term of this asymptotic solution is compared with results from direct numerical simulations, showing excellent agreement. This work not only provides a comprehensive analysis of the long-time behaviors of the two-component nonlinear Klein-Gordon equation but also offers a robust framework for future studies on similar nonlinear systems with third-order Lax pair.

nlin.SI↗

arXiv2025

Riemann-Hilbert problem and long-time asymptotics of the Yajima-Oikawa equation

The Yajima-Oikawa equation is an integrable long wave-short wave resonance interaction model arising as a deformation of the Zakharov system for Langmuir waves coupled to ion-acoustic waves. In this work, a Riemann-Hilbert approach is developed for the Cauchy problem for the Yajima-Oikawa equation with rapidly decaying initial data. A main novelty is the formulation of a direct and inverse scattering theory adapted to its third-order spectral problem, including a detailed treatment of the singular spectral point $k=0$. The associated Riemann-Hilbert problem is expressed in terms of two reflection coefficients determined by the initial data, together with possible discrete eigenvalues and norming constants. We prove a vanishing lemma which ensures the unique solvability of the Riemann-Hilbert problem under suitable positivity assumptions, and hence obtain a rigorous reconstruction formula for the solution. We also classify the admissible discrete spectrum and derive exact pure soliton solutions from the reflectionless Riemann-Hilbert problem. In the solitonless case, we apply the Deift-Zhou nonlinear steepest descent method to obtain rigorous long-time asymptotic formulas in the different regions of the upper $(x,t)$-plane. The leading oscillatory behavior of the short-wave component is described explicitly in terms of the reflection coefficients evaluated at the stationary phase points, while the long-wave component is shown to be of lower order away from the transition region. These results provide, to the best of our knowledge, the first Riemann-Hilbert framework for the long-time asymptotic analysis of the Yajima-Oikawa equation in the presence of continuous spectrum.

nlin.SI↗