The modified Korteweg--de Vries limit of the Ablowitz--Ladik system

2024/04/02 by Rowan Killip, Zhimeng Ouyang, Killip, Rowan +5 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2404.02366

openalex publication_date 2024/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For slowly-varying initial data, solutions to the Ablowitz-Ladik system have been proven to converge to solutions of the cubic Schrödinger equation. In this paper we show that in the continuum limit, solutions to the Ablowitz-Ladik system with H¹ initial data may also converge to solutions of the modified Korteweg--de Vries equation. To exhibit this new limiting behavior, it suffices that the initial data is supported near the inflection points of the dispersion relation associated with the Ablowitz-Ladik system. Our arguments employ harmonic analysis tools, Strichartz estimates, and the conservation of mass and energy. Correspondingly, they are applicable beyond the completely integrable models of greatest interest to us.

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