Asymptotics for small data solutions of the Ablowitz-Ladik equation

TL;DR

This paper investigates the long-term behavior of small data solutions to the Ablowitz-Ladik equation, revealing its connection to the complex mKdV equation and describing its modified scattering behavior.

Contribution

It establishes a link between the Ablowitz-Ladik equation and the complex mKdV equation through continuum limits and analyzes the solution's asymptotic behavior using space-time resonance methods.

Findings

Near degenerate frequencies, Ablowitz-Ladik solutions resemble complex mKdV solutions.

Identifies two regions with self-similar solution behavior.

Provides a detailed description of the modified scattering behavior.

Abstract

We study the asymptotics for the Ablowitz-Ladik equation. By taking appropriate continuum limits, it can be shown that the behavior of the equation near degenerate frequencies is well approximated by a complex modified Korteweg-de Vries equation. Using this connection, we use the method of space-time resonances to derive a description of the modified scattering behavior of the Ablowitz-Ladik equation, which includes two regions where the solution behaves like a self-similar solution to the complex mKdV equation.