Long-Time Asymptotics for the integrable discrete nonlinear Schr\"odinger equation: The Focusing Case

TL;DR

This paper analyzes the long-time behavior of solutions to the focusing integrable discrete nonlinear Schrödinger equation, showing they decompose into solitons and oscillatory decay, confirming aspects of the soliton resolution conjecture.

Contribution

It provides the first detailed asymptotic description of solutions in the focusing discrete NLS, including phase shifts and decay patterns, using the nonlinear steepest descent method.

Findings

Solutions decompose into 1-solitons and decaying oscillations.

Different phase shifts are identified in various regions.

The results support the soliton resolution conjecture.

Abstract

We investigate the long-time asymptotics for the focusing integrable discrete nonlinear Schr\"odinger equation. Under generic assumptions on the initial value, the solution is asymptotically a sum of 1-solitons. We find different phase shift formulas in different regions. Along rays away from solitons, the behavior of the solution is decaying oscillation. This is one way of stating the soliton resolution conjecture. The proof is based on the nonlinear steepest descent method.