Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schr\"{o}dinger equation

TL;DR

This paper constructs exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation, analyzing their long-period limit and convergence properties, thereby deepening understanding of soliton behavior, especially in the massless case.

Contribution

It introduces a method to construct explicit periodic solutions that approximate solitons in the long-period limit, using elliptic functions and integrals.

Findings

Exact periodic solutions converge to solitons in the long-period limit.

The regularity of convergence is established.

Elliptic functions are effectively used in the analysis.

Abstract

We study the periodic traveling wave solutions of the derivative nonlinear Schr\"{o}dinger equation (DNLS). It is known that DNLS has two types of solitons on the whole line; one has exponential decay and the other has algebraic decay. The latter corresponds to the soliton for the massless case. In the new global results recently obtained by Fukaya, Hayashi and Inui, the properties of two-parameter of the solitons are essentially used in the proof, and especially the soliton for the massless case plays an important role. To investigate further properties of the solitons, we construct exact periodic traveling wave solutions which yield the solitons on the whole line including the massless case in the long-period limit. Moreover, we study the regularity of the convergence of these exact solutions in the long-period limit. Throughout the paper, the theory of elliptic functions and elliptic integrals is used in the calculation.