# Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation

### Masayuki Hayashi

Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

## Abstract

We study the periodic traveling wave solutions of the derivative nonlinear Schrödinger 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 [15], 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.

## Cite this article

Masayuki Hayashi, Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, pp. 1331–1360

DOI 10.1016/J.ANIHPC.2018.12.003