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

Masayuki Hayashi

TL;DR

The paper develops exact periodic traveling-wave solutions for the derivative nonlinear Schrödinger equation on a torus and shows that these periodic waves converge to the two-parameter solitons on the real line in the long-period limit. By reducing the elliptic problem to an ODE for $\psi=\Phi^2$ with a quartic polynomial, the authors obtain an explicit representation of the periodic profile in terms of Jacobi elliptic functions, $\mathrm{dn}$ and $\mathrm{sn}$, parameterized by $(\omega,c)$ and the torus length $L$. They establish mass preservation in the limit and prove $L^2$- and $C^m$-convergence (for any $m$) of the periodic profiles to the corresponding solitons, using Brézis–Lieb and elliptic-integral techniques. The results connect periodic waves to the soliton landscape, highlight the role of the massless case, and provide a rigorous framework for the long-period transition from periodic to solitary waves in the DNLS context, with potential implications for stability and spectral analysis in periodic settings. The analysis hinges on the detailed elliptic-function structure and yields precise asymptotics for the modulus $k$ and the fundamental period as $L\to\infty$.

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, 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.

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

TL;DR

The paper develops exact periodic traveling-wave solutions for the derivative nonlinear Schrödinger equation on a torus and shows that these periodic waves converge to the two-parameter solitons on the real line in the long-period limit. By reducing the elliptic problem to an ODE for with a quartic polynomial, the authors obtain an explicit representation of the periodic profile in terms of Jacobi elliptic functions, and , parameterized by and the torus length . They establish mass preservation in the limit and prove - and -convergence (for any ) of the periodic profiles to the corresponding solitons, using Brézis–Lieb and elliptic-integral techniques. The results connect periodic waves to the soliton landscape, highlight the role of the massless case, and provide a rigorous framework for the long-period transition from periodic to solitary waves in the DNLS context, with potential implications for stability and spectral analysis in periodic settings. The analysis hinges on the detailed elliptic-function structure and yields precise asymptotics for the modulus and the fundamental period as .

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, 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.

Paper Structure

This paper contains 16 sections, 15 theorems, 223 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Related problems and remarks
  5. Organization of the paper
  6. Preliminaries
  7. Existence of exact periodic traveling waves
  8. Construction of exact solutions
  9. Fundamental properties of exact solutions
  10. Pointwise convergence in the long-period limit
  11. Long-period limit procedure
  12. $L^2$-convergence
  13. $L^{\infty}$-convergence
  14. Proof of Theorem \ref{['thm:1.2']} and Theorem \ref{['thm:1.3']}
  15. Monotonicity of the modulus
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(\omega ,c) \in {\mathbb R}^2$ satisfy (WC). Assume that $L>0$ satisfies where $L_0(\omega ,c)$ is a positive constant determined by $(\omega ,c)$(see Remark rem:1.2 below). Then, there exists the positive single-bump solution $\Phi_{\omega ,c}^L$ of (ELL) on ${\mathbb T}_{2L}$ such that $\Phi_{\omega ,c}^L(x) \to \Phi_{\omega , c}(x)$ for any $x \in {\mathbb R}$ as $L\t with parameters $\et

Figures (1)

  • Figure 1: The ellipse (\ref{['eq:3.19']}) for several values of $s$. Note that the ellipse moves toward upper right when one changes the parameter $s$ from $-1$ to $1$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 3.1
  • proof
  • ...and 22 more