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

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.

34 pages, 1 figure. Minor revision; updated references. To appear in Annales de l'Institut Henri Poincaré / Analyse Non Linéaire

References in corpus (4)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data
• Orbital stability of periodic waves for the nonlinear Schroedinger equation
• A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation
• Global well-posedness for the derivative nonlinear Schrödinger equation

Cited by in corpus (4)

• Modulational instability of periodic standing waves in the derivative NLS equation
• Potential well theory for the derivative nonlinear Schrödinger equation
• Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach
• Instability of stationary solutions for double power nonlinear Schrödinger equations in one dimension