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 [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.

Potential well theory for the derivative nonlinear Schrödinger equation

“…where s * := −γ/(1 − γ) and γ := 1 + 16 3 b. We note that the value b = −3/16 gives the turning point in the structure of the solitons.…”

“…There is a large literature on the Cauchy problem for (DNLS); see [38,39,19,20,36,3,8,9,42,43,15,11,16,23,24] and references therein. Here we are mainly interested in the results of energy space H 1 (R).…”

“…where γ = 1 + 16 3 b. From Φ ω,c ∈ H 1 (R) and the equation (2.5), one can show that Im Φ ω,c Φ ω,c = 0 (see [7,Lemma 2]).…”

Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities

“…The connection of solitons with two different decays for (DNLS) has been studied in [19,20], where the proofs depend on the explicit formulae of solitons, which is not applicable to at least higher-dimensional case in our setting. Here we use variational characterization of ground states effectively for the proof of Theorem 1.4.…”

Modulational instability of periodic standing waves in the derivative NLS equation

“…Such solutions are generally quasi-periodic when they are expressed in the polar form φ ω,ν = R ω,ν e iΘω,ν with R ω,ν and Θ ω,ν being periodic with the same period L. The simplest periodic standing wave solutions to the DNLS equation (1.1) can be analyzed directly by separating the variables in the polar form [15]. Convergence of periodic waves to the solitary waves (1.5) was shown in [21]. Spectral stability of periodic waves with non-vanishing φ ω,c was established with respect to perturbations of the same period in [20].…”