On the derivative nonlinear Schrödinger equation

Nakao Hayashi, Tohru Ozawa

Research output: Contribution to journalArticle

93 Citations (Scopus)

Abstract

In this paper we discuss the Cauchy problem for the derivative nonlinear Schrödinger equation: i∂tψ + 2iδ∂x(|;ψ|2ψ) = 0, ψ(0, x) = f{cyrillic}(x), where δ ≠ 0. Under an explicit smallness condition of the initial data, we prove the unique global existence of solutions to this problem in the usual Sobolev spaces, in the weighted Sobolev spaces, and in the Schwartz class. We describe the smoothing effect in detail. Furthermore, for the data decaying exponentially at infinity we prove that the above equation has unique local solutions which are analytic in the space direction.

Original languageEnglish
Pages (from-to)14-36
Number of pages23
JournalPhysica D: Nonlinear Phenomena
Volume55
Issue number1-2
DOIs
Publication statusPublished - 1992
Externally publishedYes

Fingerprint

Sobolev space
Sobolev spaces
Nonlinear equations
nonlinear equations
Nonlinear Equations
Derivatives
Smoothing Effect
Derivative
Cauchy problem
Weighted Sobolev Spaces
Local Solution
smoothing
Global Existence
infinity
Sobolev Spaces
Existence of Solutions
Cauchy Problem
Infinity

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

On the derivative nonlinear Schrödinger equation. / Hayashi, Nakao; Ozawa, Tohru.

In: Physica D: Nonlinear Phenomena, Vol. 55, No. 1-2, 1992, p. 14-36.

Research output: Contribution to journalArticle

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