### Abstract

In this paper we discuss the Cauchy problem for the derivative nonlinear Schrödinger equation: i∂_{t}ψ + 2iδ∂_{x}(|;ψ|^{2}ψ) = 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 language | English |
---|---|

Pages (from-to) | 14-36 |

Number of pages | 23 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 55 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 1992 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*55*(1-2), 14-36. https://doi.org/10.1016/0167-2789(92)90185-P

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

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 55, no. 1-2, pp. 14-36. https://doi.org/10.1016/0167-2789(92)90185-P

}

TY - JOUR

T1 - On the derivative nonlinear Schrödinger equation

AU - Hayashi, Nakao

AU - Ozawa, Tohru

PY - 1992

Y1 - 1992

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0002770583&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002770583&partnerID=8YFLogxK

U2 - 10.1016/0167-2789(92)90185-P

DO - 10.1016/0167-2789(92)90185-P

M3 - Article

VL - 55

SP - 14

EP - 36

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -