Existence and stability of standing waves of fourth order nonlinear Schrödinger type equation related to vortex filament

Masaya Maeda; Jun Ichi Segata

doi:10.1619/fesi.54.1

Existence and stability of standing waves of fourth order nonlinear Schrödinger type equation related to vortex filament

Masaya Maeda, Jun Ichi Segata

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

In this paper, we study the fourth order nonlinear Schrödinger type equation (4NLS) which is a generalization of the Fukumoto-Moffatt [5] model that arising in the context of the motion of a vortex filament. Firstly, we mention the existence of standing wave solution and the conserved quantities. We next investigate the case that the equation is completely integrable and show that the standing wave obtained in [20] is orbitally stable in Sobolev spaces H^m with m ∈ N. Further, we show that the completely integrable equation is ill-posed in H^s with s ∈(-1/2,1/2) by following Kenig-Ponce-Vega [13].

Original language	English
Pages (from-to)	1-14
Number of pages	14
Journal	Funkcialaj Ekvacioj
Volume	54
Issue number	1
DOIs	https://doi.org/10.1619/fesi.54.1
Publication status	Published - 2011
Externally published	Yes

All Science Journal Classification (ASJC) codes

Analysis
Algebra and Number Theory
Geometry and Topology

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10.1619/fesi.54.1

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abstract = "In this paper, we study the fourth order nonlinear Schr{\"o}dinger type equation (4NLS) which is a generalization of the Fukumoto-Moffatt [5] model that arising in the context of the motion of a vortex filament. Firstly, we mention the existence of standing wave solution and the conserved quantities. We next investigate the case that the equation is completely integrable and show that the standing wave obtained in [20] is orbitally stable in Sobolev spaces Hm with m ∈ N. Further, we show that the completely integrable equation is ill-posed in Hs with s ∈(-1/2,1/2) by following Kenig-Ponce-Vega [13].",

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AB - In this paper, we study the fourth order nonlinear Schrödinger type equation (4NLS) which is a generalization of the Fukumoto-Moffatt [5] model that arising in the context of the motion of a vortex filament. Firstly, we mention the existence of standing wave solution and the conserved quantities. We next investigate the case that the equation is completely integrable and show that the standing wave obtained in [20] is orbitally stable in Sobolev spaces Hm with m ∈ N. Further, we show that the completely integrable equation is ill-posed in Hs with s ∈(-1/2,1/2) by following Kenig-Ponce-Vega [13].

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Existence and stability of standing waves of fourth order nonlinear Schrödinger type equation related to vortex filament

Abstract

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