Asymptotics of solutions to the fourth order Schrödinger type equation with a dissipative nonlinearity

Junichi Segata, Akihiro Shimomura

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrödinger type equation with a cubic dissipative nonlinearity λ|u|2u, where λ is a complex constant satisfying Im λ < 0, is studied. This nonlinearity is a long-range interaction. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as t → +∞ in a suitable function space. Our modified free dynamics decays like (t log t)-1/2 as t → ∞.

Original languageEnglish
Pages (from-to)439-456
Number of pages18
JournalKyoto Journal of Mathematics
Volume46
Issue number2
DOIs
Publication statusPublished - Jan 1 2006
Externally publishedYes

Fingerprint

Order Type
Asymptotics of Solutions
Fourth Order
Nonlinearity
Long-range Interactions
Unique Solution
Function Space
Cauchy Problem
Asymptotic Behavior
Decay
Restriction
Imply

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Asymptotics of solutions to the fourth order Schrödinger type equation with a dissipative nonlinearity. / Segata, Junichi; Shimomura, Akihiro.

In: Kyoto Journal of Mathematics, Vol. 46, No. 2, 01.01.2006, p. 439-456.

Research output: Contribution to journalArticle

@article{957adbde52e94d34af90a6a8150acadb,
title = "Asymptotics of solutions to the fourth order Schr{\"o}dinger type equation with a dissipative nonlinearity",
abstract = "In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schr{\"o}dinger type equation with a cubic dissipative nonlinearity λ|u|2u, where λ is a complex constant satisfying Im λ < 0, is studied. This nonlinearity is a long-range interaction. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as t → +∞ in a suitable function space. Our modified free dynamics decays like (t log t)-1/2 as t → ∞.",
author = "Junichi Segata and Akihiro Shimomura",
year = "2006",
month = "1",
day = "1",
doi = "10.1215/kjm/1250281786",
language = "English",
volume = "46",
pages = "439--456",
journal = "Kyoto Journal of Mathematics",
issn = "0023-608X",
publisher = "Kyoto University",
number = "2",

}

TY - JOUR

T1 - Asymptotics of solutions to the fourth order Schrödinger type equation with a dissipative nonlinearity

AU - Segata, Junichi

AU - Shimomura, Akihiro

PY - 2006/1/1

Y1 - 2006/1/1

N2 - In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrödinger type equation with a cubic dissipative nonlinearity λ|u|2u, where λ is a complex constant satisfying Im λ < 0, is studied. This nonlinearity is a long-range interaction. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as t → +∞ in a suitable function space. Our modified free dynamics decays like (t log t)-1/2 as t → ∞.

AB - In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrödinger type equation with a cubic dissipative nonlinearity λ|u|2u, where λ is a complex constant satisfying Im λ < 0, is studied. This nonlinearity is a long-range interaction. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as t → +∞ in a suitable function space. Our modified free dynamics decays like (t log t)-1/2 as t → ∞.

UR - http://www.scopus.com/inward/record.url?scp=33846315496&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846315496&partnerID=8YFLogxK

U2 - 10.1215/kjm/1250281786

DO - 10.1215/kjm/1250281786

M3 - Article

VL - 46

SP - 439

EP - 456

JO - Kyoto Journal of Mathematics

JF - Kyoto Journal of Mathematics

SN - 0023-608X

IS - 2

ER -