Simple proof of stationary phase method and application to oscillatory bifurcation problems

Keiichi Kato, Tetsutaro Shibata

Research output: Contribution to journalArticle

Abstract

We consider the nonlinear eigenvalue problem −u′′(t)=λf(u(t)),u(t)>0,t∈I≔(−1,1),u(±1)=0, where f(u)=f1(u)=u3+sin(u3)∕u, f(u)=f2(u)=u+upsin(uq) (0≤p<1, 1<q≤p+2) and λ>0 is a bifurcation parameter. It is known that, in this case, λ is parameterized by the maximum norm α=‖uλ of the solution uλ associated with λ and is written as λ=λ(α). We simplify the argument of the stationary phase method and show the asymptotic formulas for λ(α) for f1(u) and f2(u) as α→∞ and α→0. In particular, the shape of bifurcation diagram of λ(α) for f1(u) seems to be new.

Original languageEnglish
Article number111594
JournalNonlinear Analysis, Theory, Methods and Applications
Volume190
DOIs
Publication statusPublished - 1 Jan 2020

Fingerprint

Stationary Phase
Maximum Norm
Nonlinear Eigenvalue Problem
Bifurcation Diagram
Asymptotic Formula
Simplify
Bifurcation

Keywords

  • Global structure
  • Nonlinear eigenvalue problems
  • Oscillatory bifurcation

Cite this

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abstract = "We consider the nonlinear eigenvalue problem −u′′(t)=λf(u(t)),u(t)>0,t∈I≔(−1,1),u(±1)=0, where f(u)=f1(u)=u3+sin(u3)∕u, f(u)=f2(u)=u+upsin(uq) (0≤p<1, 10 is a bifurcation parameter. It is known that, in this case, λ is parameterized by the maximum norm α=‖uλ‖∞ of the solution uλ associated with λ and is written as λ=λ(α). We simplify the argument of the stationary phase method and show the asymptotic formulas for λ(α) for f1(u) and f2(u) as α→∞ and α→0. In particular, the shape of bifurcation diagram of λ(α) for f1(u) seems to be new.",
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Simple proof of stationary phase method and application to oscillatory bifurcation problems. / Kato, Keiichi; Shibata, Tetsutaro.

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 190, 111594, 01.01.2020.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Simple proof of stationary phase method and application to oscillatory bifurcation problems

AU - Kato, Keiichi

AU - Shibata, Tetsutaro

PY - 2020/1/1

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N2 - We consider the nonlinear eigenvalue problem −u′′(t)=λf(u(t)),u(t)>0,t∈I≔(−1,1),u(±1)=0, where f(u)=f1(u)=u3+sin(u3)∕u, f(u)=f2(u)=u+upsin(uq) (0≤p<1, 10 is a bifurcation parameter. It is known that, in this case, λ is parameterized by the maximum norm α=‖uλ‖∞ of the solution uλ associated with λ and is written as λ=λ(α). We simplify the argument of the stationary phase method and show the asymptotic formulas for λ(α) for f1(u) and f2(u) as α→∞ and α→0. In particular, the shape of bifurcation diagram of λ(α) for f1(u) seems to be new.

AB - We consider the nonlinear eigenvalue problem −u′′(t)=λf(u(t)),u(t)>0,t∈I≔(−1,1),u(±1)=0, where f(u)=f1(u)=u3+sin(u3)∕u, f(u)=f2(u)=u+upsin(uq) (0≤p<1, 10 is a bifurcation parameter. It is known that, in this case, λ is parameterized by the maximum norm α=‖uλ‖∞ of the solution uλ associated with λ and is written as λ=λ(α). We simplify the argument of the stationary phase method and show the asymptotic formulas for λ(α) for f1(u) and f2(u) as α→∞ and α→0. In particular, the shape of bifurcation diagram of λ(α) for f1(u) seems to be new.

KW - Global structure

KW - Nonlinear eigenvalue problems

KW - Oscillatory bifurcation

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