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Research on minimization problem related with stabilizing effect for standing waves caused by non-local interaction

Research Project

Project/Area Number 18K03383
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionKyoto Sangyo University

Principal Investigator

Watanabe Tatsuya  京都産業大学, 理学部, 教授 (60549749)

Project Period (FY) 2018-04-01 – 2022-03-31
Project Status Discontinued (Fiscal Year 2021)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords楕円型微分方程式 / 変分問題 / 定在波 / 安定性解析 / 非線形解析 / 楕円型偏微分方程式
Outline of Final Research Achievements

In this research, I have investigated minimization problems related with the stabilizing effect for standing waves caused by non-local interactions. Especially, I have studied the solution structure (uniqueness, multiplicity and asymptotic profile) of the stationary problem of a quasilinear Schroedinger equation which is derived as a limit equation. One of the main results of my research is to show the uniqueness and the non-degeneracy of ground state solutions for a whole range of parameters. Moreover, I have studied the asymptotic behavior of solutions when the nonlinear term has the Sobolev critical growth. My result completely reveals how the parameter dependence of the asymptotic profiles of solutions changes with respect to the nonlinear exponent.
I have also performed the stability analysis of standing waves for the Schroedinger-Maxwell system, and investigated several other physical models.

Academic Significance and Societal Importance of the Research Achievements

本研究では、定在波の安定化効果が非局所的相互作用によって得られるということを、付随する最小化問題のラグランジュ乗数の解析から捉えることを目標としている。定在波は様々な数理モデルで現れ、その安定性解析は重要な研究課題の一つであるが、厳密な解析が行われていない数理モデルは数多く残されている。本研究成果で得られた結果や手法は他の方程式に対する最小化変分問題にも応用できる。様々な物理モデルにおける定在波の安定性解析の研究にも大きな寄与があり、物理学者による(形式的な)考察を数学的に厳密化でき、応用面でも大きな意義がある。

Report

(4 results)
  • 2021 Final Research Report ( PDF )
  • 2020 Annual Research Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (18 results)

All 2021 2020 2019 2018 Other

All Int'l Joint Research (2 results) Journal Article (10 results) (of which Int'l Joint Research: 7 results,  Peer Reviewed: 10 results,  Open Access: 1 results) Presentation (6 results) (of which Int'l Joint Research: 4 results,  Invited: 5 results)

  • [Int'l Joint Research] バーリ工科大学(イタリア)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] ボルドー大学(フランス)

    • Related Report
      2020 Annual Research Report
  • [Journal Article] Ground state solutions for quasilinear scalar field equations arising in nonlinear optics2021

    • Author(s)
      Alessio Pomponio and Tatsuya Watanabe
    • Journal Title

      Nonlinear Differential Equations and Applications Research

      Volume: 28 Issue: 3

    • DOI

      10.1007/s00030-021-00687-7

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Uniqueness of asymptotic limit of ground states for a class of quasilinear Schrodinger equation with H^1-critical growth in R^3,2020

    • Author(s)
      Shinji Adachi, Masataka Shibata, Tatsuya Watanabe
    • Journal Title

      Applicable Analysis

      Volume: - Issue: 2 Pages: 671-691

    • DOI

      10.1080/00036811.2020.1757079

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Cauchy problem for the nonlinear Schrodinger equation coupled with the Maxwell equation2020

    • Author(s)
      Colin Mathieu、Watanabe Tatsuya
    • Journal Title

      Annales Henri Lebesgue

      Volume: 3 Pages: 67-85

    • DOI

      10.5802/ahl.27

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Standing waves of modified Schrodinger equations coupled with the Chern-Simons gauge theory2020

    • Author(s)
      Pietro D'Avenia, Alessio Pomponio and Tatsuya Watanabe
    • Journal Title

      Proceedings A of the Royal Society of Edinburgh

      Volume: 150 Issue: 4 Pages: 1915-1936

    • DOI

      10.1017/prm.2019.9

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A refined stability result for standing waves of the Schrodinger-Maxwell system2019

    • Author(s)
      Colin Mathieu、Watanabe Tatsuya
    • Journal Title

      Nonlinearity

      Volume: 32 Issue: 10 Pages: 3695-3714

    • DOI

      10.1088/1361-6544/ab248c

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Asymptotic property of ground states for a class of quasilinear Schrodinger equation with H^1-critical growth2019

    • Author(s)
      Adachi Shinji、Shibata Masataka、Watanabe Tatsuya
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 58 Issue: 3

    • DOI

      10.1007/s00526-019-1527-y

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Standing waves of modified Schrodinger equations coupled with the Chern-Simons gauge theory2019

    • Author(s)
      Pietro D'Avenia, Alessio Pomponio and Tatsuya Watanabe
    • Journal Title

      Proceedings A of the Royal Society of Edinburgh

      Volume: -

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Asymptotic property of ground states for a class of quasilinear Schrodinger equations with $H^1$-critical growth2019

    • Author(s)
      Shinji Adachi, Masataka Shibata, Tatsuya Watanabe
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: - Issue: 4 Pages: 1-22

    • DOI

      10.1016/s0252-9602(18)30803-8

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the existence of ground states for a nonlinear Klein-Gordon-Maxwell type system2018

    • Author(s)
      Mathieu Colin and Tatsuya Watanabe
    • Journal Title

      Funkcialaj Ekvacioj

      Volume: 61 Pages: 1-14

    • NAID

      130006726036

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Some quasilinear elliptic equations involving multiple p-Laplacians2018

    • Author(s)
      Alessio Pomponio and Tatsuya Watanabe
    • Journal Title

      Indiana University Mathematics Journal

      Volume: 67 Pages: 2199-2224

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Ground state solutions for quasilinear scalar field equations arising in nonlinear optics2020

    • Author(s)
      Tatsuya Watanabe
    • Organizer
      NSFC-JSPS Mini-workshop on Nonlinear Analysis
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Ground state solutions for quasilinear scalar field equations arising in nonlinear optics2020

    • Author(s)
      渡辺達也
    • Organizer
      日本数学会 2020年 秋季総合分科会
    • Related Report
      2020 Annual Research Report
  • [Presentation] Convex properties of positive solutions for a class of quasi-linear elliptic problems and the associated new minimization problem2019

    • Author(s)
      Tatsuya Watanabe
    • Organizer
      Variational analysis on critical problems of nonlinear partial differential equations
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the stability of standing waves for the Maxwell-Schrodinger system and the associated minimization problem coupled with the Maxwell equation2019

    • Author(s)
      Tatsuya Watanabe
    • Organizer
      ICIAM 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A refined stability of standing waves for the Schrodinger-Maxwell system and the associated new minimization problem2019

    • Author(s)
      渡辺達也
    • Organizer
      反応拡散方程式と非線形分散型方程式の解の挙動
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Stability issues for the nonlinear Schrodinger equation coupled with the Maxwell equation2018

    • Author(s)
      Tatsuya Watanabe
    • Organizer
      12th AIMS conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2018-04-23   Modified: 2023-01-30  

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