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Existence and Nonexistence of Standing Wave Solutions of Nonlinear Schrödinger-Poisson System

Existence and Nonexistence of Standing Wave Solutions of Nonlinear Schrödinger-Poisson System

非線形 Schrödinger-Poisson 方程式の
定常波解の存在と非存在

The 39th Differential Equation Seminar at Yokohama National University
26 Aug 2016
College of Engineering Science, Yokohama National University

横浜国立大学における微分方程式セミナー (通算第 39 回)
2016 年 8 月 26日
横浜国立大学・理工学部講義棟

Kazune Takahashi

August 26, 2016
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  1. ඇઢܗ4DISÖEJOHFS 1PJTTPOํఔࣜͷ
 ఆৗ೾ղͷଘࡏͱඇଘࡏ ߴڮ࿨Ի ౦ژେֶେֶӃ਺ཧՊֶݚڀՊത࢜ޙظ՝ఔ೥

  2. ͜ͷݚڀ͸ɼٶݪ߂ߦࢯͱͷ
 ڞಉݚڀͰ͢ɽ<.5> [MT] Hiroyuki Miyahara and Kazune Takahashi.
 Existence and

    nonexistence of standing wave solutions of nonlinear schrödinger- poisson system. preprint.
  3. j໰୊ɾઌߦݚڀɾ ओఆཧ

  4. ˒ ໰୊ ͸ͷୈݻ༗஋ ׈Β͔ɼ༗քྖҬ ʹͷͱ͖ྟքɺʻͷͱ͖ྼྟքͱݺͿ

  5.  ͕ٿͷͱ͖ɼͳΒ͹ɼ ˒ ͸ඇࣗ໌ղΛ࣋ͭɽ  ͕੕ܗͷͱ͖ɼͳΒ͹ɼ ˒ ͸ ඇࣗ໌ղΛ࣋ͨͳ͍ɽ ઌߦݚڀ

    ຆͲ͕ ଟ͕͘ྼྟք <"E> 5IFPSFN ͷ߲͸ͳ͍ ͸ྟք
  6.  ͷΈͳΒͣ  ྟքͷΈͳΒͣྟքͷέʔε΋  ͷΈͳΒͣ ˒ ઌߦݚڀͱͷൺֱ

  7. ओఆཧ ղ͕ଘࡏ͢Δे෼৚݅ʜ5IFPSFN 5IFPSFN 5IFPSFN 5IFPSFN ղ͕ଘࡏ͠ͳ͍े෼৚݅ʜ5IFPSFN ഑෍ࢿྉΛࢀর

  8. ղͷଘࡏɾඇଘࡏ͕ܾఆ͢Δྫ ղ͋Γ5IN W ੕ܗղͳ͠5IN J ੕ܗղͳ͠5IN ٿղ͋Γ5IN ࢀߟ<"E>

  9. jূ໌ͷϙΠϯτ

  10. ม෼๏ ͷྟք఺ʹର͠ɼ͸ ˒ ͷղɽ ͷ།Ұղɽ ͸ ͜͜Ͱ ΛҎԼͰఆΊΔɽ

  11. 5IFPSFNͷϙΠϯτ ͷྟք఺ ಕͷิ୊ ಕͷิ୊ͷԾఆ J ɽ JJ ͕ଘࡏ͠ɼ೚ҙͷʹର͠ɼ
 JJJ ͕ଘࡏ͠ɼ


  12. J ɽ JJ ͕ଘࡏ͠ɼ೚ҙͷʹର͠ɼ
 JJJ ͕ଘࡏ͠ɼ
 ্͕ͳ͢໘

  13. 5IFPSFNͷϙΠϯτ ͷྟք఺ ಕͷิ୊ ಕͷิ୊ͷԾఆ J ɽ JJ ͕ଘࡏ͠ɼ೚ҙͷʹର͠ɼ
 JJJ ͕ଘࡏ͠ɼ


    ͜ͷதͰ͸ JJJ ͕Ұ൪ݫ͍͠ˠ5IFPSFNͷ৚݅
  14. ͸ɼ5IN J  JJ  JW ͷͱ͖ɼଈै͏ɽ 5IFPSFN 

  15. 5IFPSFN ͷͱ͖ Ͱܾ·Δɽ ͸ɼ

  16. 5IFPSFN ͷͱ͖ -FNNB Ͱ໌ࣔతʹٻ·Δɽ ˠ5IN JJJ ˠ5IN W

  17. ಕͷิ୊ͷؼ݁ ࢁΛӽ͑Δ࠷খͷߴ͞ ͷ఺͕ྟք఺ ྟք఺ʹۙͮ͘఺ྻ͕ಘΒΕΔ ऴ఺

  18. ಕͷิ୊ͷؼ݁ ྻ͕ಘΒΕΔɽ ఆٛ ͕ྻͰ͋Δͱ͸ ఆٛ NPVOUBJOQBTTMFWFM ͜͜Ͱ͸ͱ೚ҙͷऴ఺Λͭͳ͙ಓɽ

  19. 5IFPSFN ͷϙΠϯτ ͷྟք఺ ಕͷิ୊ ྻ͕ಘΒΕΔɽ ಕͷิ୊ͷؼ݁  ऑऩଋઌ͕ྟք఺Ͱ͋Δɽ ༰қ ྟքࢦ਺

    5IN  ͷ৔߹ ΛಘΔͨΊʹɼ NPVOUBJOQBTTMFWFMʹ৚͕݅ͭ͘ɽ 5BMFOUJؔ਺Λ࢖͏<#/>
  20. 5BMFOUJؔ਺ ࢖༻ํ๏ ࢁΛӽ͑Δಓͱͯ͠ํ޲ͷ൒௚ઢΛબͿɽ Λখ͘͞͠ɼΛධՁ͢Δɽ ͸্ͷDVUP⒎GVODUJPO

  21. 5IFPSFN ͷख๏ 5INɿ<#/>ͷख๏ 5INɿ<"E>ͷख๏ʜʜ<#/>Λ޻෉͢Δɽ J  JJ ͕ಘΒΕΔɽ JJJ 

    W ͸͞Βʹ޻෉͢Δɽ
  22. 5IFPSFN Λূ໌͍ͨ͠ɽഎཧ๏ɿͱԾఆ͢Δɽ ͕ྻͰ͋Δ͜ͱ͔Βɼ ࿈ཱํఔࣜղ͚Δ ͜ͷܭࢉ݁Ռͱɼ4PCPMFWෆ౳ࣜͳͲΑΓɼ ͕ඞཁɽʜ ̇ ͱ͓͘ɽ

  23. 5IFPSFN ͷͷ߲ΛධՁ͢Δɽ ৗʹ͕੒ཱ͢Δɽ མͱ͢ 5IN J  JJ ͷͱ͖͕੒ཱ͢Δɽ

  24. 5IFPSFN J  JJ ͜͜Ͱ͸ͷ།Ұͷਖ਼஋ղɽ ໌ࣔతʹॻ͚Δ

  25. 5IFPSFN J  JJ ͜͜Ͱ͸ͷ།Ұͷਖ਼஋ղɽ ໌ࣔతʹॻ͚Δ ܭࢉʹΑΓͱΘ͔Δɽ ͱ͓͘ͱɼ ̇ ʹໃ६

    5BMFOUJؔ਺ͷͷٞ࿦͔Β J  JJ ͕ಘΒΕΔɽ
  26. ͜͜Ͱ͸ͷ།Ұͷਖ਼஋ղɽ 5IFPSFN JJJ  W 5IN JJJ  W ͸ߋʹ޻෉

    ໌ࣔతʹॻ͚ͳ͍
  27. ͜͜Ͱ͸ͷ།Ұͷਖ਼஋ղɽ 5IFPSFN JJJ  W ໌ࣔతʹॻ͚ͳ͍ ͱ͜Ζ͕ɼ ͸ͷʹӨڹΛ༩͑ͳ͍

  28. 5IFPSFN JJJ  W ͸ͷʹӨڹΛ༩͑ͳ͍ -FNNB JJJ  W ͕ಘΒΕΔɽ

  29. ˒ ղ͋Γ5IN W ੕ܗղͳ͠5IN J