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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 林太家(Tai-Chia Lin) | |
| dc.contributor.author | Chiun-Chang Lee | en |
| dc.contributor.author | 李俊璋 | zh_TW |
| dc.date.accessioned | 2021-06-15T04:52:36Z | - |
| dc.date.available | 2012-08-04 | |
| dc.date.copyright | 2010-08-04 | |
| dc.date.issued | 2010 | |
| dc.date.submitted | 2010-07-30 | |
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| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/46057 | - |
| dc.description.abstract | In this thesis, we investigate two different types partial differential equations, one is the coupled nonlinear Schrödinger equations and the other is the renormalize Poisson-Boltzmann equations (the steady-state solutions of the Poisson-Nernst-Planck systems).
Recently, a rich variety of dynamical phenomena and a turbulent relaxation have been observed in rotating Bose-Einstein condensates depicted by Gross-Pitaevskii equations coupled with rotating fields and trap potentials. The dynamical phenomena range from shock-wave formation to anisotropic sound propagation. The turbulent relaxation leads to the crystallization of vortex lattices. To see the dynamical phenomena and the turbulent relaxation of two-component rotating Bose-Einstein condensates, we study the incompressible and the compressible limits of two-component systems of Gross-Pitaevskii equations. Our arguments generalize the idea of [22] and define 'H-function' a modulated energy functional which may control the propagation of densities and linear momentums under the effect of rotating fields and trap potentials. The Poisson-Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in different media and solvents. We study a renormalized Poisson-Boltzmann (RPB) equation with a small dielectric parameter ∈2 and nonlocal nonlinearity which takes into consideration of the preservation of the total amounts of each individual ion. This equation can be derived from the original Poisson-Nernst-Planck (PNP) system. Under Robin type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviors of one dimensional solutions of RPB equations as the parameter $epsilon$ approaches to zero. In particular, we show that in case of electro-neutrality, i.e., (∑N1 κ=1 ακακ=∑N2 l=1 blβl), we prove that φ∈'s solutions of 1-D RPB equations may tend to a nonzero constant c at every interior point as $epsilon$ goes to zero. The value c can be uniquely determined by ακ, bl's valences of ions, ακ, βl's total concentrations of ions and the limit of φ∈'s at the boundary x=±1. In particular, when N1=1, N2=2, a1=b1=1 and b2=2, a precise formula of the value c and the ratio β1/β2 is given in (4.1.3). Such a result can not be found in conventional 1-D Poisson-Boltzmann (PB) equations. On the other hand, as (∑N1 κ=1 ακακ≠∑N2 l=1 blβl) (non-electroneutrality), solutions of 1-D RPB equations have blow-up behavior which also may not be obtained in 1-D PB equations. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-15T04:52:36Z (GMT). No. of bitstreams: 1 ntu-99-F92221030-1.pdf: 937451 bytes, checksum: c2d9b29b60b87a9c20fe47e1be510e04 (MD5) Previous issue date: 2010 | en |
| dc.description.tableofcontents | Acknowledgements (in Chinese) i
Acknowledgements (in English) ii Abstract (in Chinese) iii Abstract (in English) iv Contents vi 1 Introduction 1.1 Two-component systems of nonlinear Schrodinger equations. . . . . . . 2 1.2 Poisson-Nernst-Planck systems and renormalized Poisson-Boltzmann equations. . . . . . . . . . . . . . . . . . . . . . 5 2 Limit Problems of Solutions for the Coupled Nonlinear Schrödinger Equations 2.1 Conservation Laws and H-function. . . . . . . . . . . . . . . . . . . . . . . . . ..9 2.2 Incompressible limits for the solution of NLS. . . . . . . . . . . . . . . . . . 20 2.3 Compressible limits for the solution of NLS. . . . . . . . . . . . . . . . . . . 23 2.4 Proofs of Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.1 Proof of Theorem 2.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 Proof of Theorem 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 2.5 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Renormalized Poisson-Boltzmann (RPB) Equations: One-Dimensional Solutions 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.1 MainTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Electroneutral cases (α=β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.1 Interior estimates of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.2 Boundary asymptotic behaviors of solutions . . . . . . . . . . . . . . . . 51 3.3 Non-electroneutral cases (α≠β). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.1 Interior estimates of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.2 Boundary estimates of solutions . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4 Comparison of RPB and PB equations . . . . . . . . . . . . . . . . . . . . . 68 3.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.6 Appendix: Some Results in High Dimension Cases . . . . . . . . . . . . 73 3.6.1 Existence and uniqueness for solutions of RPB. . . . . . . . . . . . . . 73 3.6.2 Asymptotic blow-up properties . . . . . . . . . . . . . . . . . . . . . . . . . 78 4 Renormalized Poisson-Boltzmann Equations for Multiple Species Ions 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.1 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Electroneutral cases (∑N1 κ=1 ακακ=∑N2 l=1 blβl). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2.1 Interior estimates of solutions. . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.2 The relations of c and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3 Non-electroneutral cases (∑N1 κ=1 ακακ≠∑N2 l=1 blβl). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3.1 Boundary estimates of solutions. . . . . . . . . . . . . . . . . . . . . . . . 103 4.3.2 Interior estimates of solutions. . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.4 Solutions of RPB and PB equations. . . . . . . . . . . . . . . . . . . . . . . 112 4.4.1 Proof of Theorem 4.1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.6 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5 Ongoing Problem: Energetic Variational Approaches in Multi-Spinor Bose-Einstein Condensate 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2 Derivation of the system: Variational structure. . . . . . . . . . . . . . . 124 5.3 Exact solutions in one-dimensional. . . . . . . . . . . . . . . . . . . . . . . 127 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 | |
| dc.language.iso | en | |
| dc.subject | 泊松-波茲曼方程 | zh_TW |
| dc.subject | 玻氏-愛因斯坦凝聚 | zh_TW |
| dc.subject | 薛丁格方程 | zh_TW |
| dc.subject | 可壓縮極限 | zh_TW |
| dc.subject | 不可壓縮極限 | zh_TW |
| dc.subject | 泊松-能斯特-普朗克方程 | zh_TW |
| dc.subject | Bose-Einstein condensates | en |
| dc.subject | Poisson-Boltzmann Equation | en |
| dc.subject | Poisson-Nernst-Planck Equation | en |
| dc.subject | incompressible limit | en |
| dc.subject | compressible limit | en |
| dc.subject | Schrodinger equation | en |
| dc.title | 非線性薛丁格方程組的極限問題與泊松-能斯特-普朗克方程組的穩態解 | zh_TW |
| dc.title | Limit Problems of Solutions for the Coupled Nonlinear Schrödinger Equations and Steady-state Solutions of the Poisson-Nernst-Planck Systems | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 98-2 | |
| dc.description.degree | 博士 | |
| dc.contributor.oralexamcommittee | 許世壁,柳春,陳俊全,郭忠勝,陳建隆,吳宗芳 | |
| dc.subject.keyword | 玻氏-愛因斯坦凝聚,薛丁格方程,可壓縮極限,不可壓縮極限,泊松-能斯特-普朗克方程,泊松-波茲曼方程, | zh_TW |
| dc.subject.keyword | Bose-Einstein condensates,Schrodinger equation,compressible limit,incompressible limit,Poisson-Nernst-Planck Equation,Poisson-Boltzmann Equation, | en |
| dc.relation.page | 139 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2010-08-02 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
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