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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 陳宜良 | |
| dc.contributor.author | Yung-Hsien Lu | en |
| dc.contributor.author | 呂勇賢 | zh_TW |
| dc.date.accessioned | 2021-05-14T17:49:09Z | - |
| dc.date.available | 2015-03-13 | |
| dc.date.available | 2021-05-14T17:49:09Z | - |
| dc.date.copyright | 2015-03-13 | |
| dc.date.issued | 2015 | |
| dc.date.submitted | 2015-01-23 | |
| dc.identifier.citation | [1] Weizhu Bao and Yongyong Cai. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic & Related Models, 6(1), 2013.
[2] Weizhu Bao, I-Liang Chern, and Fong Yin Lim. Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates. Journal of Computational Physics, 219(2):836–854, 2006. [3] R Carretero-Gonzalez, DJ Frantzeskakis, and PG Kevrekidis. Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques. Nonlinearity, 21(7):R139, 2008. [4] T. Cazenave and P.-L. Lions. Orbital stability of standing waves for some nonlinear Schrodinger equations. Comm. Math. Phys., 85(4):549–561, 1982. [5] Franco Dalfovo, Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari. Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys., 71:463–512, Apr 1999. [6] L.C. Evans. Partial Differential Equations. Graduate studies in mathematics. American Mathematical Society, 1998. [7] GM Fraiman. The asymptotic stability of the manifold of self-similar solutions in the presence of self-focusing. Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, 88:390–400, 1985. [8] R. T. Glassey. On the blowing up of solutions to the cauchy problem for nonlinear Schrodinger equations. Journal of Mathematical Physics, 18(9):1794–1797, 1977. [9] ManKam Kwong. Uniqueness of positive solutions of △u u + up = 0 in Rn. Archive for Rational Mechanics and Analysis, 105(3):243–266, 1989. [10] M.J. Landman, G.C. Papanicolaou, C. Sulem, P.L. Sulem, and X.P. Wang. Stability of isotropic singularities for the nonlinear Schrodinger equation. Physica D: Nonlinear Phenomena, 47(3):393 – 415, 1991. [11] C. Sulem and P.L. Sulem. The Nonlinear Schrodinger Equation: Self-Focusing and Wave Collapse. Number 139 in Applied Mathematical Sciences. Springer, 1999. [12] Terence Tao. Why are solitons stable? Bulletin of the American Mathematical Society, 46(1):1–33, 2009. [13] Michael I. Weinstein. Nonlinear Schrodinger equations and sharp interpolation estimates. Communications in Mathematical Physics, 87(4):567–576, 1982. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/4867 | - |
| dc.description.abstract | 我們用數值方法來模擬在一維及二維空間兩分量玻色-愛因斯坦凝聚中二孤立子(soliton) 的碰撞，探討孤立子在交互作用時速度和形狀的變化。我們用一個梯度下降法[2](gradient flow method) 來計算二維空間中孤立子的形狀，以及利用時間分步正弦擬譜法[1](time-splitting sine pseudospectral method) 來計算波函數隨時間的變化。數值模擬的結果顯示在一維空間中若孤立子間若有足夠強的相斥作用力，則它們的碰撞像是彈性碰撞；而在強相吸作用力下，孤立子在碰撞後將分為兩個或多個波包(wave packets)。在二維空間中，孤立子間的相吸作用力如果夠強，則會在碰撞過程中發生爆破現象(blow up phenomenon)，其它的情形下孤立子將在碰撞後成為漸漸散開(spread out) 的波包。 | zh_TW |
| dc.description.abstract | We investigate interaction of bright solitons for two-component Bose-Einstein condensates (BECs) in one and two dimensions numerically (1D, 2D). The numerical methods we adopt are: (1) Gradient flow with discrete normalization (GFDN) method for computing the profile function of solitons in 2D. We use backward Euler sine pseudospectral (BESP) method to discretize it. The algorithm is constructed by Chern and Bao [2]. (2) Timesplitting sine pseudospectral (TSSP) method for computing the evolution of wave functions. The algorithm is construct by Bao [1]. We discuss the change of velocities and shapes of the wave packets during and after the interactions between them. It is found that (1) In 1D, soliton collisions are like elastic collisions under strong repulsive interactions. When the interactions are attractive and strong enough, the wave packets may split into two or more parts after collisions. (2) In 2D, wave packets spread out after collisions when the interactions are repulsive or weak attractive. The wave functions blow up during interactions when the attractive interactions are strong enough. | en |
| dc.description.provenance | Made available in DSpace on 2021-05-14T17:49:09Z (GMT). No. of bitstreams: 1 ntu-104-R01221027-1.pdf: 1580052 bytes, checksum: 1b76b8755244577715d91227c69c06d2 (MD5) Previous issue date: 2015 | en |
| dc.description.tableofcontents | 誌謝 iii
摘要 v Abstract vii 1 Introduction 1 2 Bright soliton solutions 7 2.1 Basic properties of solutions of GPE 7 2.2 Bright soliton solutions 13 2.3 Stability/Instability of solitons in 1D and 2D 18 3 Numerical methods 21 3.1 Gradient flow method for finding solitons in 2D 21 3.2 TSSP method for computing the dynamics of BECs 24 4 Numerical examples 31 4.1 Examples of one-component BEC: perturbations of solitons 31 4.2 Examples of two-component BECs: collisions of solitons 40 5 Conclusions 65 References 67 | |
| dc.language.iso | en | |
| dc.subject | 變差等式 | zh_TW |
| dc.subject | 爆破 | zh_TW |
| dc.subject | 數值模擬 | zh_TW |
| dc.subject | 玻色-愛因斯坦凝聚 | zh_TW |
| dc.subject | 孤立子碰撞 | zh_TW |
| dc.subject | 兩分量 | zh_TW |
| dc.subject | Blow-up | en |
| dc.subject | Two-component | en |
| dc.subject | Bose-Einstein condensates (BECs) | en |
| dc.subject | Gross-Pitaevskii equation (GPE) | en |
| dc.subject | Numerical simulations | en |
| dc.subject | Soliton collisions | en |
| dc.subject | Variance identity | en |
| dc.subject | Stability | en |
| dc.title | 兩分量玻色─愛因斯坦凝聚中孤立子碰撞的數值模擬 | zh_TW |
| dc.title | Numerical Simulations of Soliton Collisions in Two-component Bose-Einstein Condensates | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 103-1 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 鄧君豪,林太家 | |
| dc.subject.keyword | 兩分量,玻色-愛因斯坦凝聚,孤立子碰撞,數值模擬,變差等式,爆破, | zh_TW |
| dc.subject.keyword | Two-component,Bose-Einstein condensates (BECs),Gross-Pitaevskii equation (GPE),Numerical simulations,Soliton collisions,Variance identity,Stability,Blow-up, | en |
| dc.relation.page | 68 | |
| dc.rights.note | 同意授權(全球公開) | |
| dc.date.accepted | 2015-01-23 | |
| 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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