在這論文裡我們研究了多種情況下的波局域性。主要著重在兩種不同系統上，第一種是我們能在非線性色散的系統中找到孤子解，是以贗勢找到的近似解析解，以及牛頓法和傳播一個高斯波函數得到的數值解。第二種為了要做彎曲空間的研究，對於多種曲線我們推導了相對應的等效位能以及曲線長度用來做為彎曲空間座標, 像是拋物線，雙曲線，S形的，正弦曲線，橢圓形，螺旋形，玫瑰形，心形，進而得到對於各種曲線的相關的局域態，同時也將此作法運用於許多拋物線組合成的三維拓撲弦，同時使拓撲弦的等效位能對應於Andre-Aubry-Harper的位能。同時如同預期的我們也能在此系統中找到局域態，非局域態，邊界態。另外使用逆參與率找到了局域-非局域的過度邊界。

In this thesis, we study the wave localizations in several cases. There are two main parts of the system. First is that we obtain the solitary solution in the system with nonlinear dispersion, approach analytical solution by pseudo-potential method, also obtain the numerical solution by Newton’s method and propagate an initial Gaussian state. Second, in order to research on the curved space, for assorted curves we derived corresponding effective potential and arc length as the curved coordinate, such as parabola, hyperbola, S-shape, sinusoidal, ellipse, spiral, rose and cardioid curves, to obtain the related localized state on those curves. And we have the application on the three-dimensional topological string by many parabolas be connected, makes the effective potential present as Andre-Aubry-Harper model. And in the meanwhile, we found localized states, delocalized states and edge states as expected in this system. In addition, find the localization-delocalization transition for the system, by inverse participation ratio.

ABSTRACT-----------------------------------------v

Chapter 1 Introduction...........................1
1.1 1D soliton...............................1
1.2 2D soliton...............................2
1.3 2D dipolar BEC...........................3
1.4 X-wave...................................4
Chapter 2 Soliton in nonlinear dispersion system.6
2.1 introduction..........................6
2.2 pseudo-potential method and asymptotic analysis.........................................7
2.3 Numerical solution...................10
2.4 Conserved density convergence........12
2.5 Phase diagram........................13
2.6 Evolution with an initial Gaussian…..14
2.7 Summary..................................15
Chapter 3 Wave localization in curved potentials17
3.1 Introduction.........................17
3.2 Parabola.............................21
3.3 Hyperbola............................23
3.4 S-shape..............................24
3.5 Sinusoidal...........................28
3.6 Ellipse..............................31
3.7 Spiral...............................34
3.8 Rose.................................35
3.9 Cardioid.............................36
3.10 Summary….............................37
Chapter 4 Wave localization in curved potentials in AAH model...........................................38
4.1 introduce the model..................38
4.2 Result...............................44
4.3 Summary..............................48
Chapter 5 Conclusion............................49
Appendix A......................................51
Reference.......................................55

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